1. Introduction. The general theory for dynamical systems ... - MSU math

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1. Introduction. The general theory for dynamical systems has been developed by many mathematicians and scientists starting with Poincar´e, Lyapunov and Birkhoff. The investigation of a particular dynamical system or a family of dynamical systems usually can be traced to an evolving physical system whose behavior one would like to understand and possibly predict. And so one of the main goal of the study of dynamical systems is to understand the long term behavior of states in the systems. In the original modeling process, empirical laws, simplifying assumptions, and even conjectured relationships are used to derive a dynamical system in the hope of then being able to approximately describe physical reality. Therefore, to have a better understanding of the physical phenomena being modeled, one needs to investigate not only the mathematical model but also the perturbations of the model. One also needs to study how the qualitative properties of the perturbed models are related to the qualitative properties of the original model. This is especially important in numerical computation. Because of round off error and imprecise numerical schemes, the model studied by numerical computations is actually a perturbation of the original model. Although bifurcation theory, static and dynamic, are natural outcomes of this line of inquiry, for physical systems being modeled, one hopes that all the flows associated with small perturbations of the given system exhibit the same qualitative behaviors. When such is the case the dynamical system is said to be structurally stable, and the study of structural stability and its necessary or sufficient conditions has been a particularly fruitful field of study. If a dynamical system is not structurally stable, one may want to know when part of the qualitative properties are preserved under small perturbation. For instance, one may ask when equilibria perturb smoothly and what becomes of the flow in a neighborhood of an equilibrium as the system is perturbed. More generally, to understand the dynamics of a system, one needs to investigate the existence of invariant sets, in particular, such as equilibrium points, periodic orbits, invariant tori, and attractors, and then to study their structures and what happens in their vicinity. This leads to the fundamental problem of the persistence of invariant manifolds under perturbation and to the study of the qualitative properties of the flow near invariant manifolds. As outlined in “Historical Background” section below, this led to the establishment of the theory of normally hyperbolic invariant manifolds for finite dimensional dynamical systems, normal hyperbolicity (defined later) being exactly the right condition for persistence. In this paper we initiate a program of extending the theory for normally hyperbolic invariant manifolds to infinite dimensional dynamical systems in a Banach space, thereby providing tools for the study of PDE’s and other infinite dimensional equations of evolution. Let X be a Banach space and let T t be a C 1 semiflow defined on X, that is, it is 1

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continuous on [0, ∞) × X, for each t ≥ 0, T t : X → X is C 1 , and T t ◦ T s (x) = T t+s (x) for all t, s ≥ 0 and x ∈ X. The typical example is the solution operator for a parabolic differential equation. Suppose that there exists a smooth compact manifold, M , embedded in X which is invariant with respect to T t , that is, T t (m) ∈ M for all m ∈ M and t ≥ 0. As examples we may think of critical points, periodic orbits or invariant tori, etc. The questions which are addressed here concern the persistence of this invariant manifold under perturbations in the semiflow and the qualitative behavior of the semiflow near the invariant manifold. In general there will be no such manifold for the perturbed flow even in finite dimensional space, see examples given by Hale [Hal3]. However, if there is a certain nondegeneracy of the linearized flow at all points of M then one expects even smooth continuation of M with sufficiently small perturbations of T t . This is as in the Implicit Function Theorem which provides smooth continuation of a zero of a function under smooth perturbation provided the linearization of the function at the zero is nonsingular. For an invariant manifold, rather than just a critical point, the nondegeneracy condition is normal hyperbolicity. This condition c u s c the tangent ⊕ Xm , with Xm gives, for each m ∈ M , a decomposition X = Xm ⊕ Xm space to M at m such that (a) This splitting is invariant under the linearized flow, DT t (m) and t u expands and does so to a greater degree than does DT (m)|X c (b) DT t (m)|Xm m t t s contracts and does so to a greater degree than does DT (m)|X c . while DT (m)|Xm m The superscripts c, u and s stand for “center,” “unstable, ” and “stable.” The precise definition of normal hyperbolicity is given in Section 2, where we give notation and collect some fundamental preliminaries. Our main results may be summarized as Theorem. Suppose that T t is a C 1 semiflow on X and M is a C 2 compact connected invariant manifold on which T t is normally hyperbolic for t sufficiently large. Suppose u is an isomorphism. Let t1 > 0 be large enough also that for each m ∈ M, DT t |Xm and be fixed and N be a fixed neighborhood of M . For any  > 0 there exists σ > 0 such that if T˜ is a C 1 map on X which satisfies kT˜ − T t1 kC 1 (N ) < σ, then (a) Persistence: T˜ has a unique C 1 compact connected normally hyperbolic in˜ near M . variant manifold M ˜ converges to M in the C 1 topology as ||T˜ − T t1 ||C 1 tends (b) Convergence: M zero;

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˜ cs () and W ˜ cu () in a tubu(c) Existence: T˜ has unique C 1 invariant manifolds W lar neighborhood N () of M , which at M are tangent to the center-stable vector ˜c ⊕ X ˜ s and the center-unstable vector bundle X ˜c ⊕ X ˜ u , respectively. bundle X (d) Characterization: n o ˜ cs () = x0 ∈ N () : T˜k (x0 ) ∈ N (), for k ≥ 1, T˜k (x0 ) → M ˜ as k → ∞ , W n ˜ cu () = x0 ∈ N () : ∃ {xk }k>0 ⊂ N (), satisfying W o ˜ ˜ T (xk ) = xk−1 for k ≥ 1, and xk → M as k → ∞ ˜ =W ˜ cs () ∩ W ˜ cu (). and M Furthermore, if T˜t is a C 1 semiflow on X which satisfies kT˜t1 − T t1 kC 1 (N ) < σ and ˜ is a normally hyperbolic invariant manikT˜t − T t kC 0 (N ) < σ for 0 ≤ t ≤ t1 , then M ˜ cs () and center-unstable manifold W ˜ cu (), fold for T˜t with center-stable manifold W respectively. Our Theorems are stated precisely in Section 3. ˜ cs consists Remark. From property (d) we see that the local center-stable manifold W of points for which all forward iterates lie in the tubular neighborhood and approach ˜ . Hence, W ˜ cs can be regarded as the stable manifold of M ˜ . Likewise, the local M ˜ cu (), consists of points for which backward orbits exist, center-unstable manifold, W ˜ . Hence, stay in the tubular neighborhood for all backward iterates, and approach M ˜. this can be regarded as the unstable manifold of M It is instructive to see what this theorem says in simple settings, where the T t is the solution operator for a linear differential equation x0 = Ax,

x ∈ Rn

where A is an n × n matrix of which all eigenvalues have either positive real part or negative real part. In this case, the equilibrium point x = 0 is normally hyperbolic, which is the same as the concept of hyperbolicity. The eigenspace corresponding the eigenvalues with negative real part is the stable space (manifold) of the equilibrium 0 and the eigenspace corresponding the eigenvalues with positive real part is the unstable space (manifold) of the equilibrium 0. For a small C 1 perturbation f (x), x0 = Ax + f (x) has a unique hyperbolic equilibrium point x ˜ near 0 by the Implicit Function Theorem and has also stable and unstable manifolds at x ˜ which are the perturbations of the

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stable space and the unstable space at 0, respectively. These invariant manifolds play a key role in the study of the structural stability of the flow near hyperbolic equilibrium points (see [Har] for instance). A more complicated example is a periodic orbit of an ordinary differential equation x0 = f (x),

x ∈ Rn .

Let p(t) be the periodic solution with period L and let Φ(t) be the fundamental matrix of x0 = Df (p(t))x. The periodic orbit M = {p(t) : 0 ≤ t ≤ L} is normally hyperbolic if all engenvalues of the time-L map Φ(L) are off the unit circle except 1, which is simple. Historical Background. The theory of invariant manifolds for discrete and continuous dynamical systems has a long and rich history. For the case of M consisting of a single equilibrium point, Hadamard [Ha] constructed the unstable manifold of a hyperbolic fixed point of a diffeomorphism of the plane by iterating the mapping applied to a curve in the plane, thereby obtaining a convergent sequence of curves. The limit of this sequence of curves gives the unstable manifold. This geometric method is now called Hadamard’s graph transform. Lyapunov [Ly] and Perron [P1], [P2], [P3] constructed the unstable manifold of an equilibrium point by formulating the problem in terms of an integral equation. This method is analytic rather than geometric and now is called the method of LyapunovPerron. Although the successful application of this approach tends to give more information about the smoothness of the manifold, certain obstacles which we explain later, lead us to use Hadamard’s approach. There is an extensive literature on the stable, unstable, center, center-stable, and center-unstable manifolds of equilibrium points for both finite and infinite dimensional dynamical systems. We do not attempt to give an exhaustive list of references. The general theory for finite dimensional dynamical systems may be found in [BDL], [Ca], [HP], [HPS], [Ir], [Ke], [Ku], [MS], [Pl], [Si], [Sm], [Va], and [VV]. For infinite dimensional dynamical systems we refer the reader to [Ba], [BJ], [CL1], [He], and [VI]. Most of these works use the approach of Lyapunov- Perron. A good treatment of center manifold theory using the Lyapunov-Perron approach for the finite dimensional case may be found in the monograph by Carr [Ca], where many applications are also set forth. Certain infinite dimensional settings are also treated. Vanderbauwhede and Van Gils [VV] also use the Lyapunov Perron method to obtain smooth center manifolds but with some important differences in technique. Ball [Ba] used the Lyapunov-Perron approach to obtain local stable, unstable and center manifolds for equilibrium points of dynamical systems in Banach space, with application to

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the beam equation. Henry [He] developed the theory for semilinear parabolic equations. Later, Chow and Lu [CL1] used this approach to prove the existence of smooth center-unstable manifolds with application to the damped wave equation. For more on center manifold theory in the infinite dimensional setting, using the LyapunovPerron method, see [VI]. The theory of invariant manifolds for an equilibrium point of finite dimensional dynamical using Hadamard’s approach may be found in [HP]. For infinite dimensional dynamical systems, we refer to [BJ], where applications are given demonstrating the stability of a pulse solution to the FitzHugh-Nagumo equations and the instability of stationary solutions to the nonlinear Klein-Gordon equation. Krylov and Bogoliubov [KB] and Mitropolsky [BM] studied time-periodic ordinary differential equations arising from the study of nonlinear oscillations. Under the assumption that the averaged equation has an asymptotically stable equilibrium point, they proved the existence of periodic integral manifolds, which gives the existence of asymptotically stable periodic orbits for a class of equations. An integral manifold is an invariant manifold in the product space of time and phase space. Levinson [Le] studied periodic perturbations of an autonomous ordinary differential equation possessing an asymptotically stable periodic orbit. He proved that if the perturbation was sufficiently small, then the perturbed equation has a periodic integral manifold, which may be viewed as a two-dimensional torus. Levinson’s results were extended to periodic surfaces by Diliberto [Di], Hufford [Hu], Marcus [Ma], and Kyner [Ky1]. Hale [Ha1] established a general theory of integral manifolds for nonautonomous ordinary differential equations and obtained more general results than those just mentioned above. An extension of Hale’s integral manifold theory to a larger class of nonautonomous ordinary differential equations was recently obtained in [Yi]. McCarthy [Mc] studied the persistence of a compact normally hyperbolic invariant manifold for a diffeomorphism, where the normal bundle is the stable normal bundle. McCarthy’s work was extended by Kyner [Ky2], Kurzweil [Ku] and Nyemark [Ny]. Sacker gave further extensions in [Sa]. The most general theory of compact normally hyperbolic invariant manifolds for finite dimensional dynamical systems were independently obtained by Hirsch, Pugh and Shub [HPS1], [HPS2] and Fenichel [F1-3]. They proved the persistence of normally hyperbolic invariant manifolds, and the existence of the center-stable and centerunstable manifolds and their invariant foliations. Ma˜ n´e [Mn1] proved that normal hyperbolicity defined in [HPS] is a necessary condition for the persistence of an invariant manifold for finite dimensional dynamical systems. Recently, Pliss and Sell [PS] studied the persistence of hyperbolic attractors for ordinary differential equations. Sell [Se] has reported that their methods also extend to the infinite dimensional setting. Henry [He] extended Hale’s theory of integral manifolds to the most general nonau-

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tonomous abstract semilinear parabolic equations. His result can also be applied to ordinary differential equations. Henry also studied compact normally hyperbolic invariant manifolds with trivial normal bundle for semilinear parabolic equations and obtained a coordinate transformation which leads to the setting of integral manifolds. His results suggest how one may obtain the persistence of compact normally hyperbolic invariant manifolds under perturbation, by employing the Lyapunov-Perron approach. There has been much recent work surrounding one class of global manifolds, namely inertial manifolds, which are global, attracting, finite dimensional manifolds for dissipative systems. As such these manifolds are rather robust objects which are large enough to contain most of the interesting dynamics of a system. That they are finite dimensional means a substantial reduction in phase space is possible when one is interested in the asymptotic behavior of solutions. A general theory may be found in [FST], [M-PS] and [CFNT]. Our results can not be directly applied to the persistence of inertial manifolds. However, our analysis may be applied to obtain results along these lines. Very recently, Li, McLaughlin, Shatah and Wiggins [LMSW] have proved the persistence of the center-stable and center-unstable manifolds of a circle of stationary solutions for the nonlinear Schr¨ odinger equation as it is perturbed. Note that for the NLS equation the solution operator is a group. Linearizing at this particular circle of stationary solutions, which happens to consist of spatially constant functions, gives trivial stable and unstable bundles. Therefore a global change of variables is possible in a neighborhood of this circle, distinguishing center, stable, and unstable directions. By using the Lyapunov-Perron approach, they proved persistence of center-stable and center-unstable manifolds of the circle. They also produce invariant foliations of center-stable (with stable leaves) and center-unstable (with unstable leaves) manifolds of perturbations of the circle. These foliations are very useful in tracking trajectories and completing a shooting argument to discover homoclinic orbits. In [LMW], Li, McLaughlin, and Wiggins studied the persistence of overflowing manifolds of finite codimension for a C r (r ≥ 2) group S t in a Hilbert space. In this case, for each fixed t ∈ R, S t is a C r diffeomorphism. Under certain assumptions such as trivial normal bundle, uniform boundedness of the second derivative, D2 S t , and that the overflowing manifold is covered by finitely many balls, they obtained the persistence of the overflowing manifold and the existence of an invariant foliation of the overflowing manifold. They did this by using the method of Hadamard’s graph transform. They also applied these results to the Schr¨odinger equation to recover the results of [LMSW]. The versions of the theorems we will establish, in the case that the time-t map is a diffeomorphism, as in the finite dimensional setting, were proved by Hirsch, Pugh and Shub [HPS]. Simultaneously and independently, N. Fenichel [F1] proved a similar result with a somewhat different definition of normal hyperbolicity in terms of a

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variant of Lyapunov number. Our definition of normal hyperbolicity, given in Section 2, is modeled on that given in [HPS]. In both [F1] and [HPS] Hadamard’s graph transform is used to develop the perturbation theory for normally hyperbolic compact invariant manifolds. Their proofs are very elaborate with many details left to the reader. Recently, Wiggins [W] provided more detailed proofs of Fenichel’s theorems while Bronstein and Kopanskii [BK] provided the detailed proof of Hirsch, Pugh and Shub’s theorem on the persistence of normally hyperbolic invariant manifolds. There are several differences between a finite dimensional dynamical system and an infinite dimensional dynamical system. The most significant difference is that the infinite dimensional dynamical systems generated, for example, by parabolic equations, define only semiflows since backwards solutions may not exist. Thus the time-one maps associated with these dynamical systems are not invertible. Furthermore, the phase spaces are not locally compact. In addition, the trivialization of the normal bundle of a manifold, the local orthogonal coordinates, and some results in differential geometry such as the results on smoothing of bundles are not available in Banach spaces. These make the study of normally hyperbolic invariant manifolds for a semiflow in a Banach space much more complicated. In our analysis, the lack of local compactness is mitigated by the compactness of the original manifold. However, the lack of inverse and global trivialization were significant obstacles which greatly added to length of this paper. The results given here lay the groundwork for constructing invariant foliations in a neighborhood of a compact normally hyperbolic invariant manifold for a semiflow. Invariant foliations for flows or semiflows are extremely useful in that they can be used to track the asymptotic behavior of solutions and to provide coordinates in which systems of differential equations may be decoupled and normal forms derived. In [HS] and in Fenichel’s work [F2], [F3], invariant foliations are obtained and some of their uses demonstrated. Since then, the applications of this theory to problems from science and engineering have flourished, especially applications to singular perturbation problems, see for example [D], [G], [GS], [HW], [JK], [KW], [Li], [Sz] and [Te]. Recently Jones [Jo] has given a clear discussion of the use of Fenichel’s theorems as they apply to singular perturbation problems. He includes proofs of these theorems and important extensions of the λ-lemma (see also [JK]). Kirchgraber and Palmer [KP] have recently given detailed results on invariant foliations and their applications to linearizations for finite dimensional systems. Ruelle [Ru] proved a result giving invariant stable and unstable fibrations almost everywhere on a compact invariant set for a semiflow in Hilbert space. It was assumed that the linearized time-t map is compact and injective with dense range. The results are therefore applicable to some parabolic PDE’s. Ma˜ n´e [Mn2] extended Ruelle’s results to semiflows in Banach space. Considering the semiflow generated by a parabolic equation, Lu in [Lu] constructed infinitely many invariant manifolds as perturbations of eigenspaces of the operator ob-

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tained by linearizing at an equilibrium point. With these and corresponding invariant foliations, a new coordinate system was constructed in a neighborhood of the equilibrium point. This facilitated a proof of a Hartman-Grobman theorem for scalar parabolic PDE’s, which yields that the flow near a hyperbolic equilibrium point is structurally stable. In [BL] a more general theorem on the existence of invariant foliations was proved. This theorem was then used to obtain a Hartman-Grobman result for both the phase-field system and the Cahn-Hilliard equation. The previous two papers are concerned with dynamics in a neighborhood of an equilibrium but Chow, Lin and Lu [CLL] proved a general result giving a stable fiber at each point of an inertial manifold, thereby giving a more global invariant foliation of infinite dimensional space. Recently, Aulbach and Garay [AG] used invariant foliations to study partial linearization for noninvertible mappings near fixed points. In the present paper we deal only with the persistence of a normally hyperbolic invariant manifold and the existence of the center-stable and center-unstable manifolds, leaving invariant foliations to a forthcoming paper [BLZ1]. However, this paper sets the stage for that development. We shall discuss application of the results obtained in this paper and in [BLZ1] elsewhere. Nontechnical Overview. Our proof for the existence and persistence of invariant manifolds is based on Hadamard’s graph transform and consists of seven main steps: Step 1. Coordinate Systems. We first build a C 1 tubular neighborhood of M . Then we introduce three coordinate systems. The first coordinate system is given by the tubular neighborhood, identifying it with a neighborhood, N (), in the normal bundle for the manifold. The second is defined by the splitting of the tangent bundle of the phase space X restricted to M , which is a Cartesian system in a neighborhood of a point on M . The third coordinate system is induced by the local trivialization of the bundles based on M . We then estimate how the coordinates differ from each other. These details are given in Section 4. The main reason for these coordinate systems is that we will be looking at Lipschitz graphs over the unstable (or stable) bundle of M . The graph is a global geometric object but the Lipschitz property requires local coordinates for its description. The local trivialization relates two nearby local coordinate systems.

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Figure 1 Step 2. Invariant Cones. In Section 2, we define various cones: The global cone with ‘vertex’ M and ‘axes’ being the stable and unstable bundles, the global fattened cone which includes a small neighborhood of the ‘vertex’, and the global gap cone in which a small neighborhood of the ‘vertex’ is deleted (see Figure 2). By using the normal hyperbolicity of M , we show that these cones are invariant for the unperturbed time−t1 map, T t1 . Furthermore, the global fattened cone and the global gap cone are invariant for a small C 1 perturbation T˜ of T t1 . These statements are proved in Section 5. The invariance of local cones moving with the semiflow is also proved in Section 5. The invariance of these cones play crucial roles in establishing the existence of invariant manifolds. Step 3. Existence of Lipschitz Center-Unstable Manifold. We regard the normal bundle X u ⊕ X s as a bundle over the unstable bundle X u and consider sections of this bundle in the tubular neighborhood N (). Define the complete metric space Γcu of Lipschitz sections, which are described in terms of the global fattened cone and local cones. The fattened cone (and the gap cone) are introduced to allow for perturbations. The key to construct a “graph transform” in the space Γcu is to show that for any given Lipschitz section h ∈ Γcu , the image of the graph of this section under T˜ restricted to the tubular neighborhood is the graph of a section in Γcu . This yields a graph transform F cu defined in Γcu . Using the invariance of the cones, one can prove that F cu is a contraction and the fixed point is the desired center-unstable manifold. This is done in Section 6, where some properties of the center-unstable manifold are also discussed. Step 4. Existence of Lipschitz Center-Stable Manifold. For a finite dimensional dynamical system, one may obtain the stable manifold as the unstable manifold for the corresponding time reversed system(see, for example, [HPS]). However, this technique does not work for infinite dimensional dynamical systems, for example those generated by parabolic equations, since backward solutions may not exist and the corresponding solution map is not invertible. In order to overcome this difficulty, a new technique is needed to establish the existence of the center-stable manifold. We again introduce a complete metric space Γcs of Lipschitz sections, this time, of the bundle X u ()⊕X s () over X s (). For each Lipschitz section h ∈ Γcs , since T˜ may not be invertible, a point on the graph of h may have no preimage or have more than one preimage. Thus the preimage of the graph of the section h under T˜ can be very complicated. In Section 7, we find a way to construct a unique Lipschitz section in Γcs whose the graph is perhaps just a part of the preimage of the graph of h under T˜. This is the key idea, this construction inducing a graph transform in Γs . Finally we show that the this graph transform is a contraction and the fixed point is the desired stable manifold. Again the invariance of the cones is used. Step 5. The Smoothness of the Center-Unstable and Center-Stable Manifolds. For the center unstable manifold, the basic idea to show its smoothness is to find a

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candidate for the tangent bundle of this manifold, which is invariant under the linearization DT˜, then to prove it indeed is tangent to the manifold. The arguments are based on the use of Lipschitz jets, which is borrowed from [HPS]. Since the trivialization of the normal bundle is not available in a Banach space, the proof is more complicated than for finite dimensional systems. We first define a space of sections of the Lipschitz jet bundle, which is different from the jet spaces introduced in [HPS]. Then we construct a graph transform based on the linearization DT˜ and show that it has a unique fixed point which gives the tangent bundle of the center-unstable manifold. A major difficulty in finding the fixed point is that the space of sections of the Lipschitz jet bundle is not complete. Finally, we prove that the tangent bundle is C 0 . For the center-stable manifold, an additional difficulty is that DT˜ may not be invertible. One can use similar arguments to those which gave the existence of the center-stable manifold to construct the graph transform in the space of sections of the Lipschitz jet bundle. Step 6. The Normal Hyperbolicity In Section 10, we show that the intersection of the center-stable manifold and center-unstable manifold is a C 1 compact connected invariant manifold. The basic ˜ is to construct idea to obtain normal hyperbolicity for the perturbed manifold M stable and unstable bundles from the tangent bundles of the center-stable and centerunstable manifolds by finding projection operators. This is done in Section 11. Summarizing all results obtained so far gives the results for the perturbed map T˜. Step 7. Results for the Perturbed Semiflow In Section 12 we show that the results for maps which are C 1 -close to the map T t1 also hold for semiflows whose time-t1 maps are C 1 -close to T t1 and whose time−t maps are C 0 close to T t for t ∈ [0, t1 ]. Acknowledgement. We would like to acknowledge several fruitful conversations with Shui-Nee Chow, Jack Hale, Chris Jones, and Xiao-Biao Lin. PWB and CZ would like to thank the faculty and staff of the Isaac Newton Institute for their hospitality and support during the final stages of this work. We also would like to thank the referee for valuable suggestions.

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2. Notation and Preliminaries. The results and proofs presented in this paper require a certain amount of technical notation which we collect in this section for future reference. Throughout, X will represent a Banach space with norm | · |. In subspaces the same norm symbol is used. The notation k · k will be reserved for the linear operator norm kLk ≡ sup{|Lx| : |x| = 1}. When nonlinear operators are considered on a bounded subset B ⊂ X we use the notation kF k0 = sup{|F (x)| : x ∈ B} and for k ≥ 1 kF kk = kF k0 +

k X

sup{kDj F (x)k : x ∈ B},

j=1

expecting the choice of B to be clear from the context. Here, D is the derivative operator. The open ball centered at x ∈ X of radius r > 0 will be denoted by B(x, r). On X we suppose that there is a semiflow, T t , for which the following conditions hold: (H1) it is continuous on [0, ∞) × X into X, and for each t ≥ 0, T t : X → X is C 1 . (H2) There exists a C 2 compact connected invariant manifold M ⊂ X, i.e., T t (M ) ⊂ M for each t ≥ 0. (H3) M is normally hyperbolic, that is (i) for each m ∈ M there is a decomposition c u s X = Xm ⊕ Xm ⊕ Xm c of closed subspaces with Xm the tangent space to M at m. (ii) For each m ∈ M and t ≥ 0, if m1 = T t (m) α α DT t (m) X α : Xm → Xm for α = c, u, s m

1

t

u u u is an isomorphism from X and DT (m)|Xm m onto Xm1 . (iii) There exists t0 ≥ 0 and λ < 1 such that for all t ≥ t0

  u c k λ inf DT t (m)xu : xu ∈ Xm , |xu | = 1 > max 1, kDT t (m) Xm   c s k , |xc | = 1 > kDT t (m) Xm λ min 1, inf DT t (m)xc : xc ∈ Xm

(2.1) (2.2)

u Condition (2.1) suggests that near m ∈ M, T t is expansive in the direction of Xm , t and at a rate greater than that on M , while (2.2) suggests that T is contractive in s c u the direction of Xm , and at a rate greater than that on M . Thus, fibers Xm , Xm and s Xm are distinguished by the growth and decay rates of the flow, much as one sees with an exponential trichotomy. A consequence of (H3) is the following

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Lemma 2.1. For all t ≥ 0, T t is a C 1 diffeomorphism on M . Proof. Suppose that for some t1 > 0 and points of M, m1 6= m2 , T t1 (m1 ) = T t1 (m2 ). ¯ Let t¯ = inf{t > 0 : T t (m1 ) = T t (m2 )} and denote T t¯(m1 ) = T t¯(m2 ) by m. t2 c Note that for t2 > t0 , DT (m) Xm is invertible by (iii), and so, by the Inverse Function Theorem, there exist neighborhoods U1 of m ¯ and U2 of T t2 (m) ¯ such that t2 T is a diffeomorphism from U1 ∩ M onto U2 ∩ M . Now take δ ∈ (0, t2 ) so small that ¯ = T t¯−δ (m1 ), T t¯−δ (m2 ) ∈ U1 ∩M , then T t2 T t¯−δ (m1 ) = T t2 −δ T t¯(m1 ) = T t2 −δ (m)  t2 t¯−δ t2 T T (m2 ) , contradicting the fact that T is one-to-one on U1 ∩ M . Therefore, for t1 > 0, T t1 |M is one-to-one. Again, using the fact that for t ≥ t0 DT t |X c is an isomorphism, T t (M ) is open in M , by the Inverse Function Theorem. Since M is compact, T t (M ) is closed also, and so T t (M ) = M . Consequently, for any t ≥ t0 , T t |M is a diffeomorphism. For t ∈ [0, t0 ] T t ◦ T t0 −t |M = T t0 −t ◦ T t |M = T t0 |M , which is a diffeomorphism, and therefore T t is also a diffeomorphism. c is invertible with inverse uniformly bounded for t in compact Similarly, DT t (m)|Xm intervals. we have

Lemma 2.2. For any t1 > 0, there exists a constant C1 such that for all m ∈ M and 0 ≤ t ≤ t1 n o −1 t c c c c c k ≤ C1 . C1 ≤ inf |DT (m)x | : x ∈ Xm , |x | = 1 ≤ kDT t (m)|Xm Proof. Let Md2 = {(m1 , m2 ) : m1 , m2 ∈ M, m1 6= m2 }. For i = 1, 2, we define functions fi from [0, t1 ] × Md2 to R+ by |T t (m1 ) − T t (m2 )| , |m1 − m2 | |m1 − m2 | f2 (t, m1 , m2 ) = t . |T (m1 ) − T t (m2 )| f1 (t, m1 , m2 ) =

For m1 , m2 ∈ M , define nZ d(m1 , m2 ) = inf

1

o |γ 0 (τ )|dτ : γ ∈ C 1 ([0, 1], M ), γ(0) = m1 , γ(1) = m2 .

0

It is easy to show that d(·, ·) is a metric on M , and induces an equivalent topology for M . In fact, we first note that d(m1 , m2 ) ≥ |m1 − m2 |, then by the compactness of M there exits a constant C such that d(m1 , m2 ) ≤ C|m1 − m2 |. For each fixed 0 ≤ t ≤ t1 , from Lemma 2.1, t t c kd(m1 , m2 ) ≥ d(T (m1 ), T (m2 )) max kDT t (m)|Xm  −1 ≥ max kD(T t |M )−1 (m)k d(m1 , m2 ),

m∈M

m∈M

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which implies that f1 (t, ·, ·) and f1 (t, ·, ·) are bounded for each fixed 0 ≤ t ≤ t1 . Let Ek = {t ∈ [0, t1 ] : fi (t, m1 , m2 ) ≤ k, i = 1, 2, (m1 , m2 ) ∈ Md2 }. 0 00 Clearly, [0, t1 ] = ∪∞ 1 Ek . By the Baire Category Theorem, there exits 0 ≤ t < t ≤ t1 0 00 2 and k > 0 such that for all t ∈ [t , t ] and (m1 , m2 ) ∈ Md

f1 (t, m1 , m2 ) ≤ k and f2 (t, m1 , m2 ) ≤ k, which implies t −1 c k ≤ k and kD(T |M ) c k ≤ k. kDT t (m)|Xm (m)|Xm

Thus, for t ∈ [0, t00 − t0 ] and m ∈ M ,  0 0 t+t0 2 c k = kDT c k ≤ k . kDT t (m)|Xm (T t |M )−1 (m) D(T t |M )−1 (m)|Xm Similarly, 2 c k ≤ k . kD(T t |M )−1 (m)|Xm

Therefore, by using the semigroup property and the fact that [0, t1 ] is compact, there exists a constant C1 such that Lemma 2.2 holds. This completes the proof. One may ask if (H3) (iii) is need for all t ≥ t0 . In fact, we have the following Lemma 2.3. Let T and M satisfy (H1), (H2) and (i), (ii) in (H3). If there exists 0 < t0 < t1 such that (2.1) and (2.2) hold for all t ∈ [t0 , t1 ], then (2.1) and (2.2) also 0 0 hold for all t ≥ (1 + [ t1t−t ])t0 , where [ t1t−t ] denotes the integer part of the number 0 0 t0 t1 −t0 . t

Proof. Let k be a positive integer. For t ∈ [kt0 , kt1 ] and m ∈ M , let mi = T i k (m) for i = 0, 1, 2, · · · , k − 1. Observe t0 ≤ t/k ≤ t1 and t

t

t

DT t (m) = DT k (mk−1 )DT k (mk−2 ) · · · DT k (m0 ). Thus, by the assumption of this lemma, we have that (2.1) and (2.2) hold for all 0 t ∈ [kt0 , kt1 ]. On the other hand, for k ≥ 1 + [ t1t−t ], 0 kt1 − (k + 1)t0 = k(t1 − t0 ) − t0 ≥ (1 + [

t0 ])(t1 − t0 ) − t0 ≥ 0 t1 − t0

which implies [(1 + [

t0 ])t0 , ∞) ⊂ ∪∞ k=1 [kt0 , kt1 ]. t1 − t0

14

This completes the proof. Associated with the decomposition of X introduced in (H3) are vector bundles α X α ≡ {(m, Xm ) : m ∈ M } , α = c, u, and s.

In the usual way we define the direct sum of bundles, e.g., u s X u ⊕ X s ≡ {(m, xu + xs ) : xu ∈ Xm and xs ∈ Xm , m ∈ M} .

For  > 0 we shall use the notation α X α () ≡ {(m, xα ) : xα ∈ Xm , m ∈ M, |xα | < } , α = c, u, and s,

and u s X u () ⊕ X s () ≡ {(m, xu + xs ) : xu ∈ Xm , xs ∈ Xm , m ∈ M, |xu | < , |xs | < } ,

the latter of which may be identified with a tubular neighborhood of M , as will be seen in Section 4. The mapping which does this is Θ : X u ⊕ X s → X defined by Θ(m, xu +xs ) = m+xu +xs . For m ∈ M we shall also use projections, Πα m , associated with the decomposition of X given in (H3), for α = c, u, and s. We assume (H4) The mapping from M ⊂ X → L(X), the continuous linear operators on X, 1 defined by m → Πα m is C for α = c, u, and s. In [BLZ2], we remove the assumption on the smoothness of the bundles, that is, (H4) can be removed and the assumption that the manifold M is C 2 can be relaxed to require only C 1 . Remark. If | · |X α is defined by |(m, x)|X α = |x| with | · | the norm in X, then the continuity in m of the above mapping gives the bundle X α a so-called Finsler structure. To show that Θ is a diffeomorphism we will require neighborhoods to be so small that Πα m changes very little as m varies in the neighborhood. We therefore give the following: Definition: For m0 ∈ M and η ∈ (0, 1), a neighborhood U of m0 in X is said to be α an η-neighborhood if kΠα m − Πm0 k ≤ η for all m ∈ U ∩ M , for α = c, u, and s. Remark. Because M is compact, for any η > 0 there exists r > 0 independent of m ∈ M such that B(m, r) is an η-neighborhood. With these projections we can define local trivializations of the bundles: For m0 ∈ M, U a neighborhood of m0 , and for α = c, u, and s, define α α Φα m0 : (U ∩ M ) × Xm0 → X

15

by α α α Φα m0 (m, x ) = (m, Πm x )

and  u s u s Φus m0 : (U ∩ M ) × Xm0 ⊕ Xm0 → X ⊕ X by u s u u s s Φus m0 (m, x + x ) = (m, Πm x + Πm x ) .

In Section 4 we show that for  > 0 and sufficiently small Θ is a diffeomorphism from X u () ⊕ X s () onto a tubular neighborhood, V , of M . As such, we may define −1 Ψ ≡ Θ X u ()⊕X s () : V → X u () ⊕ X s () with component functions denoted through Ψ(x) = (ψ(x), ψ u (x) + ψ s (x)) and ψ α (x) = Πα ψ(x) (x − ψ(x)). In section 5 we prove that certain cone-like sets have invariance properties as a result of the normal hyperbolicity of M . These geometrical properties of the flow will then be used to establish the main results. Here we define and illustrate the various “cones” to be used. For  > δ > 0 and small and µ > 0 we define: The global cone o n u s u s u s u s K(, µ) = m + x + x : (m, x + x ) ∈ X () ⊕ X () and µ|x | ≥ |x | the fattened cone u

s

Kf (, µ, δ) = {m + xu + xs : (m, xu + xs ) ∈ X () ⊕ X () and µ|xu | ≥ |xs | or |xs | ≤ δ, } and the gap-cone u

s

Kg (, µ, δ) = {m + xu + xs : (m, xu + xs ) ∈ X () ⊕ X (), µ|xu | ≥ |xs | and |xu | ≥ δ}. The fattened cone and the gap cone are introduced to allow for perturbations. The closures of their complements in the -neighborhood of M will be denoted with a prime. For instance, u

s

Kf0 (, µ, δ) = {m + xu + xs : (m, xu + xs ) ∈ X () ⊕ X (), µ|xu | ≤ |xs | and |xs | ≥ δ}. Some of these cones are illustrated below.

16

17

Figure 2

18

§3. Statements of Theorems. The principal result of this paper is the persistence of compact, smooth, normally hyperbolic invariant manifolds under C 1 perturbations of the semiflow. We shall, however, prove more than this by first considering maps which are C 1 perturbations, T˜, of the time-t map associated with the original semiflow. Secondly, we obtain, ˜ , the perturbed compact invariant manifold by showing the existence of centerM unstable and center-stable manifolds for T˜, in a neighborhood of M , and then taking their intersection. Furthermore, we give characterizations of the center-unstable and center-stable manifolds both for maps and perturbations of the semiflow. Let B be a fixed neighborhood of M in X containing the tubular neighborhood Θ(X u (0 )) ⊕ X s (0 )). We consider a C 1 perturbation T˜ of the time-t map T t . Let T˜ : B → X be a C 1 map. Recall that kT˜ − T t k0 = sup |T˜(x) − T t (x)| x∈B

and kT˜ − T t k1 = sup |T˜(x) − T t (x)| + sup kDT˜(x) − DT t (x)k. x∈B

x∈B

Theorem A. Let t > t0 be fixed. For each small  > 0, there exists σ > 0 such that if ˜ cu () in the tubular neighborhood, ||T˜ − T t ||1 ≤ σ, then T˜ has a unique C 1 manifold W Θ(X u () ⊕ X s ()) which satisfies ˜ cu ()) ∩ Θ(X u () ⊕ X s ()) = W ˜ cu () (i) T˜(W ˜ cu () ∩ T˜−1 (W ˜ cu ()) → W ˜ cu () is a diffeomorphism, (ii) T˜ : W ˜ cu () = ∩∞ Ak , where Ak is defined by induction, Ak = T˜(Ak−1 ) ∩ (iii) W k=1 Θ(X u () ⊕ X s ()) and A0 = Θ(X u () ⊕ X s ()). ˜ cu (), consists From property (iii) we see that the local center-unstable manifold, W of points for which backward orbits exist and stay in the tubular neighborhood for all backward iterates. Likewise, the local center-stable manifold consists of points for which all forward iterates lie in the tubular neighborhood, as described in (ii) below. These, then, can be taken as definitions of the center-unstable and centerstable manifolds. Theorem B. Let t > t0 be fixed. For each small  > 0, there exists σ > 0 such that if ˜ cs () in the tubular neighborhood, ||T˜ − T t ||1 ≤ σ, then T˜ has a unique C 1 manifold W u s Θ(X () ⊕ X ()) which satisfies ˜ cs ()) ⊂ W ˜ cs (), (i) T˜(W n cs ∞ ˜ (ii) W () = ∩k=1 A−k , where A−k is defined by induction, A−k = x : T˜(x) ∈ o A1−k ∩ Θ(X u () ⊕ X s ()) and A0 = Θ(X u () ⊕ X s ()).

19

˜ , an invariant manifold The above theorems combine to yield the existence of M which is a perturbation of M . Theorem C. Let t > t0 be fixed. For each small  > 0, there exists σ > 0 such that if ||T˜ − T t ||1 ≤ σ, then T˜ has a unique C 1 compact connected normally hyperbolic ˜ in the tubular neighborhood, Θ(X u () ⊕ X s ()) which satisfies invariant manifold M ˜ =W ˜ cs () ∩ W ˜ cu (), (i) M ˜ →M ˜ is a C 1 diffeomorphism, (ii) T˜ : M ˜ which satisfies (iii) There exists a C 1 diffeomorphism K = K ˜ : M → M T

||KT˜ − I||C 1 (M,X) → 0,

as

||T˜ − T t ||1 → 0.

˜ cs () and W ˜ cu () at M ˜ are tangent to X ˜s ⊕ X ˜ c and X ˜u ⊕ X ˜ c , respectively, (iv) W ˜u ⊕ X ˜s ⊕ X ˜ c is the invariant decomposition associated with the where X = X normal hyperbolicity. ˜ cs () and W ˜ cu () the (local) center-stable and center-unstable We now may call W manifolds for T˜. ˜ cu () and W ˜ cs () given in Theorem A In addition to the characterizations of W ˜ cu () is the unstable (iii) and Theorem B (ii), respectively, we actually have that W ˜ and W ˜ cs () is the stable manifold of M ˜: manifold of M Theorem D. n cu ˜ W () = x0 ∈ Θ(X u () ⊕ X s ()) : there exists {xk }k>0 ⊂ Θ(X u () ⊕ X s ()), o ˜ as k → ∞ , such that T (xk ) = xk−1 , for k ≥ 1, and xk → M and n ˜ cs () = x0 ∈ Θ(X u () ⊕ X s ()) : T˜k (x0 ) ∈ Θ(X u () ⊕ X s ()), k ≥ 1, W o ˜ as k → ∞ . and T˜k (x0 ) → M

We now consider the perturbed semiflow T˜t of T t . Assume the perturbed semiflow T˜t is continuous on [0, ∞) × X into X, and for each t ≥ 0, T˜t : X → X is C 1 . ˜ cs () ˜ cu (), W For t1 > t0 we apply theorems A–C to the time-t1 map T˜t1 , to obtain W ˜ if ||T˜t1 − T t1 ||1 is sufficiently small. The following results indicate that they and M are the center-stable, center-unstable, and normally hyperbolic invariant manifolds for the the semiflow T˜t .

20

Theorem A0 . Let t1 > t0 be fixed. For each small  > 0, there exists σ > 0 such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 then ˜ cu ()) ∩ Θ(X u () ⊕ X s ()) ⊂ W ˜ cu () for 0 ≤ t ≤ t1 , (i) T˜t (W ˜ cu (), if T˜t (x) ∈ Θ(X u () ⊕ X s ()) for 0 ≤ t ≤ t2 , then T˜t (x) ∈ (ii) For x ∈ W ˜ cu () for 0 ≤ t ≤ t2 , W ˜ cu () ⊃ ∩t≥0 A˜t , where (iii) W  A˜t = m0 + xu0 + xs0 ∈ Θ(X u () ⊕ X s ()) : there exists (m1 , xu1 + xs1 ) ∈ X u () ⊕ X s () such that T˜t (m1 + xu1 + xs1 ) = m0 + xu0 + xs0 , and T˜τ (m1 + xu1 + xs1 ) ∈ Θ(X u () ⊕ X s ()), for all 0 ≤ τ ≤ t . Theorem B0 . Let t1 > t0 be fixed. For each small  > 0, there exists σ > 0 such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 then u s ˜t ˜ cs ˜ cs (), (i) For ⊂W  all t u≥ 0, sT (Wt ()) ∩ uΘ(X s() ⊕ X ()) u ˜ cs (). (ii) m0 + x0 + x0 : T˜ (m0 + x0 + x0 ) ∈ Θ(X () ⊕ X s ()), t ≥ 0 ⊂ W

Theorem C0 . Let t1 > t0 be fixed. For each small  > 0, there exists σ > 0 such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 ˜ is a C 1 compact connected normally hyperbolic invariant manifold for M ˜ and then M t 1 ˜ onto M ˜. for each t ≥ 0, T˜ is a C diffeomorphism from M Theorem D0 . ˜ cu (), limt→∞ d(T˜−t (x), M ˜ ) = 0, uniformly on W ˜ cu () (i) For each x ∈ W and ˜ cs (), limt→∞ d(T˜t (x), M ˜ ) = 0, uniformly on W ˜ cs () (ii) For each x ∈ W Remark. In trying to apply these theorems it is sometimes helpful to know how σ depends upon . In the case that T t is C 2 for t ≥ 0, then one can show that σ can be taken to be of order s for any s > 1.

21

§4. Local Coordinate Systems. In this section, we first prove that the bundles introduced in Section 2 are C 1 and that we can find a tubular neighborhood of M which is C 1 −diffeomorphic to open set in X u ⊕ X s . Then we introduce three coordinate systems. The first coordinate system is given by the tubular neighborhood. The second is defined by the splitting of the tangent bundle of the phase space X restricted to M into its center, stable, and unstable components, as mentioned in (H3), which is a Cartesian coordinate system in a neighborhood of a point on M . The third coordinate system is induced by the local trivialization of the bundles based on M . Then we estimate how the coordinates differ from each other. α Let us first establish the relations among corresponding fibers Xm , α = u, s, and c, as the base point moves through M . α α for α = u, s, c. is isomorphic to Xm Lemma 4.1. For m1 , m2 ∈ M, Xm 2 1

Proof. From the hypothesis (H4), Πα m is continuous in m, thus for fixed m and 0 < η < 1 there exists an η-neighborhood Vm of m in X, that is, α kΠα m ¯ − Πm k ≤ η, α = u, s, c

(4.1)

for any m ¯ ∈ Vm ∩ M . √ α α We first claim that if η < 2 − 1, then Xm is isomorphic to Xm ¯ ∈ Vm ∩ M . ¯ for m α To see this, let x ∈ Xm , then (4.1) implies α α |Πα m ¯ x − x| = |Πm ¯ x − Πm x| ≤ η|x|,

(4.2)

|Πα m ¯ x| ≤ (1 + η)|x|.

(4.3)

which yields α Similarly, for y ∈ Xm ¯ α α |Πα m y − y| = |Πm y − Πm ¯ y| ≤ η|y|. α Thus, for x ∈ Xm α |Πα m Πm ¯ x − x| α α α ≤ |Πα m Πm ¯ x − Πm ¯ x| + |Πm ¯ x − x|

≤ η|Πα m ¯ x| + η|x| ≤ η(2 + η)|x|, which implies α α k ≤ η(2 + η) . kI − Πα m Πm ¯ Xm

22

√ α α α α α Π is invertible. Hence Π Π It follows that, as long as η < 2−1, Πα X m m ¯ m m ¯ Xm is an m α α is an isomorphism. isomorphism. Interchanging m and m, ¯ we obtain that Πm ¯ Πm Xm ¯ α α α α Therefore Πm Xm¯ and Πm ¯ Xm are isomorphisms. α Note that (4.2) also yields for x ∈ Xm |Πα m ¯ x| ≥ (1 − η)|x|.

(4.4)

Therefore we have the estimates −1 α α α k ≤ 1 + η and k Π k≤ kΠα X m ¯ m ¯ Xm m

1 1−η

(4.5)

α Since M is compact, we conclude that for any m1 , m2 ∈ M , Xm is isomorphic to 1 α Xm2 . This completes the proof.

As we mentioned in Section 2, for (m, xα ) ∈ X α , α = u, s, c, we may define a “norm” for the bundle X α for α = u, s, c, by k(m, xα )k = |xα |. Under this norm, we have Lemma 4.2. X α for α = u, s, c, is a C 1 bundle with the local trivialization Φα m0 (defined in Section 2). Proof. For each m1 , m√ 2 ∈ M , let Vm1 and Vm2 be η−neighborhoods of m1 and of m2 , respectively, with η < 2 − 1. Let V = Vm1 ∩ Vm2 ∩ M and suppose V 6= φ. Consider Φα m2

−1

α α Φα m1 : V × Xm1 → V × Xm2 .

−1 α We first notice that from (4.5), Φα Φm1 is well-defined. m2 −1 α α α α to Xm and the Φm1 (m, ·) is an isomorphism from Xm Furthermore, Φm2 2 1 α image of (m, x1 ) is given by −1 α (m, xα 2 ) = Φm2 Φm1 (m, x1 ) ,

where

 −1   α α α α α α = x (m) = Π xα Π 2 2 m Xm2 m Xm1 x1 .

(4.6)

Let us first formally compute the derivative of xα 2 (m) in m. Applying the projection to (4.6), we obtain α α α Πα (4.7) m x2 (m) = Πm x1 .

Πα m

23

Formally computing the derivative of (4.7) in m, we have α α α α α Πα m Dx2 (m) + (DΠm )x2 (m) = DΠm x1 .

Thus Dxα 2 (m)

=





α Πα m Xm 2

−1

α α (DΠα m ) (x1 − x2 (m)) .

(4.8)

It is clear from (4.5) that the right hand side of (4.8) is well-defined. Next we shall show that the right hand side of (4.8) indeed is the derivative of xα (m). 2 0 α Observe that xα 2 (m) is C in m uniformly with respect to unit vectors x1 . In fact, from (4.7) we have α α α α α Πα ¯ − xα m ¯ (x2 (m) 2 (m)) = (Πm ¯ − Πm ) (x1 − x2 (m)) . α α is invertible, it Thus, from the continuity of Πα ¯ |Xm m in m and the fact that Πm 2 α 0 follows that x2 (m) is C in m uniformly with respect to unit vectors xα 1. α 1 2 In order to prove x2 (m) is C , since M is a C manifold, we choose a C 2 local chart (Um ∩ M, φ), where

φ : B(0, 1) ⊂ Rn → Um ∩ M and φ(0) = m. We first note that α ∗ α Πα φ(0) (x2 (φ(τ v )) − x2 (φ(0))) α ∗ α = Πα φ(τ v ∗ ) (x2 (φ(τ v )) − x2 (φ(0)))   α α ∗ α − Πα φ(τ v ∗ ) − Πφ(0) (x2 (φ(τ v )) − x2 (φ(0))) .

Thus, by using (4.7), we find the identity α ∗ α α α α α Πα φ(τ v ∗ ) (x2 (φ(τ v )) − x2 (φ(0))) = −(Πφ(τ v ∗ ) − Πφ(0) )(x2 (φ(0)) − x1 ). 1 Since Πα m is C in m by hypotheses (H4), using the Taylor expansion, we obtain



Πα φ(τ v ∗ )



Πα φ(0)



α (xα 2 (φ(0)) − x1 )

∗ α α α α = τ DΠα φ(0) (Dφ v )(x2 (φ(0)) − x1 ) + τ |x2 (φ(0)) − x1 | O(τ ),

24

where the quantity O(τ ) does not depend on v ∗ or xα 1 and O(τ ) → 0 as τ → 0. Hence, 1 α (x (φ(τ v ∗ )) − xα 2 (φ(0))) τ 2 ∗ α α α α = −DΠα φ(0) (Dφ v ) (x2 (φ(0)) − x1 ) − |x2 (φ(0)) − x1 | O(τ )  1 α α ∗ α Πφ(τ v∗ ) − Πα − φ(0) (x2 (φ(τ v )) − x2 (φ(0))) . τ

Πα φ(0)

(4.9)

∗ α ∗ α α 1 Since xα 2 (φ(τ v )) → x2 (φ(0)) as τ → 0 uniformly in v and x1 , and Πm is C in m, letting τ → 0 in (4.9), we find

Πα φ(0)

1 α ∗ α α α (x (φ(τ v ∗ )) − xα 2 (φ(0))) → −DΠφ(0) Dφ v (x2 (φ(0)) − x1 ) τ 2

α uniformly in v ∗ and xα 1 . Therefore, x2 (m) is differentiable and (4.8) holds since α Πα φ(0) |Xm2 is invertible. Finally, we show that Dxα 2 (m) is continuous in m. We first notice that (4.8) may also be written as

 −1 α α α α α Dxα (m) = Π Πα 2 m Xm m (DΠm ) (x1 − x2 (m)) . 2

α α Since DΠα m is continuous in m from (H4) and (x1 − x2 (m)) is continuous in m uniformly with respect to unit vectors xα 1 , it is sufficient to prove that





α Πα m Xm 2

−1

Πα m

is continuous in m. Let x ∈ X and set  −1 α Πα y = Πα mx m Xm2

 −1 α and y¯ = Πα Πα m ¯ Xm2 m ¯ x.

We obtain α |Πα m x − Πm ¯ x| α = |Πα ¯| m y − Πm ¯y α ≥ |Πα ¯)| − kΠα m ¯ (y − y m − Πm ¯ k|y|.

Thus, α |Πα ¯)| ≤ kΠα m ¯ (y − y m − Πm ¯ k(|x| + |y|)

(4.10)

25

On the other hand, from (4.5) it follows that |y − y¯| = |



≤k ≤



−1



−1

α Πα m ¯ Xm 2



α Πα m ¯ Xm 2

Πα ¯)| m ¯ (y − y k|Πα ¯)| m ¯ (y − y

1 |Πα¯ (y − y¯)|. 1−η m

We also note that |y| ≤

C |x|, 1−η

where C is a bound for kΠα m k. Therefore, |y − y¯| ≤

1+C −η α kΠm − Πα m ¯ k|x| (1 − η)2

which implies k





α Πα m Xm 2

−1

Πα m







α Πα m ¯ Xm 2

−1

Πα m ¯k ≤

1+C −η α kΠm − Πα m ¯ k. (1 − η)2

(4.11)

 −1 α This gives the continuity of Πα Πα m Xm2 m in m and completes the proof. From Lemma 4.2 one easily sees that X u ⊕ X s is a C 1 bundle. The next result implies that the image of X u () ⊕ X s () is a tubular neighborhood of M for  sufficiently small. Lemma 4.3. There exists  > 0 such that Θ is a C 1 diffeomorphism from X u () ⊕ X s () onto a neighborhood of M . Proof. For m0 ∈ M , let B(m0 , r) be an η-neighborhood of m0 with η < Consider the map



2 − 1.

 u s ΘΦus m0 : B(m0 , r) ∩ M × Xm0 ⊕ Xm0 → X. Let (xc1 , xu1 + xs1 ) be a tangent vector at (m, 0) to the manifold B(m0 , r) ∩ M × u s Xm ⊕ Xm . A simple computation gives 0 0 c u s c u u s s D(ΘΦus m0 ) (m,0) (x , x1 + x1 ) = x + Πm x1 + Πm x1 .

26

u us s s and Π Since Πum Xm m Xm0 are isomorphisms from (4.5), D(ΘΦm0 ) is an isomor0 c u s phism from Xm × (Xm ⊕ Xm ) to X. Therefore, by the Inverse Function Theo0 0 0 us a diffeomorphism from a neighborhood of (m, 0) ∈ B(m0 , r) ∩ M × rem, ΘΦm0 is  u s Xm0 ⊕ Xm0 onto a neighborhood of m in X. Hence, Θ is a diffeomorphism from a neighborhood of (m, 0) ∈ X u ⊕ X s onto a neighborhood of m in X. Because of the compactness of M , there is a finite covering of M ,

B(mi , i ) = {m ∈ M : |m − mi | < i } , i = 1, 2, · · · , k, where i are positive constants chosen such that Θ restricted on each subset E(mi , i ) = {(m, xu + xs ) ∈ X u (i ) ⊕ X s (i ) : m ∈ B(mi , 5i )} is a diffeomorphism. Let  =min{i }. To show that Θ is a diffeomorphism from X u () × X s () onto a neighborhood of M in X, it is sufficient to prove that Θ is one-to-one. Let (m, xu + xs ), (m, ¯ x ¯u + x ¯s ) ∈ X u () ⊕ X s (). We first notice that m ∈ B(mi , i ) for some i. Suppose Θ(m, xu + xs ) = Θ(m, ¯ x ¯u + x ¯s ), i.e., m + xu + xs = m ¯ +x ¯u + x ¯s . Then |m − m| ¯ = |¯ xu − xu + x ¯ s − xs | < 4, which gives m ¯ ∈ B(mi , 5i ) and (m, xu + xs ), (m, ¯ x ¯u + x ¯s ) ∈ E(mi , i ), hence m = m, ¯ xu = x ¯ u , xs = x ¯s . This completes the proof. Obviously, Lemma 4.3 holds for any 0 < 1 < . The next result concerns about the projection Πcm . Lemma 4.4. For any  > 0, there exists β ∗ > 0 such that if m0 , m1 , m2 ∈ M satisfy |m1 − m0 | < β ∗ and |m2 − m0 | < β ∗ then

|m1 − m2 − Πcm0 (m1 − m2 )| < . |m1 − m2 |

(4.12)

27

Proof. For m0 ∈ M , let φm0 : B(0, 1) ⊂ Rn → Um0 ∩ M be the C 2 coordinate map with φm0 (0) = m0 . Choose β0∗ > 0 such that if |m1 − m0 | < β0∗ and |m2 − m0 | < β0∗ then m1 , m2 ∈ Um0 ∩ M . Let vi = φ−1 m0 (mi ), i = 1, 2. We find 1

Z m1 − m2 =

Dφm0 (τ v1 + (1 − τ )v2 ) (v1 − v2 )dτ 0

= o(1)(v1 − v2 ) + Dφm0 (0)(v1 − v2 )

(4.13)

as v1 , v2 → 0. Applying the projection Πcm0 to (4.13), we obtain Πcm0 (m1 − m2 ) = Dφm0 (0)(v1 − v2 ) + Πcm0 (o(1)(v1 − v2 )). Subtracting the above identity from (4.13), we have |m1 − m2 − Πcm0 (m1 − m2 )| = o(1)|v1 − v2 | = o(1)|m1 − m2 |. as m1 , m2 → m0 . From the fact that M is compact and Πcm is continuous in m, by a standard compactness argument, it follows that there exists β ∗ > 0 such that (4.12) holds. The proof is complete. Let  > 0 be the same as in Lemma 4.3. For m0 ∈ M , let B(m0 , r) be an ηneighborhood of m0 , where r is chosen so that this is an η-neighborhood with η
0 be fixed such that Θ(X u (0 ) ⊕ X s (0 )) is a tubular neighborhood contained in the fixed neighborhood B of M introduced in Section 3. We shall also work with a smaller tubular neighborhood Θ(X u () ⊕ X s ()) such that its image under the time-t map T t stays in the fixed tubular neighborhood Θ(X u (0 ) ⊕ X s (0 )), which is assured by the next lemma. In fact, we shall work in such a neighborhood with  even smaller. Lemma 5.1. Let 0 ≤ t1 < ∞. For the fixed 0 , there exists ∗ > 0 such that for 0 <  < ∗  T t Θ(X u () ⊕ X s ()) ⊂ Θ(X u (0 ) ⊕ X s (0 )) for 0 ≤ t ≤ t1 .

(5.1)

The proof of this lemma, which we omit, is based on the continuity of T t and the compactness of M . Throughout the remainder of this paper we shall assume that  satisfies (5.1) The next lemma gives the invariance of the cones under the unperturbed time-t map T t . Lemma 5.2. Let t ≥ t0 be fixed. For any λ1 ∈ (λ, 1) and µ > 0, there exists ∗ > 0 such that for each  ∈ (0, ∗ ) and δ ∈ (0, ), the following statements hold:  (i) T t K(, µ) ∩ Θ(X u () ⊕ X s ()) ⊂ K(, λ21 µ),  (ii) T t Kg (, µ, δ) ∩ Θ(X u () ⊕ X s ()) ⊂ Kg (, λ21 µ, λ−1 1 δ),  t 2 u s (iii) T Kf (, µ, δ) ∩ Θ(X () ⊕ X ()) ⊂ Kf (, λ1 µ, λ1 δ). Proof. We first define a function F :X ×X →X by F (x, y) =

T t (y) − T t (x) − DT t (x)(y − x) , |y − x|

and F (x, x) = 0.

for x 6= y,

31

From the smoothness of T t it follows that F (x, y) is a continuous function. Let (m0 , xu0 + xs0 ) ∈ X u () ⊕ X s (). By Lemma 5.1, we have that for any fixed 2 < 0 as long as  is sufficiently small, then  T t Θ(X u () ⊕ X s ()) ⊂ Θ(X u (2 ) ⊕ X s (2 )). we may write T t (m0 + xu0 + xs0 ) = m1 + xu1 + xs1 , α where xα 1 ∈ Xm1 (2 ). By the Taylor expansion, we find

m1 + xu1 + xs1 = T t (m0 ) + DT t (m0 )(xu0 + xs0 ) + F (m0 , xu0 + xs0 + m0 )|xu0 + xs0 |. (5.2) Therefore, |m1 − T t (m0 )| ≤ |xu1 | + |xs1 | + kDT t (m0 )k(|xu0 | + |xs0 |) + |F (m0 , xu0 + xs0 + m0 )||xu0 + xs0 |) ≤ 22 + 2C + 2|F (m0 , xu0 + xs0 + m0 )|. The fact that F is continuous in a neighborhood of the compact manifold M × M implies that |F (m0 , xu0 + xs0 + m0 )| = O() uniformly with respect to m0 , , xu0 , and xs0 . In order to apply the estimates of Lemma 4.5, we first need to show that m1 is in an η-neighborhood of T t (m0 ). By Lemma 4.4, for 1 < 0 there exists β ∗ > 0 such that (4.12) holds with 1 instead of . Let η, r satisfy (4.14). 1 Let 1 < 2C where C is the constant in Lemma 4.5. Choose 2 such that 2 ≤ 14 r and choose ∗ to satisfy the requirement that Lemma 5.1 holds and for any  ∈ (0, ∗ ) 2C + 2|F (m0 , xu0 + xs0 + m0 )| ≤

r . 2

(5.3)

Thus, we obtain |m1 − T t (m0 )| ≤ r,

(5.4)

which implies that m1 is in an η-neighborhood of m ¯ = T t (m0 ). Hence, we may write m1 + xu1 + xs1 as m1 + xu1 + xs1 = m1 + Πum1 x ˜u1 + Πsm1 x ˜s1 , (5.5) α where x ˜α ¯ for α = u, s. 1 ∈ Xm Applying Lemma 4.5 to points m ¯ and m1 + xu1 + xs1 and using (5.2) and (5.4), we obtain u s u s |Πcm ¯ ≤ C1 |m1 − m| ¯ ¯ F (m0 , x0 + x0 + m0 )|x0 + x0 | − (m1 − m)|

32

which, since 1
λ and inf |ˆxu0 |=1 |DT t (m0 )ˆ xu0 | > λ1 , we further require that ∗ is chosen to ∗ satisfy for  <  (λ21 − λ2 ) inf |DT t (m0 )ˆ xu0 | − O() > 0, u |ˆ x0 |=1

which implies that (i) holds. To see that (ii) holds, it is enough to show that |xu1 | ≥ λ−1 1 δ. From (5.9) and the normal hyperbolicity we have |xu1 | ≥ λ−1 |xu0 | − O()(1 + µ)|xu0 |  = λ−1 − O() |xu0 |. Thus by choosing smaller ∗ if necessary, we obtain |xu1 | ≥ λ−1 1 δ To see that (iii) holds, since (i) holds, it is enough to show that if |xs0 | ≤ δ and µ|xu1 | < |xs1 | then |xs1 | ≤ λ1 δ. This directly follows from (5.10) and the normal hyperbolicity. In fact |xs1 |



λ|xs0 |

  1 |xs0 |. + O() 1 + µ

Thus, requireing ∗ to be smaller if necessary, the proof is complete. Recall that B is a fixed neighborhood of M in X containing the tubular neighborhood Θ(X u (0 ) ⊕ X s (0 )). We consider a C 1 perturbation T˜ of the time-t map T t . We set kT˜ − T t k0 = sup |T˜(x) − T t (x)| x∈B

and kT˜ − T t k1 = sup |T˜(x) − T t (x)| + sup kDT˜(x) − DT t (x)k x∈B

x∈B

A direct consequence of Lemma 5.2 is Lemma 5.3. Let λ2 ∈ (λ1 , 1) and let µ, , and δ be the same as in Lemma 5.2. Then there exists a σ > 0 such that if kT˜ − T t k1 < σ, we have (i) (ii)

 T˜ Kg (, µ, δ) ∩ Θ(X u () ⊕ X s ()) ⊂ Kg (, λ22 µ, λ−1 2 δ),  2 u s ˜ T Kf (, µ, δ) ∩ Θ(X () ⊕ X ()) ⊂ Kf (, λ2 µ, λ2 δ).

34

The invariance of the gap cone Kg (, µ, δ) and the fattened cone Kf (, µ, δ) is preserved under small C 1 perturbation of the time-t map, however the invariance of the regular cone K(, µ) is not preserved under the perturbation since M will no longer be invariant. This is the reason we introduce the gap and fattened cones. Finally, we establish the invariance of moving cones, which will be stated in terms of inequalities. Before we state this moving cone lemma, we need a lemma which is a generalization of Lemma 5.1. Lemma 5.4. For 2 < 0 , there exist positive constants ∗ , ˜∗ and σ such that if  < ∗ , ˜ < ˜∗ and kT˜ − T t k0 < σ, then  T˜ Θ(X u () ⊕ X s ()) ⊂ Θ(X u (2 ) ⊕ X s (2 )) α (˜ and for all (m, xu + xs ) ∈ X u () ⊕ X s () and x ˆα ∈ Xm ), α = u, s, c,

m + xu + xs + x ˆu + x ˆs + x ˆc ∈ Θ(X u (2 ) ⊕ X s (2 ))

(5.11)

T˜(m + xu + xs + x ˆu + x ˆs + x ˆc ) ∈ Θ(X u (2 ) ⊕ X s (2 )).

(5.12)

and

Again the proof of this lemma, which we omit, is based on the compactness of M and the continuity of T t and T˜. Now, for each (m, xu + xs ) ∈ X u () ⊕ X s () we may write T˜(m + xu + xs ) as T˜(m + xu + xs ) = m1 + xu1 + xs1 α where xα 1 ∈ Xm1 (2 ), α = u, s. We may also write

T˜(m + xu + xs + x ˆu + x ˆs + x ˆc ) = m1 + xu1 + xs1 + x ˆu1 + x ˆs1 + x ˆc1 .

(5.13)

α where x ˆα 1 ∈ Xm , α = u, s, c. The next results state the “invariance” of moving cones in terms of inequalities.

Lemma 5.5. There exist positive constants ∗ , ˜∗ and σ such that if  < ∗ , ˜ < ˜∗ and kT˜ − T t k1 < σ then the following statements hold for all (m, xu + xs ) ∈ X u () ⊕ α (˜ X s (), x ˆα ∈ Xm ), α = u, s, c: (i) if |ˆ xs | ≤ µ(|ˆ xu | + |ˆ xc |) then |ˆ xs1 | ≤ λ1 µ(|ˆ xu1 | + |ˆ xc1 |), u s c u s c x1 |. (ii) if |ˆ x | + |ˆ x | ≤ µ|ˆ x | then |ˆ x1 | + |ˆ x1 | ≤ λ1 µ|ˆ

35

Remark. If we define the following local cones, this lemma says that they are invariant under T˜ relative to the evolving vertex (See Figure 3). For (m, xu + xs ) ∈ X u () ⊕ X s (), and for µ > 0, define Ku (m + xu + xs , µ, ˜) n = m + xu + xs + x ˜u + x ˜s + x ˜c : α

x ˜ ∈

α Xm (˜ )

o for α = u,s, and c, µ|˜ x | > (|˜ x | + |˜ x |) u

c

s

and Kcu (m + xu + xs , µ, ˜) n = m + xu + xs + x ˜u + x ˜s + x ˜c : o α x ˜α ∈ Xm (˜ ) for α = u,s, and c, µ(|˜ xu | + |˜ xc |) > |˜ xs | .

36

Figure 3. Moving Local Cones Proof of Lemma 5.5. Applying the Taylor expansion to (5.13), we have m1 + xu1 + xs1 + x ˆu1 + x ˆs1 + x ˆc1 = T˜(m + xu + xs ) + DT˜(m + xu + xs )(ˆ xu + x ˆs + x ˆc ) u

s

(5.14)

c

+ O(˜  +  + σ)|ˆ x +x ˆ +x ˆ |, where O(˜  +  + σ) → 0 as ˜ +  + σ → 0 uniformly in m ∈ M . From (5.14) and the continuity of DT t it follows that m1 + xu1 + xs1 + x ˆu1 + x ˆs1 + x ˆc1 = m1 + xu1 + xs1 + DT t (m)(ˆ xu + x ˆs + x ˆc ) (5.15) t u s u s c u s c + (DT˜ − DT )(m + x + x )(ˆ x +x ˆ +x ˆ ) + O(˜  +  + σ)|ˆ x +x ˆ +x ˆ |. Similarly, m1 + xu1 + xs1 = T t (m) + DT t (m)(xu + xs ) + (T˜(m) − T t (m)) + (DT˜(m) − DT t (m))(xu + xs ) + O( + σ)|xu + xs |.

(5.16)

Thus, |m1 − T t (m)| ≤ |xu1 | + |xs1 | + kDT t (m)k|xu + xs | + kT˜ − T t k1 (1 + |xu + xs |) + O( + σ)|xu + xs | ≤ 22 + (C + O( + σ)) + σ(1 + 2). 1 , by Lemma 4.4 there exists β ∗ > 0 As in the proof of Lemma 5.2, letting 1 < 2C such that (4.12) holds with 1 instead of . Let η and r satisfy (4.14). Choose 2 such that 2 ≤ 41 r and choose ∗ and σ such that

(C + O( + σ)) + σ(1 + 2) ≤

r , 2

which implies that m1 is in an η-neighborhood of T t (m). For simplicity, we denote T t (m) by m. ¯ Applying Lemma 4.5 to points m ¯ and m1 + xu1 + xs1 and using (5.16), we obtain

37

|m1 − T t (m)| c 1 t ˜ Πm ≤ ¯ T (m) − T (m) 1 − C1  + (DT˜(m) − DT t (m))(xu + xs ) + O( + σ)(|xu + xs |)  ≤ 2(CkT˜ − T t k0 + kT˜ − T t k1 + O( + σ) |xu + xs |).

(5.17)

Applying the projection Πα m1 to (5.15), we find t α t x ˆα xα + (Πα xα 1 = DT (m)ˆ m1 − Πm ¯ )DT (m)ˆ u u t + Πα xs + x ˆc ) m1 (Πm1 − Πm ¯ )DT (m)(ˆ s s t + Πα xu + x ˆc ) m1 (Πm1 − Πm ¯ )DT (m)(ˆ

(5.18)

c c t + Πα xu + x ˆs ) m1 (Πm1 − Πm ¯ )DT (m)(ˆ t u s ˜ + Πα xu + x ˆs + x ˆc ) m (D T − DT )(m + x + x )(ˆ 1

+

Πα + m1 O(˜

 + σ)|ˆ xu + x ˆs + x ˆc |

for α = u, s, c. Note that α α kΠα ¯ m 1 − Πm ¯ k ≤ Lipm (Πm )|m1 − m|.

(5.19)

Thus, from (5.17) and (5.18) there exists a positive constant C, which does not depend on m, , ˜, T˜, x ˆα ˆα , such that 1 and x |ˆ xu1 | ≥ |DT t (m)ˆ xu | − C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |), |ˆ xs1 | ≤ |DT t (m)ˆ xs | + C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |), |ˆ xc1 | ≥ |DT t (m)ˆ xc | − C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |). In the following, as in Section 4, C is a generic constant whose value may change from line to line, but it depends only on the projections and the time-t map T t . Thus, λ1 µ(|ˆ xu1 | + |ˆ xc1 |) − |ˆ xs1 |  ≥ λ1 µ u inf |DT t (m)xu ||ˆ xu | + u u x ∈Xm ,|x |=1

inf

c ,|xc |=1 xc ∈Xm

 |DT (m)x ||ˆ x | t

c

c

s || |ˆ − ||DT t (m)|Xm xs | − (λ1 µ + 1)C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |)

38

which, by (H3), is larger than   λ1 µ u t c c s inf |DT (m)x | |ˆ x | + λ1 µ|ˆ x | − λ|ˆ x | λ |xc |=1 − (λ1 µ + 1)C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |)   λ1 µ u t c c u c ≥ inf |DT (m)x | |ˆ x | + λ1 µ|ˆ x | − λµ(|ˆ x | + |ˆ x |) λ |xc |=1 − (λ1 µ + 1)C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |)   t c t ˜ ≥ µ(λ1 − λ) inf |DT (m)x | − (λ1 µ + 1)C(kT − T k1 + O(˜  +  + σ))(1 + µ) c u

|x |=1 c

(|ˆ x | + |ˆ x |),

where the assumption in (i) of this lemma was used. Since inf |xc |=1 |DT t (m)xc | > 0 and λ < λ1 < 1, one may choose ∗ , ˜∗ and σ sufficiently small so that µ(λ1 − λ) inf |DT t (m)xc | − (λ1 µ + 1)C(kT˜ − T t k1 + O(˜  +  + σ))(1 + µ) > 0. |xc |=1

This proves (i). From (5.18) we also obtain |ˆ xu1 | ≥ |DT t (m)ˆ xu | − C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |) and t |ˆ xα xα | + C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |) 1 | ≤ |DT (m)ˆ

for α = s, c. Thus, by using the normal hyperbolicity and the assumption in (ii) of this lemma, we obtain λ1 µ|ˆ xu1 | − (|ˆ xs1 | + |ˆ xc1 |) λ1 c ||( ≥ ||DT t (m)|Xm µ|ˆ xu | − λ|ˆ xs | − |ˆ xc |) λ − (λ1 µ + 1)C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |)   λ1 s t c s c c || (|ˆ x | + |ˆ x |) − λ|ˆ x | − |ˆ x | ≥ ||DT (m)|Xm λ − (λ1 µ + 1)C(kT˜ − T t k1 + O(˜  +  + σ))(|ˆ xu | + |ˆ xs | + |ˆ xc |)     λ1 t t c || ≥ ||DT (m)|Xm − 1 − (λ1 µ + 1)C(kT˜ − T k1 + O(˜  +  + σ))(1 + µ) |ˆ xs | + |ˆ xc | λ ≥ 0, which is achieved by choosing ∗ , ˜ and σ smaller if necessary. This completes the proof.

39

6. Center-Unstable Manifold. We now have the machinery to construct the manifolds given in Theorems A and B, starting with the local center-unstable manifold (see Section 3 for the definition). The idea is to take Lipschitz graphs over X u () in X u ()⊕X s () and map them under T˜, the perturbation of T t for some t > t0 . Using the cone lemmas we show that in a certain sense, this “graph transform” is a contraction on the space of Lipschitz graphs. The fixed point is the desired invariant manifold. We regard the normal bundle X u ⊕ X s as a bundle over the unstable bundle X u and consider sections of this bundle in the tubular neighborhood. Define the complete metric space Γcu of Lipschitz sections, which are described in terms of the global fattened cone and local cones. The key to construct a “graph transform” in the space Γcu is to show that for any given Lipschitz section h ∈ Γcu , the image of its graph under T˜ restricted to the tubular neighborhood is the graph of a Lipschitz section in Γcu . This is given in Proposition 6.4 and yields a graph transform F cu defined in Γcu . Using the invariance of the cones, one can prove that F cu is a contraction and the fixed point is the desired center-unstable manifold. Because we do not have a global Cartesian coordinate system, we must interpret the Lipschitz property in the various local coordinates. This is the point of the technical lemmas 6.1 and 6.1. Another difficulty is that T t is not a diffeomorphism (not even a homeomorphism, in fact neither one-to-one nor onto) and therefore Lipschitz graphs are not automatically mapped into Lipschitz graphs. Estimates due to the normal hyperbolicity are used to overcome this difficulty, and allow us to define the graph transform. Throughout this paper, we shall encounter several positive parameters:  – the size of a tubular neighborhood; ˜ – the size of the perturbation of the tubular neighborhood; δ – the size of the neck of the fat and gap cones; σ – the C 1 size of the perturbation from the time-t map T t . We use ∗ , ˜∗ and δ ∗ to denote the upper bounds for the parameters , ˜ and δ, respectively. The values of ∗ , ˜∗ and δ ∗ may change from lemma to lemma, but they are equal or smaller than those in the previous related lemmas. Without loss of generality, we assume that max{∗ , ˜∗ , δ ∗ } ≤ 1. As before, we will use C for a generic constant whose value may change from line to line, but it depends only on the projections and the time-t map. The quantity O() also depends only the projections and the time-t map. We shall discuss sections of this bundle, h : X u () → X u () ⊕ X s (). With an obvious abuse of notation, for such a section we will usually write h(m, xu ) for the u s s () ⊕ Xm () which is the image of (m, xu ) () rather than the point in Xm point in Xm under h. By the “graph” of the section h we shall mean

gr(h) ≡ {(m, xu + xs ) ∈ X u () ⊕ X s () : xs = h(m, xu )} .

40

Define, for fixed µ ∈ (0, 1),  > 0, 0 < δ < , and ˜ > 0 the space of sections Γcu = Γcu (, µ, δ, ˜) n ≡ h : X u () → X u () ⊕ X s () : gr(h) ⊂ Kf (, µ, δ) and for all (m, xu ) ∈ X u () if Θ−1 (m + xu + h(m, xu ) + xu1 + xs1 + xc1 ) ∈ gr(h) o s u c α (˜ X  ), then |x | ≤ µ(|x | + |x |) . for xα ∈ 1 m 1 1 1 We may define a norm on Γcu by o n khk = sup |h(m, xu )| : (m, xu ) ∈ X u () , which makes Γcu into a complete metric space. Observe that from Lemma 4.5, there exist positive constants ∗ and ˜∗ such that if  < ∗ and ˜ < ˜∗ then Γcu is not empty for δ ∈ (0, ) , in fact the zero section is in Γcu . In what follows we take such a value of ˜∗ but we will take ˜∗ possibly smaller than that indicated here.

Figure 4 Lipschitz sections h ∈ Γ induce mappings in the local coordinate systems near M . The next two lemmas give the properties of these induced mappings. cu

Lemma 6.1. Let µ ∈ (0, 1) and ρ ∈ (1, 2). For each ˜ < ˜∗ there exists ∗ > 0 such that for each  ∈ (0, ∗ ), δ ∈ (0, ) and for all h ∈ Γcu and m0 ∈ M , if u s g : (M ∩ B(m0 , ρ)) × Xm (ρ−1 ) → Xm 0 0

is defined by Πsm g(m, x ˜u ) = h(m, Πum x ˜u ),

(6.1)

41

then |g(m1 , x ˜u1 ) − g(m2 , x ˜u2 )| ≤ ρµ (|m1 − m2 | + |˜ xu1 − x ˜u2 |) .

(6.2)

Proof. Let 1 < 0 , where 0 is introduced in Section 5. By Lemma 4.4, there exists β ∗ > 0 such that (4.12) holds with 1 instead of . Let η and r satisfy (4.14) and u η < ρ−1. For m0 ∈ M choose ∗ < min{1 , r/2}. For  < ∗ by (4.5) Πum Xm (ρ−1 ) ⊂ 0 u Xm () for m ∈ B(m0 , ρ) and Πsm X s is an isomorphism. Thus g is well-defined on m0

u (M ∩ B(m0 , ρ)) × Xm (ρ−1 ). 0 u (ρ−1 ) and To show that (6.2) holds, for i = 1, 2 let mi ∈ B(m0 , ρ), x ˜ui ∈ Xm 0 α x ˜si = g(mi , x ˜ui ). We represent points in different coordinate systems: with x ¯α i ∈ Xm0 , for α = u, s, c

mi + Πumi x ˜ui + Πsmi x ˜si = m0 + x ¯ui + x ¯si + x ¯ci = mi + xui + xsi ,

(6.3)

where xui = Πumi x ˜ui and xsi = Πsmi x ˜si = Πsmi g(mi , x ˜ui ) = h(mi , Πumi x ˜ui ). u (ρ−1 ) we have |xui | <  and so |xsi | = Since mi ∈ B(m0 , ρ) and x ˜ui ∈ Xm 0 u cu u ˜i )| ≤  since h ∈ Γ . By (4.5) and (6.1) it follows that |˜ xsi | = |g(mi , x ˜ui )| ≤ |h(mi , Πmi x  1−η < 2. From (6.3) we have |¯ xui + x ¯si + x ¯ci | ≤ |m0 − mi | + |xui | + |xsi | < 4 and so |¯ xα i | ≤ C for some constant C, for α = u, s, c. By Lemma 4.5, for a possibly larger value of C |˜ xs1 − x ˜s2 − (¯ xs1 − x ¯s2 )| ≤ C1 |m1 − m2 | + C|m1 − m0 | (|˜ xu1 − x ˜u2 | + |˜ xs1 − x ˜s2 |) and so (1 − 2C)|˜ xs1 − x ˜s2 | ≤ C1 |m1 − m2 | + 2C|˜ xu1 − x ˜u2 | + |¯ xs1 − x ¯s2 |.

(6.4)

To obtain (6.2) we must estimate |¯ xs1 − x ¯s2 |. First we use (4.15) to write m2 + xu2 + xs2 = m1 + x ˆu2 + x ˆs2 + x ˆc2 α where x ˆα 2 ∈ Xm1 for α = u, s, c. Note that for some constant C > 0 α u s |ˆ xα 2 | ≤ kΠm1 k|m2 − m1 + x2 + x2 | ≤ C. Also, m1 + xu1 + h(m1 , xu1 ) + (ˆ xu2 − xu1 ) + (ˆ xs2 − xs1 ) + x ˆc2

= m2 + xu2 + xs2 = m2 + xu2 + h(m2 , xu2 ) ∈ Θ(gr(h)). By choosing ∗ sufficiently small, we find that |ˆ xu2 − xu1 | ≤ C < ˜, |ˆ xs2 − xs1 | ≤ C < ˜ and |ˆ xc2 | ≤ C < ˜, hence from the fact that h ∈ Γcu , |ˆ xs2 − xs1 | ≤ µ (|ˆ xu2 − xu1 | + |ˆ xc2 |) .

(6.5)

42

Now x ¯s1 − x ¯s2 =Πsm0 (m1 + xu1 + xs1 − (m2 + xu2 + xs2 )) = Πsm0 (xu1 − x ˆu2 + xs1 − x ˆs2 − x ˆc2 ) so, by (4.5)  |¯ xs1 − x ¯s2 | ≤ | Πsm0 − Πsm1 (xu1 − x ˆu2 − x ˆc2 ) | + (1 + η)|xs1 − x ˆs2 | and (6.5) with (H4) gives |¯ xs1 − x ¯s2 | ≤ (C|m0 − m1 | + (1 + η)µ)(|xu1 − x ˆu2 | + |ˆ xc2 |) ≤ (C + (1 + η)µ)(|xu1 − x ˆu2 | + |ˆ xc2 |).

(6.6)

Using (6.3) and the definition of x ˆu2 we have ˆu2 | = |Πum1 (¯ ¯u2 + x ¯s1 − x ¯s2 + x ¯c1 − x ¯c2 ) | |xu1 − x xu1 − x u k|¯ ≤ kΠum1 |Xm xu1 − x ¯u2 | + kΠum1 − Πum0 k (|¯ xs1 − x ¯s2 | + |¯ xc1 − x ¯c2 |) 0

≤ (1 + η)|¯ xu1 − x ¯u2 | + C (|¯ xs1 − x ¯s2 ) + |¯ xc1 − x ¯c2 |) .

(6.7)

|ˆ xc2 | ≤ (1 + η)|¯ xc1 − x ¯c2 | + C (|¯ xs1 − x ¯s2 | + |¯ xu1 − x ¯u2 |) .

(6.8)

Similarly, Combining (6.6)–(6.8) gives |¯ xs1 − x ¯s2 | ≤ (O() + (1 + η)2 µ)(|¯ xu1 − x ¯u2 | + |¯ xc1 − x ¯c2 |).

(6.9)

Now (4.16) and (4.17) may be used for (6.3) to get, respectively, |¯ xc1 − x ¯c2 | ≤ (1 + C1 )|m1 − m2 | + C(|˜ xu1 − x ˜u2 | + |˜ xs1 − x ˜s2 |), and |¯ xu1 − x ¯u2 | ≤ C1 |m1 − m2 | + (1 + C)|˜ xu1 − x ˜u2 | + C|˜ xs1 − x ˜s2 |, which, with (6.9) give |¯ xs1 − x ¯s2 |   ≤ O() + (1 + η)2 µ (1 + C1 )|m1 − m2 | + (1 + C)|˜ xu1 − x ˜u2 | + C|˜ xs1 − x ˜s2 | . Using this in (6.4) yields, |˜ xs1 − x ˜s2 |   ≤ (1 + η)2 µ + O() + O(1 ) |m1 − m2 | + |˜ xu1 − x ˜u2 | .

43

Let η also satisty that η < such that

q

1+ρ 2

− 1. Then one may choose 1 and ∗ small enough

(1 + η)2 µ + O() + O(1 ) < ρµ for  < ∗ . Thus |g(m1 , x ˜u1 ) − g(m2 , x ˜u2 )| = |˜ xs1 − x ˜s2 | ≤ ρµ(|m1 − m2 | + |˜ xu1 − x ˜u2 |). The proof is complete. The next lemma is similar but set in a local Cartesian coordinate system. Here ˜∗ is as before. Lemma 6.2. Let µ ∈ (0, 1) and ρ ∈ (1, 2). Then for each ˜ < ˜∗ there exists ∗ = ∗ (˜ ) and δ ∗ = δ ∗ () <  such that for each  < ∗ and δ < δ ∗ if h ∈ Γcu (, µ, δ, ˜) and m0 ∈ M , then there exists u (ρ−2 ) × X c () → X s () f : Xm m0 m0 0

such that |f (xu1 , xc1 ) − f (xu2 , xc2 )| ≤ ρµ (|xu1 − xu2 | + |xc1 − xuc |) and u (ρ−2 ) × X c (). Θ−1 (m0 + xc0 + xu0 + f (xu0 , xc0 )) ∈ gr(h) for all (xu0 , xc0 ) ∈ Xm m0 0

Furthermore, for all m ∈ M ∩ B(m0 , ρ−1 ) and xu ∈ Xnu (ρ−3 ) m + xu + h(m, xu ) = m0 + xc0 + xu0 + f (xu0 , xc0 ) u (ρ−2 ) × X c (). for some (xc0 , xu0 ) ∈ Xm m0 0

Proof. Fix 1 , η and r to satisfy the requirements in the proof of Lemma 6.1. Take δ ∈ (0, ) fixed but restrict both δ and 1 to satisfy further conditions to be specified later. Take ∗ as given by Lemma 6.1 and also require ∗ < 1 . Choose  ≤ ∗ . Thus we may apply Lemma 6.1 and some estimates obtained in its proof to establish this lemma. u (ρ−2 )×X c () To construct f for h ∈ Γcu we must show that for all (xu0 , xc0 ) ∈ Xm m0 0 s s c u s there exists a unique x ∈ Xm0 () such that m0 + x0 + x0 + x ∈ Θ(gr(h)). s () → X s as follows: Define a map ξ : Xm m0 0

44

For 0 < 2 < 1 , by Lemma 5.4, we may choose a ∗ sufficiently small such that for α () for α = u, s, c and  < ∗ , Θ−1 (m + xu + xs + xc ) ∈ X u ( ) ⊕ X s ( ). x ∈ Xm 0 2 2 0 s u s u s s Thus, for each x ∈ Xm0 () there exists a unique point (m, x ¯ +x ¯ ) in X (2 )⊕X (2 ) such that m0 + xc0 + xu0 + xs = m + x ¯u + x ¯s (6.10) α

and |m − m0 | ≤ 52 . Choosing 2 small enough, m is in the η-neighborhood of m0 and so we may apply Lemma 4.5. Taking the two points as m + x ¯u + x ¯s and m0 in (4.16). (4.17) gives |m − m0 | ≤ (1 − C1 )−1 |xc0 | < 2

(6.11)

and u

|¯ x | ≤ (1 + η)|



u Πum |Xm 0

−1

x ¯u | ≤ (1 + η)(|xu0 | + C1 |m − m0 |) < ρ−1 

ρ−1 α ˆα ∈ Xm ,α = provided that 1 satisfies 1 < 2C(ρ+1)ρ 2 . Consequently, there exist x 0 u, s, c, such that m+x ¯u + h(m, x ¯u ) = m0 + x ˆu + x ˆs + x ˆc .

Define ξ(xs ) = x ˆs . s (). We claim that ξ is a contraction on Xm 0

We first show that ξ maps

s () Xm 0

to

s (). Xm 0

u

Letting x ˜ =



u Πum |Xm 0

−1

x ¯u , (6.10)

and (4.17) give |˜ xu | ≤ |xu0 | + C1 |m − m0 | ≤ ρ−2  + C1  < ρ−1 , ρ−1 Cρ2 .

Thus, by

m + Πum x ˜u + Πsm g(m, x ˜u ) = m + x ¯u + h(m, x ¯ u ) = m0 + x ˆu + x ˆs + x ˆc .

(6.12)

where the last estimate is obtained by choosing 1 to satisfy 1 < Lemma 6.1, we have

Let x ˜s = g(m, x ˜u ). By (4.17), with (6.12), using m0 as the second point, |ˆ xs − g(m, x ˜u )| ≤ C1 |m − m0 | and so, by (6.11) |ˆ xs − g(m, x ˜u )| ≤ C1 .

45

Using Lemma 6.1 and the relationship between g and h ∈ Γcu (, µ, δ, ˜), |g(m, x ˜u )| ≤ ρµ|˜ xu | + |g(m, 0)| ≤ ρµ|˜ xu | + Cδ. Hence, |ξ(xs )| = |ˆ xs | ≤ µ + Cδ + C1  < , by choosing 1 and δ such that 1
0, kΠum0 − Πum ˆ 2k → 0 and

 −1 u k Πum0 |Xm k→1 ˆ 2

as σ +  → 0. Therefore, from (6.42) and (H3) |¯ xu | ≤ λ|DT t (m ˜ 1 )¯ xu |  −1 u u ≤ λk Πm0 |Xm k|Πum0 DT t (m ˜ 1 )¯ xu | ˆ 2

= O(σ +  + ˜)(|¯ xu | + |¯ xs | + |¯ xc |)

(6.43)

55

as ρ +  → 0. A similar estimate holds for |¯ xc | and consequently, |¯ xu | + |¯ xc | = O(σ +  + ˜)|¯ xs |.

(6.44)

Write m ˜2 +x ˜u2 + x ˜s2 = m ˜1 +x ˜u1 + h(m ˜ 1, x ˜u1 ) + x ¯u + [˜ xs1 + x ¯s − h(m ˜ 1, x ˜u1 )] + x ¯c By choosing 1 , ∗ and σ sufficiently small, we obtain |¯ xu | < ˜, |˜ xs1 + x ¯s −h(m ˜ 1, x ˜u1 )| < ˜ c and |¯ x | < ˜. Thus, using the fact that h ∈ Γcu we have µ(|¯ xu | + |¯ xc |) ≥ |˜ xs1 + x ¯s − h(m ˜ 1, x ˜u1 )| ≥ |¯ xs | − |˜ xs1 − h(m ˜ 1, x ˜u1 )|. This with (6.44) gives |¯ xs | ≤ (1 + O(σ +  + ˜))|˜ xs1 − h(m ˜ 1, x ˜u1 )|. (6.41) also yields, in a manner similar to that which produced (6.43), |xs1 − xs2 | < (λ + O(σ +  + ˜))|¯ xs |. Consequently, |xs1 − xs2 | < (λ + O(σ +  + ˜))(1 + C(σ +  + ˜))|˜ xs1 − h(m ˜ 1, x ˜u1 )| ≤ λ1 |˜ xs1 − h(m ˜ 1, x ˜u1 )| for ∗ and σ sufficiently small. This completes the proof. Proposition 6.9. There exist positive constants ˜∗ , ∗ = ∗ (˜ ), δ ∗ = δ ∗ () <  and ∗ ∗ ∗ σ = σ(, δ) such that if ˜ < ˜ ,  <  , δ < δ , and T˜ satisfies ||T˜ − T t ||1 < σ, then F cu is a contraction on Γcu . Proof. With h1 , h2 ∈ Γcu and (m0 , xu0 ) ∈ X u (), for i = 1, 2 let xsi = (F˜ cu hi )(m0 , xu0 ). From the definition of F˜ cu , there exists (m ˜ i, x ˜ui ) ∈ X u () such that T˜(m ˜1 +x ˜u1 + h1 (m ˜ 1, x ˜u1 )) = m0 + xu0 + xs1

56

and T˜(m ˜2 +x ˜u2 + h1 (m ˜ 2, x ˜u2 )) = m0 + xu0 + xs2 . Regarding h1 (m ˜ 1, x ˜u1 ) as x ˜s1 in (6.39) and using Lemma 6.8, we obtain |xs1 − xs2 | ≤ λ1 kh1 − h2 k. Since λ1 < 1, F˜ cu is a contraction. This completes the proof. Proof Theorem 6.3. From Proposition 6.9, F˜ cu is a contraction, hence has a fixed ˜ cu . Clearly, point in Γcu which we denote by h ˜ cu ))) ∩ Kf (, µ, δ) = Θ(gr(h ˜ cu )). T˜(Θ(gr(h ˜ cu )) is a local invariant center-unstable manifold for T˜. This ˜ cu () ≡ Θ(gr(h Thus W completes the proof. ˜ cu (). Define by induction a seNow we consider another characterization of W quence of sets Ak , k = 1, 2, · · · by Ak = T˜(Ak−1 ) ∩ Θ(X u () ⊕ X s ()) for k ≥ 1 and A0 = Θ(X u () ⊕ X s ()). It is clear that we may also write Ak ={x0 ∈ Θ(X u () ⊕ X s ()) : there exists xk ∈ Θ(X u () ⊕ X s ()) such that xl = T˜(k−l) xk ∈ Θ(X u () ⊕ X s ()) for 0 ≤ l ≤ k}. Proposition 6.10. ˜ cu () = ∩∞ W k=1 Ak ˜ cu (m0 , xu ) ∈ ˜ cu () ⊂ ∩∞ Ak . For each m0 + xu + h Proof. We first show that W 0 0 k=1 ˜ cu is the fixed point of F, there exists (m1 , xu ) ∈ X u () such that ˜ cu (), since h W 1 ˜ cu (m0 , xu ) = T˜(m1 + xu + h ˜ cu (m1 , xu )), m0 + xu0 + h 0 1 1 ˜ cu (m0 , xu ) ∈ Ak for all k ≥ 1. Next we show ∩∞ Ak ⊂ which yields that m0 + xu0 + h 0 k=1 ˜ cu (m, xu ). For each k ≥ 1, ˜ cu (). Let m + xu + xs ∈ ∩∞ Ak . We claim that xs = h W k=1 since m + xu + xs ∈ Ak , there exists m0 + xu0 + xs0 ∈ Θ(X u () ⊕ X s ()) such that T˜k (m0 + xu0 + xs0 ) = m + xu + xs

57

and T˜i (m0 + xu0 + xs0 ) ∈ Θ(X u () ⊕ X s ()), 1 ≤ i ≤ k. For i = 1, 2, · · · , k, let mi + xui + xsi = T˜i (m0 + xu0 + xs0 ).

(6.45)

˜ cu is the fixed point of F˜ cu , there Note that mk + xuk + xsk = m + xu + xs . Since h u u exists (m ¯ 0, x ¯0 ) ∈ X () such that ˜ cu (m ˜ cu (m1 , xu ). T˜(m ¯0 +x ¯u0 + h ¯ 0, x ¯u0 )) = m1 + xu1 + h 1 Let i = 1 in (6.45) T˜(m0 + xu0 + xs0 ) = m1 + xu1 + xs1 . From Lemma 6.8, we obtain ˜ cu (m1 , xu )| ≤ λ1 |xs − h ˜ cu (m0 , xu )|. |xs1 − h 1 0 0 Repeating this procedure gives ˜ cu (m, xu )| = |xs − h ˜ cu (mk , xu )| ≤ λk |xs − h ˜ cu (m0 , xu )| ≤ 2λk , |xs − h k k 1 0 0 1

(6.46)

˜ cu (m, xu ) by letting k → ∞. The proof is complete. which yields xs = h From Theorem 6.3 there exists a center-unstable manifold for the time-t map T t given by W cu () = Θ(gr(hcu )) where hcu ∈ Γcu . The next result states that the perturbed center-unstable manifold ˜ cu () is close to the unperturbed one, W cu (). W Proposition 6.11. There exists ∗ such that for  < ∗ , ˜ cu − hcu k → 0 kh

as

kT˜ − T t k0 → 0.

1 Proof. We denote the ∗ in Theorem 6.3 temporarily by ∗1 . Let 2 < min{∗1 , 2C } be ∗ fixed, where C is the constant in Lemma 4.5. From Lemma 4.4, there is a β such that (4.12) holds with 1 instead of . Let η and r satisfy (4.14). By Lemma 4.3, we may choose ∗ sufficiently small such that for  < ∗ , (m1 , xu1 + xs1 ) ∈ X u () ⊕ X s (), α and for x ∈ X if |m1 + xu1 + xs1 − x| < , then x = m2 + xu2 + xs2 , where xα 2 ∈ X (2 ) and m2 is in an η-neighborhood of m1 . ˜ cu () ⊂ W ˜ cu (2 ) for all  < 2 < ∗ . In Note that Proposition 6.10 implies that W 1 ˜ cu by h ˜ cu , then we have h ˜ cu = h ˜ cu on X u (). other words, if we denote h   2

58

Let F cu denote the graph transform for the time-t map T t and F˜ cu that for T˜. From Proposition 6.9, F cu is a contraction in Γcu (2 , µ, δ, ˜) and hcu is its fixed point. ˜ cu ∈ Γcu , and for small E > 0, there is a positive integer k such that Thus, for any h  k ˜ cu ) − hcu k ≤ E. k F cu (h

(6.47)

For such fixed k, it is easy to see that there exists β > 0 such that if kT˜ − T t k0 ≤ β then for i = 1, 2, · · · , k kT˜i − (T t )i k0 ≤ E. (6.48) ˜ cu as the fixed point of F˜ cu in Γcu (, µ, δ, ˜). For Let ∗ also satisfy ∗ < 2 . Consider h  k  k u u s cu cu u s cu ˜ ˜ cu (m0 , xu ). ˜ each (m0 , x0 ) ∈ X (), let x0 = F h (m0 , x0 ) and x ˆ0 = F h 0 We want to estimate |xs0 − x ˆs0 |. From the definition of F˜ cu there exists (m1 , xu1 ) ∈ u X () such that ˜ cu (m1 , xu )) = m0 + xu + xs . T˜k (m1 + xu1 + h 1 0 0

(6.49)

Using (6.48) we may write ˜ cu (m1 , xu )) = m (T t )k (m1 + xu1 + h ¯0 +x ¯u0 + x ¯s0 , 1

(6.50)

α cu where x ¯α it follows that ¯ 0 (2 ). From (6.27) and the definition of F 0 ∈ Xm

 k ˜ cu (m x ¯s0 = F cu h ¯ 0, x ¯u0 ).

(6.51)

m ¯0 +x ¯u0 + x ¯s0 = m0 + xu0 + xu + xs0 + xs + xc .

(6.52)

Write Then we have xα = Πα ¯0 +x ¯u0 + x ¯s0 − (m0 + xu0 + xs0 )). m0 (m By (6.48) we obtain |xα | ≤ CE ≤ ρ−3 2 − 

(6.53)

for α = c, s, u. Thus, we have xs0 + xs = f (xu0 + xu , xc ), where f is the representation  k ˜ cu at m0 . Also x of F cu h ˆs0 = f (xu0 , 0), and so by Lemma 6.2 |ˆ xs0 − (xs + xs0 )| ≤ Cµ(|xu | + |xc |). Hence, |ˆ xs0 − xs0 | ≤ CE.

(6.54)

59

Using (6.47) and (6.54) we obtain ˜ cu (m0 , xu ) − hcu (m0 , xu )| |h 0 0  k k k   ˜ cu )(m0 , xu ) − F cu (h ˜ cu )(m0 , xu )| + | F cu (h ˜ cu )(m0 , xu ) − hcu (m0 , xu )| ≤ | F˜ cu (h 0 0 0 0 ≤ CE, which completes the proof. Proposition 6.12. There exists ∗ and σ such that if  < ∗ and T˜ satisfies ||T˜ − T t ||1 < σ, then ˜ cu () ∩ T˜−1 (W ˜ cu ()) → W ˜ cu () T˜ : W is a homeomorphism. Proof. From Proposition 6.4, T˜ is onto. We need to show that it is one-to-one and ˜ cu (m0 , xu ), m + xu + h ˜ cu (m, xu ) ∈ its inverse is continuous. Consider m0 + xu0 + h 0 ˜ cu () ∩ T˜−1 (W ˜ cu ()). We may write W ˜ cu (m0 , xu )) = m ˜ cu (m T˜(m0 + xu0 + h ¯0 +x ¯u0 + h ¯ 0, x ¯u0 ) 0

(6.55)

˜ cu (m, xu )) = m ˜ cu (m, T˜(m + xu + h ¯ +x ¯u + h ¯ x ¯u ),

(6.56)

and where (m ¯ 0, x ¯u0 ), (m, ¯ x ¯u ) ∈ X u (). Let ζ > 0 be given and small. Let ˜ cu (m ˜ cu (m, ∆=m ¯0 +x ¯u0 + h ¯ 0, x ¯u0 ) − (m ¯ +x ¯u + h ¯ x ¯u )) and assume that |∆| ≤ ζ. From (6.38) it follows that |m − m0 | ≤ C( + σ + ζ)

(6.57)

Thus, we may choose ∗ , σ and ζ sufficiently small such that m is in an η-neighborhood of m0 . Write ˜ cu (m, xu ) = m0 + xu + h ˜ cu (m0 , xu ) + x m + xu + h ˜u + x ˜s + x ˜c , 0 0

(6.58)

α where xα ∈ Xm . Then applying Lemma 4.5 to m0 and (6.58), we have 0

|˜ xα | ≤ C( + σ + ζ) < ˜, ˜ cu ∈ Γcu (, µ, δ, ˜) provided ∗ , σ and ζ are sufficiently small. Now, because h |˜ xs | ≤ µ(|˜ xu | + |˜ xc |).

(6.59)

60

˜ cu (m0 , xu ), Using (6.58) and applying the Taylor expansion to (6.56) at m0 + xu0 + h 0 we obtain ˜ cu (m0 , xu ))(˜ DT˜(m0 + xu0 + h xu + x ˜s + x ˜c ) = o(|˜ xu | + |˜ xs | + |˜ xc |) + ∆. 0

(6.60)

Note that ˜ cu (m0 , xu )) − DT t (m0 )|| ≤ C(σ + ) ||DT˜(m0 + xu0 + h 0 for some constant C. Hence, from (6.59) and (6.60), we obtain for α = u, c, |DT t (m0 )˜ xα | ≤ O(σ +  + ζ)(|˜ xu | + |˜ xc |) + Cζ. u c From (H3), we know that both ΠuT t (m0 ) DT t (m0 )|Xm and ΠcT t (m0 ) DT t (m0 )|Xm have 0 0 bounded inverses, thus

|˜ xu | + |˜ xc | ≤ O(σ +  + ζ)(|˜ xu | + |˜ xc |) + Cζ. Choosing ∗ , σ and ζ sufficiently small, we obtain |˜ xu | + |˜ xc | ≤ Cζ, ˜ cu () ∩ T˜−1 (W ˜ cu ()) → W ˜ cu () is one-to-one and which with (6.58) implies that T˜ : W has a continuous inverse. This completes the proof.

61

§7. Center-stable manifold. In section 6, we proved the existence of a Lipschitz center-unstable manifold. In addition to the difficulties addressed in Section 6 we have a further difficulty in constructing the center-stable manifold. For flows which exist in backward as well as forward time, such as for finite dimensional dynamical systems or hyperbolic PDEs, one may now obtain the center-stable manifold simply by reversing time (See [F] and [HPS]). However, this trick does not work for infinite dimensional dynamical systems in general, like, for example, those generated by parabolic PDE’s, since backward solutions may not exist and the corresponding solution map is not invertible. In order to overcome this difficulty, a new technique is needed to establish the existence of the center-stable manifold. We show that for any Lipschitz graph over the stable bundle and any unstable fiber, there is a unique point on the fiber which is mapped into this graph. We then show that the collection of such points is a Lipschitz graph. As in Section 6, we shall first introduce a space of Lipschitz sections of the bundle u X () ⊕ X s () over X s (), then construct a graph transform defined on this space, which is induced by T˜. In fact this involves a type of inverse of T˜ even though it is not everywhere defined. We then show that this graph transform is a contraction and its fixed point gives the center-stable manifold. Let h be a section of the bundle X u () ⊕ X s () over X s (), i.e., h : X s () → X u () ⊕ X s () (m, xs ) → (m, xu + xs ). With an obvious abuse of notation, we denote xu by h(m, xs ), namely, xu = h(m, xs ). The graph of h is denoted by gr(h) = {(m, xu + xs ) ∈ X u () ⊕ X s () : xu = h(m, xs )}. Define, for fixed µ ∈ (0, 1),  > 0, δ > 0 and ˜ > 0 the space of Lipschitz sections Γcs = Γcs (, µ, δ, ˜) n ≡ h : X s () → X u () ⊕ X s () : gr(h) ⊂ Kg0 (, 1/µ, δ) and for all (m, xs ) ∈ X s () if Θ−1 (m + xs + h(m, xs ) + xu1 + xs1 + xc1 ) ∈ gr(h) α ), α = u, s, c, then for xα 1 ∈ X (˜ o |xu1 | ≤ µ(|xs1 | + |xc1 |) .

62

We may define a norm on Γcs by ¯ ||h|| = sup{|h(m, xs )| : (m, xs ) ∈ X()}, which makes Γcs into a complete metric space. Note that from Lemma 4.5, there exist positive constants ∗ and ˜∗ such that if  < ∗ and ˜ < ˜∗ then Γcs is not empty, in fact, the zero section is in Γcs .

Figure 6 Before we state the main theorem of this section, we want to introduce two fundamental lemmas which give the Lipschitz properties of each section in Γcs under two different coordinate systems which were introduced in Section 4. Lemma 7.1. Let µ ∈ (0, 1) and ρ ∈ (1, 2). For each ˜ < ˜∗ there exists ∗ > 0 such that for each  ∈ (0, ∗ ) and for all h ∈ Γcs and m0 ∈ M , if s u g : (M ∩ B(m0 , ρ)) × Xm (ρ−1 ) → Xm 0 0

is defined by Πum g(m, x ˜s ) = h(m, Πsm x ˜s )

(7.2)

|g(m1 , x ˜s1 ) − g(m2 , x ˜s2 )| ≤ ρµ(|m1 − m2 | + |˜ xs1 − x ˜s2 |)

(7.3)

then

Note that g is the local Lipschitz representative of h under the third coordinate system introduced in Section 4 by trivializing the bundle. The next lemma gives the the local coordinate representative of h under the second coordinate system (Cartesian coordinate system) introduced in section 4. Lemma 7.2. Let µ ∈ (0, 1) and ρ ∈ (1, 2). Then for each ˜ < ˜∗ there exists ∗ = ∗ (˜ ) and δ ∗ = δ ∗ () <  such that for each  < ∗ and δ < δ ∗ if h ∈ Γcs (, µ, δ, ˜) and m0 ∈ M , then there exists s c u f : Xm (ρ−2 ) × Xm () → Xm () 0 0 0

63

such that |f (xs1 , xc1 ) − f (xs2 , xc2 ) ≤ ρµ(|xs1 − xs2 | + |xc1 − xc2 |)

(7.4)

and Θ−1 (m0 + xs0 + xc0 + f (xs0 , xc0 )) ∈ gr(h) s c for all (xs0 , xc0 ) ∈ Xm (ρ−2 ) × Xm (). Furthermore, for all m ∈ M ∩ B(m0 , ρ−1 ) 0 0 s s −3 and x ∈ Xm (ρ ),

m + xs + h(m, xs ) = m0 + xs0 + xc0 + f (xs0 , xc0 )

(7.5)

s c for some (xs0 , xc0 ) ∈ Xm (ρ−2 ) × Xm (). 0 0

The proofs of Lemma 7.1 and Lemma 7.2 follow the same lines as Lemma 6.1 and Lemma 6.2. We omit them. √ Theorem 7.3. Let λ1 ∈ (λ, 1), ρ ∈ (1, 1/ λ), and µ ∈ (0, 1). Then there exist positive constants ˜∗ , ∗ = ∗ (˜ ), δ ∗ = δ ∗ () <  and σ = σ(, δ) such that if ˜ < ˜∗ ,  < ∗ , δ < δ ∗ , and T˜ satisfies ||T˜ − T t ||1 < σ, then T˜ has a Lipschitz center-stable manifold with Lipschitz constant ρµ ˜ cs )), ˜ cs () = Θ(gr(h W ˜ cs ∈ Γcs (, µ, δ, ˜). where h Note that the Lipschitz constant ρµ is for the local coordinate representatives g and f . The proof of this theorem is built on the following proposition and lemmas. Throughout this section we shall assume that λ1 , ρ, and µ satisfy the requirements stated in Theorem 7.3. The next proposition is the core of this section, resulting in the graph transform. Proposition 7.4. There exist positive constants ˜∗ , ∗ = ∗ (˜ ), δ ∗ = δ ∗ () <  and σ = σ(, δ) such that if ˜ < ˜∗ ,  < ∗ , δ < δ ∗ , and T˜ satisfies ||T˜ − T t ||1 < σ, then ˜ ∈ Γcs such that for each h ∈ Γcs there exists a unique h ˜ T˜(Θ(gr(h))) ⊂ Θ(gr(h)) This will be proved through a sequence of lemmas. Lemma 7.5. There exist positive constants ˜∗ , ∗ = ∗ (˜ ), δ ∗ = δ ∗ () <  and σ = σ(, δ) such that if ˜ < ˜∗ ,  < ∗ , δ < δ ∗ , and T˜ satisfies ||T˜ − T t ||1 < σ, then for u () such all h ∈ Γcs (, µ, δ, ˜) and (m0 , xs0 ) ∈ X s (), there exists a unique xu0 ∈ Xm 0 that T˜(m0 + xu0 + xs0 ) ∈ Θ(gr(h)). (7.6)

64 u (), by ¯ = T t (m0 ). For any xu ∈ Xm Proof. Let h ∈ Γcs and (m0 , xs0 ) ∈ X s (). Set m 0 Lemma 5.4, we write T˜(m0 + xu + xs0 ) as

T˜(m0 + xu + xs0 ) = m ¯ +x ¯u + x ¯s + x ¯c = m1 + xu1 + xs1

(7.7)

α where xα 1 ∈ X (2 ) for α = u, s. To show Lemma 7.5, it is enough to prove that there u is a unique xu ∈ Xm () such that x ¯u = f (¯ xs , x ¯c ). To find such xu , we consider a 0 u () to X u defined by map ξ from Xm m0 0 u

u



t

ξ(x ) = x − DT (m0 ) X u

−1

(¯ xu − f (¯ xs , x ¯c ))

m0

u (). We claim that ξ is a contraction on Xm 0 u u We first prove that ξ is well-defined. Note that DT t (m0 ) : Xm → Xm ¯ has a 0 bounded inverse from hypothesis (H3). It is enough to show that there exists ∗ , δ ∗ = δ ∗ () <  and σ such that if  < ∗ , δ < δ ∗ , and T˜ satisfies ||T˜ − T t ||1 < σ, then s (ρ−2 ) × X c (). (¯ xs , x ¯c ) ∈ Xm ¯ Applying the projection Πsm ¯ to (7.7) we have u s ˜ x ¯s = Πsm ¯ ¯ (T (m0 + x + x0 ) − m).

(7.8)

Using the Taylor expansion of T t (m0 + xu + xs0 ) at m0 and hypothesis (H3), we obtain t s t u s u s ˜ |¯ xs | ≤ kΠsm ¯ k ||T − T ||0 + |Πm ¯ DT (m0 )(x + x0 )| + O()(|x | + |x0 |) ≤ Cσ + ||DT t (m0 ) X s || |xs0 | + O() m0  ≤ λ + O()  + Cσ < ρ−2 ,

(7.9)

√ provided ρ < 1/ λ and ∗ and σ are sufficiently small. Similarly, we have u s ˜ x ¯c = Πcm ¯ ¯ (T (m0 + x + x0 ) − m) and |¯ xc | ≤ CkT˜ − T t k0 + O() ≤ Cσ + O() < 

(7.10)

by choosing smaller ∗ and σ if necessary. Therefore, ξ is well-defined. u u Next we show that ξ : Xm () → Xm is a contraction. 0 0 u (), we write For xu1 , xu2 ∈ Xm 0 T˜(m0 + xui + xs0 ) = m ¯ +x ¯ui + x ¯si + x ¯ci , i = 1, 2.

(7.11)

65

For α = s, c, estimate α ˜ u s u s ˜ |¯ xα ¯α 1 −x 2 | = |Πm ¯ (T (m0 + x1 + x0 ) − T (m0 + x2 + x0 ))| Z 1  α u u s u u DT˜(m0 + τ x1 + (1 − τ )x2 + x0 )(x1 − x2 )dτ | = |Πm ¯



0 α ˜ kΠm ¯ k||T

+

|Πα m ¯

Z

− T t ||1 |xu1 − xu2 | 1

DT t (m0 + τ xu1 + (1 − τ )xu2 + xs0 )(xu1 − xu2 )dτ |

0

(7.12)

≤ Cσ + O() |xu1 − xu2 |. 

In order to show ξ is a contraction we also need to estimate |xu1 − xu2 − (DT t (m0 ) X u )−1 (¯ xu1 − x ¯u2 )| m0

−1

t

u ) = |(DT (m0 )|Xm DT t (m0 )(xu1 − xu2 ) 0 Z 1  u − Πm DT˜(m0 + τ xu1 + (1 − τ )xu2 + xs0 )(xu1 − xu2 )dτ | ¯ 0  ≤ Cσ + O() |xu1 − xu2 |.

(7.13)

Thus, from (7.4), (7.12) and (7.13) we obtain |ξ(xu1 ) − ξ(xu2 )|

(7.14)

 = |xu1 − xu2 − DT t (m0 ) X u )−1 (¯ xu1 − x ¯u2 − f (¯ xs1 , x ¯c1 ) + f (¯ xs2 , x ¯c2 ) | m0  u u ≤ Cσ + O() |x1 − x2 |. This shows that ξ is a contraction by choosing ∗ and σ sufficiently small. To complete u () into itself. We first estimate ξ(0). Again the proof, we have to show ξ maps Xm 0 using (7.7), we have s t ˜ x ¯u = Πum ¯ (T (m0 + x0 ) − T (m0 )), and so |¯ xu | ≤ Cσ + O(). Hence, from (7.4), (7.9) and (7.10) |ξ(0)| = |(DT t (m0 ) X u )−1 (¯ xu − f (¯ xs , x ¯c )| m0

≤ λ Cσ + O() + |f (0, 0)| + ρµ(|¯ xs | + |¯ xc |) ≤ λ Cσ + O() + δ) < λ



66

provided ∗ , δ ∗ and σ are sufficiently small. u For any xu ∈ Xm (), from (7.14) it follows that 0 |ξ(xu )| ≤ |ξ(0)| + (Cσ + O()) < (λ + Cσ + C()) µ(|xs | + |xc |).

(7.15)

m0 + xu0 + xs0 + xu + xs + xc

(7.16)

Write

=m ¯0 +

x ¯u0

+

x ¯s0 ,

67 α (). Let where x ¯α ¯0 0 ∈ Xm

T˜(m0 + xu0 + xs0 ) = m1 + xu1 + xs1 ,

(7.17)

and T˜(m ¯0 +x ¯u0 + x ¯s0 ) = m ¯1 +x ¯u1 + x ¯s1 = m1 +

xu1

+

xs1

(7.18) u

s

c

+x ˜ +x ˜ +x ˜ .

1 , where C is the constant given in Lemma 4.5. Then applying Lemma 4.4, Let 1 < 2C there exits β ∗ such that (4.12) holds with 1 instead of . Let η and r satisfy (4.14). By choosing ∗ and ˜ sufficiently small and using (7.16), m ¯ 0 is in an η-neighborhood of m0 . Thus, applying Lemma 4.5 to the points m0 and (7.16), we obtain

|m0 − m ¯ 0| ≤

1 |xc | ≤ 2˜ , 1 − C1

and for α = u, s, −1 α α α |xα x ¯0 | 0 + x − Πm ¯ 0 Xα m0

≤ C1 |m0 − m ¯ 0| ≤ C1 ˜. This yields 1 |¯ xα | + C1 ˜ 1−η 0 ≤ C( + 1 ˜)

|xα | ≤ |xα 0|+

−1 where (4.5) and (4.14) are used to bound k Πα k. Thus, using the assumption m ¯ 0 Xα m0

(7.15), we find |xc | ≤

1 C( + 1 ˜). µ

Set γ(τ ) = m0 + xu0 + xs0 + τ (xu + xs + xc ). Then from (7.16)–(7.18), it follows that u s u s c ˜ |˜ x α | = Πα m1 T (m0 + x0 + x0 + x + x + x )  − T˜(m0 + xu0 + xs0 ) Z 1 α = Πm1 DT˜(γ(τ ))(xu + xs + xc )dτ 0

1 ≤ C( + 1 ˜) < ˜, µ

(7.19)

68

provided that 1
0, there is a positive integer k such that  k ˜ cs ) − hcs k ≤ E. k F cs (h (7.24) We may also choose k such that λk1 ≤ E. For such fixed k, it is easy to see that there exists β > 0 such that if kT˜ − T t k0 ≤ β then for i = 1, 2, · · · , k kT˜i − (T t )i k0 ≤ E. (7.25) Let ∗ also satisfy ∗ < 2 . For each (m0 , xu0 ) ∈ X s (), let  k u cs ˜ cs (m0 , xs ) x ˆ0 = F h 0 and

 k ˜ cs (m0 , xs ). xu0 = F˜ cs h 0

For 0 ≤ i ≤ k, we write (T t )i (m0 + xu0 + xs0 ) = mi + xui + xsi and let

 k−i ˜ cs (mi , xs ) x ˆui = F cs h i

Using Lemma 7.8, we obtain |ˆ xui − xui | ≤ λ1 |ˆ xui+1 − xui+1 |, which yields |ˆ xu0 − xu0 | ≤ λk1 |ˆ xuk − xuk | ≤ 21 E. Therefore, ˜ cs (m0 , xs ) − hcs (m0 , xs )| |h 0 0  k  k  k cs cs s cs cs s cs ˜ ˜ ˜ cs )(m0 , xs ) − hcs (m0 , xs )| ˜ (h )(m0 , x0 ) − F (h )(m0 , x0 )| + | F (h ≤| F 0 0 ≤ CE, which completes the proof.

72

8. Smoothness of Center-Stable Manifolds. We shall see that the center-stable manifold obtained in Section 7 indeed is C 1 by choosing ∗ and σ smaller than those in Theorem 7.1. More precisely, we have √ Theorem 8.1. Let λ1 ∈ (λ, 1), ρ ∈ (1, 1/ λ), and µ ∈ (0, 1) such that µρ < 1/2. Then there exist positive constants ˜∗ , ∗ = ∗ (˜ ), δ ∗ = δ ∗ () <  and σ = σ(, δ) such ˜ cs () is a C 1 that if ˜ < ˜∗ ,  < ∗ , δ < δ ∗ , and T˜ satisfies ||T˜ − T t ||1 < σ, then W manifold. The basic idea to show its smoothness is to find a candidate for the tangent bundle of this manifold, which is invariant under the linearization DT˜, then to prove it indeed is tangent to the manifold. The arguments are based on the use of Lipschitz jets, which is borrowed from [HPS]. Since the trivialization of the normal bundle of M is not available in a Banach space, the proof is more complicated than for finite dimensional systems. We first define a space of sections of the Lipschitz jet bundle, which is different from the jet spaces introduced in [HPS]. Then we construct a graph transform based on the linearization DT˜ and show that it has a unique fixed point which gives the tangent bundle of the center-stable manifold. A major difficulty in finding the fixed point is that the space of sections of the Lipschitz jet bundle is not complete. Finally, we prove that the tangent bundle is C 0 . Let Y and Z be Banach spaces. For y0 ∈ Y and z0 ∈ Z, consider two local maps gi : Ui → Z, gi (y0 ) = z0 , i = 1, 2 where Ui is a neighborhood of y0 . Define |g1 (y) − g2 (y)| . y→y0 |y − y0 |

d(g1 , g2 ) = lim

We shall see that d(g1 , g2 ) defines a metric. If d(g1 , g2 ) = 0, we say that g1 is equivalent to g2 . The equivalence class of all local maps equivalent to g1 is called the Lipschitz jet of g1 at y0 , which is denoted by j1 = [g1 ]. We use J(Y, Z; y0 , z0 ) to denote the set of all jets at y0 carrying y0 to z0 . For j1 , j2 ∈ J(Y, Z; y0 , z0 ), we define |g1 (y) − g2 (y)| y→y0 |y − y0 |

d(j1 , j2 ) = lim

where g1 and g2 are representatives of j1 and j2 respectively. It is not hard to see that d(j1 , j2 ) does not depend on the choices of the representatives.

73

Consider the jet spaces J b = {j ∈ J(Y, Z; y0 , z0 ) : d(j, [z0 ]) < ∞} J c = {j ∈ J b : j has a representative which is continuous in a neighborhood of y0 } J d = {j ∈ J b : j has a differentiable representative} J a = {j ∈ J b : j has an affine representative} Theorem on Lipschitz Jets. Let y0 = 0, z0 = 0. Then J b is a Banach space with norm kjk = d(j, 0). The sets J c , J d and J a are closed subspaces of J b and J b ⊃ J c ⊃ J d = J a. The above results are borrowed from [HPS]. For each m ∈ M . Set s c u J b (m) = J b (Xm × Xm , Xm ; 0, 0), s c u J c (m) = J c (Xm × Xm , Xm ; 0, 0), s c u J d (m) = J d (Xm × Xm , Xm ; 0, 0), s c u J a (m) = J a (Xm × Xm , Xm ; 0, 0).

Let θ = µρ < 1/2. For each fixed m ∈ M , we define a Lipschitz jet space s c u Jθl (m) = Jθl (Xm × Xm , Xm ; 0, 0) u = {j ∈ J b (m) : j has a representative g : U → Xm satisfying Lip (g|U ) ≤ θ} c s and × Xm where U is a neighborhood of 0 in Xm

 |g(xs , xc ) − g(¯ xs , x ¯c )| : (xs , xc ), (¯ xs , x ¯c ) ∈ U, (xs , xc ) 6= (¯ xs , x ¯c )}. Lip g|U = sup{ s s c c |x − x ¯ | + |x − x ¯ | Let Jθl = Jθl (X s × X c , X u ; 0, 0) denote the Lipschitz jet bundle over M with fiber Jθl (m). Set Jθa = Jθl ∩ J a . ˜ cs (m, xs ) to jets ˜ cs () to J l which map points m + xs + h Consider all maps from W θ s c u × Xm , Xm ; 0, 0). j = j(m, xs ) in Jθl (Xm Define  ˜ cs () → Jθl : kγk < ∞ Σcs,l = γ:W θ where ˜ cs (m, xs ) ∈ W ˜ cs ()}, kγk = sup{kj(m, xs )k : m + xs + h

74

 ˜ cs (m, xs ) . Similarly, we may define Σcs,a . and we identify j(m, xs ) with γ m + xs + h θ ˜ cs (), from the invariance of W ˜ cs (), we may write For m1 + xu1 + xs1 ∈ W T˜(m1 + xu1 + xs1 ) = m2 + xu2 + xs2

(8.1)

α u s ˜ cs where xα 2 ∈ Xm2 () for α = u, s and xi = h (mi , xi ), i = 1, 2. Let γ ∈ Σcs,l be fixed and set for i = 1, 2 θ

ji = γ(xi ),

(8.2)

u be a Lipschitz where xi = mi + xui + xsi . Let g2 : B2s (0, r2 ) × B2c (0, r2 ) → Xm 2 α α α representative of j2 such that for x , x ¯ ∈ B2 (0, r2 ), α = c, s

|g2 (xs , xc ) − g2 (¯ xs , x ¯c )| ≤ θ(|xs − x ¯s | + |xc − x ¯c |),

(8.3)

α where B2α (0, r2 ) is the ball in Xm with radius r2 > 0 centered at 0. 2 We shall construct γ˜ such that γ is the image of γ˜ under DT˜ in a certain sense, which will be stated precisely later. For the given g2 , we want to find a Lipschitz map g˜1 defined on B1s (0, r1 ) × B1c (0, r1 ) for some r1 > 0 such that for each (xs , xc ) ∈ B1s (0, r1 ) × B1c (0, r1 ) there exists (¯ xs , x ¯c ) ∈ B2s (0, r2 ) × B2c (0, r2 ) satisfying

DT˜(x1 )(xs + xc + g˜1 (xs , xc )) = x ¯s + x ¯c + g2 (¯ xs , x ¯c ).

(8.4)

To see this, for (xs , xc ) ∈ B1s (0, r1 ) × B1c (0, r1 ), we define a map E from B1u (0, r1 ) to u Xm as follows 1 E(xu )  −1 u = Πum2 DT˜(x1 )|Xm (g2 (Πsm2 DT˜(x1 )(xu + xs + xc ), Πcm2 DT˜(x1 )(xu + xs + xc )) 1

− Πum2 DT˜(x1 )(xs + xc )).

(8.5)

We shall see that E is a contraction and its fixed point gives a solution of (8.4) Before we study the properties of E, we state a technical lemma providing fundamental properties for T˜, which are inherited from the time-t map T t . ˜ cs () and Lemma 8.2. There exist ∗ > 0 and σ > 0 such that for  < ∗ if x1 ∈ W kT˜ − T t k1 < σ, then the following estimates hold  −1 u (i) k Πum2 DT˜(x1 )|Xm k 1 1 λ

(8.6)

Let m ¯ = T t (m1 ), then  −1 t u k Πum DT (m )| k < λ. 1 Xm ¯ 1 From (8.1) and (H1), we obtain |m ¯ − m2 | = |T t (m1 ) − T˜(m1 + xu1 + xs1 ) + xu2 + xs2 | ≤ |T t (m1 ) − T t (m1 + xu1 + xs1 )| + |T t (m1 + xu1 + xs1 ) − T˜(m1 + xu1 + xs1 )| + |xu2 + xs2 | ≤ C + σ.

(8.7)

We write u Πum2 DT˜(m1 + xu1 + xs1 )|Xm 1  u u u s ˜ u = Πm 2 − Πm ¯ D T (m1 + x1 + x1 )|Xm 1   u u s t u s ˜ u + Πm ¯ D T (m1 + x1 + x1 ) − DT (m1 + x1 + x1 ) |Xm 1  u t u s t u t u u . + Πm + Πm ¯ DT (m1 + x1 + x1 ) − DT (m1 ) |Xm ¯ DT (m1 )|Xm 1 1

Thus, from (H1), (H4), (8.6) and (8.7) it follows that u inf{|Πum2 DT˜(m1 + xu1 + xs1 )xu | : xu ∈ Xm , |xu | = 1} 1  u u s ˜ u k ≥ inf |DT t (m1 )xu | : xu ∈ Xm , |xu | = 1 − kΠum2 − Πum ¯ kkD T (m1 + x1 + x1 )|Xm 1 1 t u t u s t ˜ − kΠum ¯ kkD T − DT k0 − kΠm ¯ kkDT (m1 + x1 + x1 ) − DT (m1 )k  1 u u ≥ inf |DT t (m1 )xu | : xu ∈ Xm , |x | = 1 − (Cσ + O()) > 1 λ

provided that ∗ and σ are sufficiently small. To complete the proof of (i) we must u u u . We estimate the is an isomorphism from Xm onto Xm show that Πum2 DT˜(x1 )|Xm 1 2 1 difference u u k kΠum2 DT˜(x1 )|Xm − Πum2 DT t (m1 )|Xm 1

≤ Cσ + O().

1

76 ∗ Choose ¯ where √  and σ sufficiently small such uthat m2 is in an η-neighborhood of m, u is an isomorphism. Furthermore, from η < 2 − 1. Thus from (4.5), we have Πm2 |Xm ¯ u u u that the fact that DT t (m1 )|Xm is an isomorphism from Xm onto Xm ¯ it follows that 1 1 Πum DT t (m1 )|X u is an isomorphism. Hence Πum DT˜(x1 )|X u is also an isomorphism 2

2

m1

m1

as long as ∗ and σ are sufficiently small so (i) holds. To see that (ii) and (iii) hold, we notice that the hypothesis (H3) also gives  −1 t t u c k < λ k Πum DT (m )| kkΠcm 1 Xm ¯ ¯ DT (m1 )|Xm 1 1 and

 −1 t t c s k < λ. k Πcm DT (m )| kkΠsm 1 X ¯ ¯ DT (m1 )|Xm m1 1

Thus one may prove (ii) and (iii) in the same fashion as (i). This completes the proof There is a simple estimate which we shall use frequently u s α t ˜ kΠα m2 D T (m1 + x1 + x1 ) − Πm ¯ DT (m1 )k ≤ Cσ + O().

(8.8)

In fact (8.8) follows from (H1), (H4) and (8.7), since u s α t ˜ kΠα m2 D T (m1 + x1 + x1 ) − Πm ¯ DT (m1 )k  α u s ˜ ≤ k Πα m2 − Πm ¯ D T (m1 + x1 + x1 )k u s t u s ˜ + kΠα m ¯ kkD T (m1 + x1 + x1 ) − DT (m1 + x + x )k 1

+

t kΠα m ¯ kkDT (m1

+

xu1

+

xs1 )

1

t

− DT (m1 )k ≤ Cσ + O().

The next lemma shows that the map E given by (8.5) is well-defined and is a contraction. Lemma 8.3. There exist ∗ > 0 and σ > 0 such that for  < ∗ and T˜ satisfying ˜ cs () and g2 satisfies (8.3), then there exists kT˜ − T t k1 < σ if m1 + xu1 + xs1 ∈ W r1∗ = r1∗ (r2 ) > 0 such that for r1 < r1∗ , E : B1u (0, r1 ) → B1u (0, r1 ) is a contraction. Proof. We first show that E is well-defined from B1u (0, r1 ) into itself. Observe that for α = s, c, and (xu , xs , xc ) ∈ B1u (0, r1 ) × B1s (0, r1 ) × B1c (0, r1 ) u s u s c ˜ |Πα m2 D T (m1 + x1 + x1 )(x + x + x )| ≤ Cr1 ≤ r2

provided that r1∗ < r2 /C. Thus, g2 (Πsm2 DT˜(x1 )(xu + xs + xc ), Πcm2 DT˜(x1 )(xu + xs + cc )) is well-defined. Note again that m1 + xu1 + xs1 is denoted by x1 . From Lemma 8.2  −1 Πum DT˜(x1 )|X u exists. Hence, E(xu ) is defined for xu ∈ B u (0, r1 ). 2

m1

1

77

Next, we show |E(xu )| ≤ r1 . Using (8.3), (8.8) and Lemma 8.2, we obtain |E(xu )|  −1  u ˜ u = | Πm2 DT (x1 )|Xm g2 (Πsm2 DT˜(x1 )(xu + xs + xc ), Πcm2 DT˜(x1 )(xu + xs + xc )) 1  − Πum2 DT˜(x1 )(xs + xc ) |  −1    u s u s c c u s c ˜ ˜ ˜ u ≤ k Πm2 DT (x1 )|Xm1 k θ |Πm2 DT (x1 )(x + x + x )| + |Πm2 DT (x1 )(x + x + x )|  + |Πum2 DT˜(x1 )(xs + xc )|  −1   s c ˜ ˜ u s c ≤ k Πum2 DT˜(x1 )|Xm k Cσ + O() + θ(kΠ D T (x )| k + kΠ D T (x )| k) r1 1 Xm 1 Xm m2 m2 1 1 1  ≤ Cσ + O() + θλ(λ + 1) r1 . Since 0 < λ < 1 and θ < 1/2, by choosing ∗ and σ sufficiently small, we have that Cσ + O() + θλ(λ + 1) < 1. Finally, we show that E is a contraction. For xu , x ¯u ∈ B1u (0, r1 ), from (8.3), (8.5) and (8.8) and Lemma 8.2 |E(xu ) − E(¯ xu )|  −1  u ˜ u = | Πm2 DT (x1 )|Xm g2 (Πsm2 DT˜(x1 )(xu + xs + xc ), Πcm2 DT˜(x1 )(xu + xs + xc )) 1  − g2 (Πsm2 DT˜(x1 )(¯ xu + xs + xc ), Πcm2 DT˜(x1 )(¯ xu + xs + xc )) |  −1   u ≤ k Πum2 DT˜(x1 )|Xm kθ |Πsm2 DT˜(x1 )(xu − x ¯u )| + |Πcm2 DT˜(x1 )(xu − x ¯u )| 1   t c c t ˜ ≤ λθ kΠsm2 DT˜(x1 ) − Πsm DT (m )k + kΠ D T (x ) − Π DT (m )k |xu − x ¯u | 1 1 1 ¯ m2 m ¯ ≤ (Cσ + O())|xu − x ¯u |

(8.9)

which yields that E is a contraction by choosing ∗ and σ sufficiently small. This completes the proof. By the contraction mapping theorem, we obtain that for each (xs , xc ) ∈ B1s (0, r1 )× point xu ∈ B1u (0, r1 ), which defines a map from We denote it by xu = g˜1 (xs , xc ). Clearly g˜1 satisfies (8.4). Furthermore, this function is Lipschitz with Lipschitz constant θ.

B1c (0, r1 ), E has a unique fixed B1s (0, r1 ) × B1c (0, r1 ) to B1u (0, r1 ).

Lemma 8.4. There exist ∗ > 0 and σ > 0 such that for  < ∗ if m1 + xu1 + xs1 ∈ ˜ cs () and kT˜ − T t k1 < σ, then for (xs , xc ), (¯ W xs , x ¯c ) ∈ B1s (0, r1 ) × B1c (0, r1 ), |˜ g1 (xs , xc ) − g˜1 (¯ xs , x ¯c )| ≤ θ (|xs − x ¯s | + |xc − x ¯c |)

(8.10)

78

Proof. From the definition of g˜1 , using Lemma 8.2 and (8.8), it follows that |˜ g1 (xs , xc ) − g˜1 (¯ xs , x ¯c )| = |xu − x ¯u |  −1  u = | Πum2 DT˜(x1 )|Xm g2 (Πsm2 DT˜(x1 )(xu + xs + xc ), Πcm2 DT˜(x1 )(xu + xs + xc )) 1 − g2 (Πsm2 DT˜(x1 )(¯ xu + x ¯s + x ¯c ), Πcm2 DT˜(x1 )(¯ xu + x ¯s + x ¯c ))  − Πum2 DT˜(x1 )(xs − x ¯ s + xc − x ¯c ) |  −1  u ˜ u ≤ k Πm2 DT (x1 )|Xm k θ(|Πsm2 DT˜(x1 )(xu − x ¯ u + xs − x ¯ s + xc − x ¯c )| 1 + |Πcm2 DT˜(x1 )(xu − x ¯ u + xs − x ¯ s + xc − x ¯c )|)  u s s c c ˜ + |Πm2 DT (x1 )(x − x ¯ +x −x ¯ )| ≤ (Cσ + O())|xu − x ¯u | + (Cσ + O())(|xs − x ¯s | + |xc − x ¯c |)  −1   s s s c c c ˜ ˜ u s c + k Πum2 DT˜(x1 )|Xm kθ kΠ D T (x )| k|x − x ¯ | + kΠ D T | k|x − x ¯ | 1 Xm Xm m2 m2 1 1 1 ≤ (Cσ + O())|xu − x ¯u | + (λθ + Cσ + O())(|xs − x ¯s | + |xc − x ¯c |) which implies |xu − x ¯u | λθ + Cσ + O() (|xs − x ¯s | + |xc − x ¯c |) 1 − (Cσ + O()) ≤ θ(|xs − x ¯s | + |xc − x ¯c |) ≤

(8.11)

by choosing ∗ and σ sufficiently small. The proof is complete. Next we want to show that the jet equivalence class [˜ g1 ] of g˜1 does not depend on the choice of g2 ∈ j2 . Let f2 be another Lipschitz representative of j2 satisfying |f2 (xs , xc ) − f2 (¯ xs , x ¯c )| ≤ θ(|xs − x ¯s | + |xc − x ¯c |),

xα , x ¯α ∈ B2α (0, r¯2 ), α = s, c.

Let f˜1 denote the Lipschitz function given by Lemma 8.3 and defined on B1s (0, r¯1 ) × B1c (0, r¯1 ).

79

Lemma 8.5. [˜ g1 ] = [f˜1 ]. Proof. Let (xs , xc ) ∈ B1s (0, rˆ1 ) × B1c (0, rˆ1 ), where rˆ1 = min{r1 , r¯1 }. Set xu = g˜1 (xs , xc ),

x ˜u = f˜1 (xs , xc ).

We want to show k[˜ g1 ] − [f˜1 ]k =

|˜ g1 (xs , xc ) − f˜1 (xs , xc )| = 0. |xs | + |xc | )→(0,0)

lim c

(xs ,x

Note that from [g2 ] = [f2 ], |g2 (xs , xc ) − f2 (xs , xc )| = o(|xs | + |xc |) as (xs , xc ) → 0. From the definition of g˜1 and f˜1 and (8.8), Fromwe obtain |xu − x ˜u |  −1  u = | Πum2 DT˜(x1 )|Xm g2 (Πsm2 DT˜(x1 )(xu + xs + xc ), Πcm2 DT˜(x1 )(xu + xs + xc )) 1 −

f2 (Πsm2 DT˜(x1 )(˜ xu

s

+x +x

c

), Πcm2 DT˜(x1 )(˜ xu

s

c



+ x + x )) |



 ≤ (Cσ − O())|xu − x ˜u | + o |Πsm2 DT˜(x1 )(˜ xu + xs + xc )| + |Πcm2 DT˜(x1 )(˜ xu + xs + xc )| which implies together with |˜ xu | ≤ θ(|xs | + |xc |) |xu − x ˜u | = o(|xs | + |xc |). The proof is complete. We observe that for the given g2 , the function g˜1 satisfying (8.4) is locally unique, even within the class of continuous functions. We record it here as a lemma but the proof follows from the definition of E and the uniqueness of the fixed point. u Lemma 8.6. Let g1 : B1s (0, r¯1 ) × B1s (0, r¯1 ) → Xm be a continuous map satisfying 1 s c s s g1 (0, 0) = 0 and for (x , x ) ∈ B1 (0, r¯1 ) × B1 (0, r¯1 ) there exists (¯ xs , x ¯c ) ∈ B2s (0, r2 ) × s B2 (0, r2 ) such that

DT˜(x1 )(xs + xc + g1 (xs , xc )) = x ¯s + x ¯c + g2 (¯ xs , x ¯c ).

(8.12)

Then g1 (xs , xc ) = g˜1 (xs , xc ) on B1s (0, rˆ1 ) × B1c (0, rˆ1 ) for some rˆ1 > 0, where g˜1 is obtained from Lemma 8.4. Thus, for each given γ ∈ Σcs,l one may define γ˜ ∈ Σcs,l by θ θ γ˜ (m1 + xu1 + xs1 ) = [˜ g1 ]. The following summarizes what we have so far:

(8.13)

80

Proposition 8.7. There exist ∗ > 0 and σ > 0 such that if  < ∗ and kT˜ −T t k1 < σ, then for each γ ∈ Σcs,l there exists a unique γ˜ ∈ Σcs,l such that DT˜ maps γ˜ to γ in θ θ ˜ cs () and x2 = T˜(x1 ), for any Lipschitz representative g2 the sense that when x1 ∈ W of γ(x2 ), there exists a Lipschitz representative g˜1 of γ˜ (x1 ) such that (8.4) holds, i.e., DT˜(x1 )(xs + xc + g˜1 (xs , xc )) = x ¯s + x ¯c + g2 (¯ xs , x ¯c ) locally in the coordinates based at m1 and m2 , respectively. Define F(γ) = γ˜ . We claim Lemma 8.8. F is a contraction on Σcs,l θ . u s ˜ cs Proof. Let γ, γ ∗ ∈ Σcs,l θ . For m1 + x1 + x1 ∈ W (), we again write

T˜(m1 + xu1 + xs1 ) = m2 + xu2 + xs2 and m ¯ = T t (m1 ). From Proposition 8.7, there exist Lipschitz functions g˜1 , g˜1∗ : B1s (0, r1 ) × B1c (0, r1 ) → B1u (0, r1 ) such that (8.10) holds and [˜ g1 ] = F(γ)(m1 + xu1 + xs1 ), [˜ g1∗ ] = F(γ ∗ )(m1 + xu1 + xs1 ). Let xu = g˜1 (xs , xc ) and x ¯u = g˜1∗ (xs , xc ). From the definition of g˜1 and g˜1∗ g˜1 (xs , xc ) − g˜1∗ (xs , xc ) = xu − x ¯u  −1  u ˜ u = Πm2 DT (x1 )|Xm g2 (Πsm2 DT˜(x1 )(xu + xs + xc ), Πcm2 DT˜(x1 )(xu + xs + xc )) 1  − g2∗ (Πsm2 DT˜(x1 )(¯ xu + xs + xc ), Πcm2 DT˜(x1 )(¯ xu + xs + xc )) where g2 and g2∗ are Lipschitz representatives of γ and γ ∗ , respectively, at m2 +xu2 +xs2 which satisfy (8.3). Using (8.8),(8.10) and Lemma 8.2 we obtain |xu − x ¯u |  −1  u ≤ k Πum2 DT˜(x1 )|Xm k θ(|Πsm2 DT˜(x1 )(xu − x ¯u )| + |Πcm2 DT˜(x1 )(xu − x ¯u )|) 1 + k[g2 ] − [g2∗ ]k(|Πsm2 DT˜(x1 )(¯ xu + xs + xc )| + |Πcm2 DT˜(x1 )(¯ xu + xs + xc )|)   s u s c c u s c ˜ ˜ + o |Πm2 DT (x1 )(¯ x + x + x )| + |Πm2 DT (x1 )(¯ x + x + x )| ≤ (Cσ + O())|xu − x ¯u | + (λ + Cσ + O())k[g2 ] − [g2∗ ]k(|xs | + |xc |) + o(|xs | + |xc |),

81

here |¯ xu | ≤ θ(|xs | + |xc |) is used. Therefore, |xu − x ¯u | ≤

λ + Cσ + O() k[g2 ] − [g2∗ ]k(|xs | + |xc |) + o(|xs | + |xc |), 1 − (Cσ + O())

which yields k[˜ g1 ] − [˜ g1∗ ]k ≤

λ + Cσ + O() k[g2 ] − [g2∗ ]k. 1 − (Cσ + O())

Since 0 < λ < 1, for λ1 ∈ (λ, 1) we may choose ∗ and σ so small that λ + Cσ + O() < λ1 . 1 − (Cσ + O()) Thus kF(γ) − F(γ ∗ )k ≤ λ1 ||γ − γ ∗ ||.

(8.14)

This completes the proof. Our goal is to find a unique fixed point of F in Σcs,l θ . The difficulty here is that cs,l Σθ may not be a complete space. On the other hand, from (8.14), we have ||F (k) (γ) − F (k) (γ ∗ )|| ≤ λk1 ||γ − γ ∗ ||, ˜ cs () which yields that at each m1 + xu1 + xs1 ∈ W F k (γ)(m1 + xu1 + xs1 ) is a Cauchy sequence in J c (m1 ). Since J c (m1 ) is a Banach space, we have F k (γ)(m1 + xu1 + xs1 ) → γ0 (m1 + xu1 + xs1 ), s c u where γ0 (m1 + xu1 + xs1 ) ∈ J c (Xm × Xm , Xm ; 0, 0). Clearly, the limit γ0 is unique 1 1 1 and does not depend on the initial choice of γ. We shall show that γ0 ∈ Σcs,l θ . cs ˜ Recall that the center-stable manifold W () obtained in Section 7 is given by cs ˜ cs )), which has local coordinate representatives g˜cs and f˜cs by lem˜ W () = Θ(gr(h ˜ cs (). From mas 7.1 and 7.2. However, g˜cs and f˜cs do not represent all points on W Lemma 7.1 one obtains that for all m ∈ M ∩ B(m0 , ρ) and xs ∈ X s ((1 − η)ρ−1 ), ˜ cs (m, xs ) may be written as h

˜ cs (m, xs ) = Πu g˜cs (m, Πs |X s h m m m0

−1

xs )

82 s u where g˜cs is a Lipschitz map from (M ∩ B(m0 , ρ)) × Xm (ρ−1 ) to Xm . And from 0 0 −1 s s −3 Lemma 7.2 one has that for all m ∈ M ∩ B(m0 , ρ ) and x ∈ Xm (ρ )

˜ cs (m, xs ) = m0 + xs + xc + f˜cs (xs , xc ) m + xs + h 0 0 0 0 s c s (ρ−2 )× () and f˜cs is a Lipschitz map from Xm (ρ−2 )×Xm for some (xs0 , xc0 ) ∈ Xm 0 0 0 c u Xm () to Xm (). On the other hand from the Proposition 7.9, we have for  < ˆ < ∗ 0 0

˜ cs () ⊂ W ˜ cs (ˆ W ), ˜ cs (m, xs ) ∈ W ˜ cs () is thus one may choose  smaller such that each point m + xs + h expressed as ˜ cs (m, xs ) = m + xs + f˜cs (xs , 0) m + xs + h  ˜ cs (m, xs ) is equal to Πu g˜cs (m, Πs |X s −1 xs ). where f˜cs is determined by ˆ and m, and h m m m0 For the sake of convenience, we shall drop the superscript cs and tilde on f˜cs in our ˜ cs (). From the above discussion, there is a Lipschitz proofs. Let m1 + xu1 + xs1 ∈ W s c u () × Xm () → Xm () such that m1 + xu1 + xs1 = m1 + xs1 + f (xs1 , 0). function f : Xm 1 1 1 s c s s c s Let f1 (x , x ) = f (x + x1 , x ) − f (x1 , 0). We have f1 (0, 0) = 0 and |f1 (xs , xc ) − f1 (¯ xs , x ¯c )| ≤ θ(|xs − x ¯s | + |xc − x ¯c |), for (xs , xc ), (¯ xs , x ¯c ) ∈ B1s (0, r1 )×B1c (0, r1 ) for some r1 > 0. Thus f1 induces γ˜0 ∈ Σcs,l θ by γ˜0 (m1 + xu1 + xs1 ) = [f1 ]. Lemma 8.9. F(˜ γ0 ) = γ˜0 and hence γ0 = γ˜0 . ˜ cs () we write m2 + xu + xs = T˜(x1 ) and denote Proof. For x1 = m1 + xu1 + xs1 ∈ W 2 2 ˜ cs () at m2 + xu + xs by the local coordinate representative of W 2 2 m2 + x ¯s + x ¯c + f¯(¯ xs , x ¯c ). Let f2 (¯ xs , x ¯c ) = f¯(¯ xs + xs2 , x ¯c ) − f¯(xs2 , 0). Let f˜1 be given by Lemma 8.3 from f2 . We want to show [f1 ] = [f˜1 ]. For (xs , xc ) ∈ B1s (0, r1 ) × B1c (0, r1 ) let xu = f1 (xs , xc ) x ˜u = f˜1 (xs , xc )

83

From the definition of f˜1 ,  −1  u x ˜u = Πum2 DT˜(x1 )|Xm f2 (Πsm2 DT˜(x1 )(˜ xu + xs + xc ), Πcm2 DT˜(x1 )(˜ xu + xs + xc )) 1  u s c ˜ − Πm2 DT (x1 )(x + x ) . (8.15) s On the other hand, from the invariance of W cs (), there exists (¯ xs , x ¯c ) ∈ Xm (ρ−2 )× 2 c Xm () such that 1

T˜(m1 + xu1 + xs1 + xu + xs + xc ) = m2 + xu2 + xs2 + x ¯u + x ¯s + x ¯c where xu = f1 (xs , xc ) and x ¯u = f2 (¯ xs , x ¯c ). By the Taylor expansion, we obtain x ¯u + x ¯s + x ¯c = DT˜(x1 )(xu + xs + xc ) + o(|xu | + |xs | + |xc |). Note that |xu | ≤ θ(|xs | + |xc |). Hence, x ¯u + x ¯s + x ¯c = DT˜(x1 )(xu + xs + xc ) + o(|xs | + |xc |) and for α = u, s, c u s c s c ˜ x ¯α = Πα m2 D T (x1 )(x + x + x ) + o(|x | + |x |).

In particular, x ¯u = Πum2 DT˜(x1 )(xu + xs + xc ) + o(|xs | + |xc |) = Πum DT˜(x1 )(xu ) + Πum DT˜(x1 )(xs + xc ) + o(|xs | + |xc |) 2

2

which yields  −1   u u ˜(x1 )(xs + xc ) + o(|xs | + |xc |) u xu = Πum2 DT˜(x1 )|Xm x ¯ − Π D T m2 1  −1   u s c u s c ˜ ˜ u = Πm2 DT (x1 )|Xm f2 (¯ x ,x ¯ ) − Πm2 DT (x1 )(x + x ) + o(|xs | + |xc |) 1  −1  u ˜ u = Πm2 DT (x1 )|Xm f2 (Πsm2 DT˜(x1 )(xu + xs + xc ) + o(|xs | + |xc |), 1 Πcm2 DT˜(x1 )(xu + xs + xc ) + o(|xs | + |xc |))  − Πum2 DT˜(x1 )(xs + xc ) + o(|xs | + |xc |).

84

Using (8.5), (8.8), (8,15) and Lemma 8.2, we obtain |˜ xu − xu |  −1   s u u c u u ˜(x1 )(˜ ˜ u ≤ k Πum2 DT˜(x1 )|Xm kθ |Π D T x − x )| + |Π D T (x )(˜ x − x )| 1 m2 m2 1 + o(|xs | + |xc |) ≤ (Cσ + O())|˜ xu − xu | + o(|xs | + |xc |). Thus, by choosing ∗ and σ so small that Cσ + O() < 1/2, we obtain |˜ xu − xu | ≤ o(|xs | + |xc |), namely, |f1 (xs , xc ) − f˜1 (xs , xc )| ≤ o(|xs | + |xc |) and hence [f1 ] = [f˜1 ]. The uniqueness of the limit γ0 implies that γ0 = γ˜0 . This completes the proof. We are now ready to show Proposition 8.10. f (xs , xc ) is differentiable. Proof. It is clear that from the definition of F, particularly the definition of E, we have that if γ ∈ Σcs,a then F(γ) ∈ Σcs,a . Since J a (m) = J d (m) is closed, γ0 (m1 +xu1 +xs1 ) ∈ θ θ J a (m1 ), where γ0 is the limit of the Cauchy sequence F k (γ). Since γ0 = γ˜0 is unique, ˜ cs (), we have [f1 ] ∈ J a (m1 ), that is, f is differentiable at (xs , 0). for m1 +xu1 +xs1 ∈ W 1 s s c Next, we show f (x , xc ) is differentiable in Xm () × X (). m1 1 s c s c For (xs0 , xc0 ) ∈ Xm () × X () write m + x + x + f (xs0 , xc0 ) as 1 m1 0 0 1 m1 + xs0 + xc0 + f (xs0 , xc0 ) = m2 + x ¯u2 + x ¯s2 .

(8.16)

˜ cs () near this point may be written as Any point on W m1 + xs0 + xc0 + xs + xc + f (xs0 + xs , xc0 + xc ) = m2 + x ¯s + xs + xc + f¯(¯ xs + xs , xc ), 2

2

2

2

2

2

(8.17)

˜ cs () at m2 . Subtracting (8.16) from (8.17), we get where f¯ is the representative of W xs + xc + f (xs0 + xs , xc0 + xc ) − f (xs0 , xc0 ) = xs + xc + f¯(¯ xs + xs , xc ) − f¯(¯ xs , 0). 2

2

2

2

2

2

(8.18)

85

To simplify computation, let f1 (xs , xc ) = f (xs0 + xs , xc0 + xc ) − f (xs0 , xc0 ), and f2 (xs2 , xc2 ) = f¯(¯ xs2 + xs2 , xc2 ) − f¯(¯ xs2 , 0). Thus (8.18) has the form xs + xc + f1 (xs , xc ) = xs2 + xc2 + f2 (xs2 , xc2 )

(8.19)

Applying the projection Πα m1 to (8.19) for α = s, c, we obtain s c |xα | ≤ kΠα m1 k(1 + θ)(|x2 | + |x2 |).

Similarly, we have α s c |xα 2 | ≤ kΠm2 k(1 + θ)(|x | + |x |).

(8.20)

Note that f2 is differentiable at (0, 0). Hence from (8.19), we obtain f1 (xs , xc ) = Πum1 (xs2 + xc2 ) + Πum1 Df2 (0, 0)(xs2 , xc2 ) + o(|xs2 | + |xc2 |)  = Πum1 Πsm2 + Πcm2 (xs + xc + f1 (xs , xc )) + Πum1 Df2 (0, 0) Πsm2 (xs + xc + f1 (xs , xc )), Πcm2 (xs + xc + f1 (xs , xc )



+ o(|xs | + |xc |).   Let G ≡ I − Πum1 Πsm2 + Πcm2 Πum1 − Πum1 Df2 (0, 0) Πsm2 Πum1 , Πcm2 Πum1 . Thus the above identity may be written as  Gf1 (xs , xc ) = Πum1 Πsm2 + Πcm2 (xs + xc )  + Πum1 Df2 (0, 0) Πsm2 (xs + xc ), Πcm2 (xs + xc ) + o(|xs | + |xc |) Note that G may also be written as  G = I − Πum1 Πsm2 − Πsm1 + Πcm2 − Πcm1 Πum1  − Πum1 Df2 (0, 0) (Πsm2 − Πsm1 )Πum1 , (Πcm2 − Πcm1 )Πum1 . Hence,  kI − Gk ≤ kΠum1 k2 kΠsm2 − Πsm1 k + kΠcm2 − Πcm1 k  + kΠum1 k2 θ kΠsm2 − Πsm1 k + kΠcm2 − Πcm1 k  = (1 + θ)kΠum1 k2 kΠsm2 − Πsm1 k + kΠcm2 − Πcm1 k .

(8.21)

86

Note that from (8.16) |m2 − m1 | < 5. Thus, from (H4) there exists ∗ > 0 such that if  < ∗ , then  1 (1 + θ)kΠum1 k2 kΠsm2 − Πsm1 k + kΠcm2 − Πcm1 k < , 2 u u which yields kI − Gk ≤ 21 and that G is an isomorphism from Xm onto Xm . Denote 1 1 −1 the inverse by G , which is a bounded linear operator. From (8.21), we obtain

f1 (xs , xc ) = G−1 R(xs + xc ) + o(|xs | + |xc |),   where R = Πum1 Πsm2 + Πcm2 −Πum1 Df2 (0, 0) Πsm2 Πum1 , Πcm2 Πum1 is a bounded linear u operator from Xm into itself. Therefore f1 is differentiable at (xs , xc ) = (0, 0). The 1 proof is complete. Finally, we want to show Df is C 0 . s c u Let L(X s ⊕ X c , X u ) denote the vector bundle over M with fiber L(Xm ⊕ Xm , Xm ), u c s ) is a Banach space√of all bounded linear operators from ⊕ Xm , Xm where L(Xm c u s . For each m0 ∈ M , let η < 2 − 1 and U be an η-neighborhood of to Xm Xm ⊕ Xm m0 . Thus, from (4.5) we may define a trivialization s c u s c u ˜ cs Ψ m0 : U ∩ M × L(Xm0 ⊕ Xm0 , Xm0 ) → L(X ⊕ X , X )

by   cs u s −1 c −1 ˜ s ) c ) Ψm0 (m, L(m0 )) = Πm L(m0 ) (Πm |Xm ⊕ (Πm |Xm . 0 0

(8.22)

It is not hard to see that L(X s ⊕ X c , X u ) is a Finsler bundle with the trivialization ˜ cs which gives an isomorphism from L(X s ⊕ X c , X u ) onto L(X s ⊕ X c , X u ), Ψ m0 m0 m0 m0 m m m ˜ → (Πum |X u )−1 L(Π ˜ sm Πsm + Πcm Πcm ). the inverse of which is L 0 0 m0 Define the space n cs ˜ cs () → L(X s ⊕ X c , X u ) is a C0 section and Λθ = ` : W k`k ≤ θ} with the norm

n o cs ˜ k`k = sup k`(x)k : x ∈ W () .

α Note that ||`(x)|| = sup{|`(x)(˜ xs , x ˜c )| : x ˜α ∈ Xm , α = s, c, |˜ xs | + |˜ xc | ≤ 1}. It is easy cs,l cs to check Λcs θ is a Banach space. The space Λθ may be regarded as a subset of Σθ by identifying ` and [`]. cs We shall show that F(Λcs θ ) ⊂ Λθ .

87

˜ cs () be fixed. Consider a point x Let x1 = m1 + xu1 + xs1 ∈ W ¯1 = m ¯1 +x ¯u1 + x ¯s1 ∈ ˜ () sufficiently close to x1 . We write W cs

T˜(x1 ) = m2 + xu2 + xs2 = x2 and T˜(¯ x1 ) = m ¯2 +x ¯u2 + x ¯s2 = x ¯2 . Let B(x1 , r) be a ball in X. We may choose r small enough that if x ¯1 ∈ B(x1 , r) ∩ cs ˜ W (), then m ¯ 1 is in an η-neighborhood of m1 and m ¯ 2 is in an η-neighborhood of m2 . cs ˜ ˜ Let ` ∈ Λcs θ and denote F(`) by `. To show ` ∈ Λθ , it is enough to show Proposition 8.11. There exist ∗ > 0 and σ > 0 such that if  < ∗ and kT˜ − T t k < σ, then  −1  ˜ 1 ) − Πu | X u ˜ x1 ) Πs Πs + Πc Πc k = 0. lim k`(x `(¯ m ¯ 1 m1 m ¯ 1 m1 m ¯ 1 m1 x ¯1 →x1

In order to prove this proposition, we need some lemmas. Let  −1 u S(x1 ) = Πum2 DT˜(x1 )|Xm Πum2 1  −1  −1 u ¯ x1 ) = Πum ˜ u u S(¯ | Π D T (¯ x )| Πum 1 Xm ¯ 1 Xm1 m ¯2 ¯ 2. ¯1

(8.23) (8.24)

From Lemma 8.2, S and S¯ are well-defined. Furthermore, Lemma 8.12. ¯ x1 ) − S(x1 )k = 0 lim kS(¯

x ¯1 →x1

Proof. We first notice that from the continuity of DT˜ and Lemma 4.3 it follows that x ¯ 2 → x2 , m ¯ 1 → m1 and m ¯ 2 → m2 as x ¯1 → x1 . We consider only x ¯1 ∈ B(x1 , r). Thus m ¯ 1 and m ¯ 2 are in an η-neighborhoods of m1 and m2 , respectively. For x ∈ X, let u ¯ x1 )x. Note that y, z ∈ Xm . Hence, y = S(x1 )x and z = S(¯ 1 Πum2 DT˜(x1 )y = Πum2 x

(8.25)

u ˜ x1 )Πum Πum ¯ 2 D T (¯ ¯ 1 z = Πm ¯ 2 x.

(8.26)

Subtracting (8.26) from (8.25), we obtain Πum2 DT˜(x1 )(y − z)   u ˜ + Πum2 DT˜(x1 )Πum1 − Πum D T (¯ x )Π 1 ¯2 m ¯1 z  = Πum2 − Πum ¯ 2 x.

(8.27)

88

Using Lemma 8.2 and (4.5), we have from (8.26)  −1 λ(1 + η) u |z| ≤ λk Πum |x|. | k|Πum ¯ 1 Xm1 ¯ 2 x| ≤ (1 − η)

(8.28)

Also from (8.27) |y − z| u u ˜ x1 )Πum ≤ λkΠum2 DT˜(x1 )Πum1 − Πum ¯ 2 D T (¯ ¯ 1 k|z| + λkΠm2 − Πm ¯ 2 k|x|,

which yields ¯ x1 )k kS(x1 ) − S(¯ 1+η u u u u ˜ x1 )Πum kΠm2 DT˜(x1 )Πum1 − Πm ≤ λ2 ¯ 2 D T (¯ ¯ 1 k + λkΠm2 − Πm ¯ 2k 1−η

(8.29)

¯1 → x1 in (8.29), we complete the Using the continuity of Πum and DT˜ and letting x proof. s c For xs + xc ∈ Xm ⊕ Xm with |xs | + |xc | ≤ 1 and let 1 1

˜ 1 )(xs + xc ) xu = `(x  ˜ x 1 ) Πs x s + Π c x c . x ¯u = `(¯ m ¯1 m ¯1 u Proof of Proposition 8.11. Since m ¯ 1 is in an η-neighborhood of m1 , by (4.5), Πum ¯ 1 |Xm 1 u u u u u is invertible. Thus there exists x ˜u ∈ Xm such that x ¯ = Π x ˜ . By the m ¯ 1 Xm1 1 ˜ definition of `,

u

x =



u Πum2 DT˜(x1 )|Xm 1

−1 

`(x2 )((Πsm2 + Πcm2 )DT˜(x1 )(xu + xs + xc ))  − Πum2 DT˜(x1 )(xs + xc ) (8.30)

and  −1  −1 u ˜ u u x ˜u = Πum | Π D T (¯ x )| 1 Xm ¯ 1 Xm1 m ¯2 ¯1   c s c c ˜ `(¯ x2 )((Πsm + Π )D T (¯ x ) Πum ˜u + Πsm 1 ¯2 m ¯2 ¯ 1x ¯ 1 x + Πm ¯ 1 x ))   s s c c ˜(¯ − Πum D T x ) Π x + Π x ) . 1 ¯2 m ¯1 m ¯1

89

Because ` ∈ Λcs θ , we have

|xu | ≤ θ (|xs | + |xc |)

and using (4.5), |˜ xu | ≤

θ(1 + η) (|xs | + |xc |) . 1−η

Next we want to estimate |xu − x ˜u |. Let   −1 s s c c u I1 = S(x1 ) `(x2 )(Πsm2 + Πcm2 ) − Πum | `(¯ x )(Π Π + Π Π ) X 2 ¯ 2 m2 m ¯ 2 m2 m ¯ 2 m2

I2 I3

I4

I5

DT˜(x1 )(xu + xs + xc )  −1   s s c c u s c ¯ x1 )) Πum ˜ u = (S(x1 ) − S(¯ | `(¯ x ) Π Π + Π Π 2 ¯ 2 Xm2 m ¯ 2 m2 m ¯ 2 m2 D T (x1 )(x + x + x )  s c c ¯ x1 )`(¯ = S(¯ x2 ) (Πsm ¯ 2 Πm 2 + Π m ¯ 2 Πm 2 )  u s s s c c c DT˜(x1 )(xu − Πum x ˜ + x − Π x + x − Π x ) ¯1 m ¯1 m ¯1  s s c c ¯ x1 )`(¯ ˜ ˜ x1 ))(Πum = S(¯ x2 ) Πsm2 (Πsm ˜u + Πsm ¯ 2 D T (x1 ) − Πm ¯ 2 D T (¯ ¯ 1x ¯ 1 x + Πm ¯ 1 x ))  c c c u u s s c c ˜ ˜ + Πm x1 ))(Πm ˜ + Πm ¯ 2 (Πm2 D T (x1 ) − Πm ¯ 2 D T (¯ ¯ 1x ¯ 1 x + Πm ¯ 1x )   s c ¯ x1 )Πum ˜ x1 ) Πsm = S(¯ ¯ 1 xc − S(x1 )Πum2 DT˜(x1 )(xs + xc ). ¯ 2 D T (¯ ¯ 1x + Π m It is straightforward to check that xu − x ˜u = I1 + I2 + I3 + I4 + I5 .

(8.31)

From the continuity of `, we have that I1 → 0 as x ¯1 → x1 . Lemma 8.12 yields that I2 , I5 → 0 as x ¯1 → x1 . The continuity of the projections and DT˜ gives that I4 → 0 as x ¯1 → x1 . Using (8.8), we may choose ∗ and σ sufficiently small such than   1 s s c c ˜u | + C kΠum1 − Πum k + kΠ − Π k + kΠ − Π k . |I3 | ≤ |xu − x ¯1 m1 m ¯1 m1 m ¯1 2 Therefore, from (8.31) we have |xu − x ˜u | → 0 as x ¯1 → x1 . This completes the proof. Proposition 8.13. There exists positive constant ∗ such that for  < ∗ , Df (xs , xc ) is continuous in (xs , xc ). cs Proof. As we mentioned before, Proposition 8.11 yields that F(Λcs θ ) ⊂ Λθ . Since F cs ˜ is a contraction and Λcs θ is complete F has a unique fixed point ` ∈ Λθ . On the other ˜ which yields hand, γ˜0 is the unique fixed point of F in Σcs,l ˜0 = [`] θ . Therefore, γ

m1 + xu1 + xs1 → Df (xs1 , 0)

90

is continuous, that is,  −1  s Πm s ¯1 ¯ u | D f (¯ x , 0) lim kDf (xs1 , 0) − Πum ¯ 1 Xm1 1 x ¯1 →x1 0

 0 k = 0, Πcm ¯1

(8.32)

˜ cs (). where x1 = m1 + xu1 + xs1 , x ¯1 = m ¯1 +x ¯u1 + x ¯s1 ∈ W We shall show that Df (xs , xc ) is continuous at (xs0 , xc0 ). We write m1 + xs0 + xc0 + f (xs0 , xc0 ) = m2 + xs2 + xu2

(8.33)

˜ cs () nearby and let f2 be the representative at m2 , so xu2 = f2 (xs2 , 0). Any point in W may uniquely be expressed as m1 + xs0 + xc0 + xs + xc + f (xs0 + xs , xc0 + xc ) = m2 + xs2 + x ¯s2 + x ¯c2 + f2 (xs2 + x ¯s2 , x ¯c2 ).

(8.34)

Subtracting (8.33) from (8.34) and letting f˜(xs , xc ) = f (xs0 + xs , xc0 + xc ) − f (xs0 , xc0 ) and f˜2 (¯ xs2 , x ¯c2 ) = f2 (xs2 + x ¯s2 , x ¯c2 ) − f2 (xs2 , 0) we obtain xs + xc + f˜(xs , xc ) = x ¯s2 + x ¯c2 + f˜2 (¯ xs2 , x ¯c2 ),

(8.35)

(xs , xc ) = (Πsm1 , Πcm1 )(¯ xs2 + x ¯c2 + f˜2 (¯ xs2 , x ¯c2 ))

(8.36)

f˜(xs , xc ) = Πum1 (¯ xs2 + x ¯c2 + f˜2 (¯ xs2 , x ¯c2 )).

(8.37)

which gives and Let G2 (¯ xs2 , x ¯c2 ) = (Πsm1 , Πcm1 )(¯ xs2 + x ¯c2 + f˜2 (¯ xs2 , x ¯c2 )). Differentiating (8.37) with respect to (¯ xs2 , x ¯c2 ) and evaluating at (¯ xs2 , x ¯c2 ) = (0, 0), we have Df (xs0 , xc0 )DG2 (0, 0) = Πum1 ⊕ Πum1 + Πum1 Df2 (xs2 , 0), and  DG2 (0, 0)(¯ xs2 , x ¯c2 ) = Πsm1 (¯ xs2 + x ¯c2 ) + Πsm1 Df2 (xs2 , 0)(¯ xs2 , x ¯c2 ),  Πcm1 (¯ xs2 + x ¯c2 ) + Πcm1 Df2 (xs2 , 0)(¯ xs2 , x ¯c2 ) .

91

Let (˜ xs0 , x ˜c0 ) be a point near (xs0 , xc0 ). Then we may write m1 + x ˜s0 + x ˜c0 + f (˜ xs0 , x ˜c0 ) = m3 + xs3 + f3 (xs3 , 0). Similarly, we have Df (˜ xs0 , x ˜c0 )DG3 (0, 0) = Πum1 ⊕ Πum1 + Πum1 Df3 (xs3 , 0) and  DG3 (0, 0)(¯ xs3 , x ¯c3 ) = Πsm1 (¯ xs3 + x ¯c3 ) + Πsm1 Df3 (xs3 , 0)(¯ xs3 , x ¯c3 ),  xs3 + x Πcm1 (¯ ¯c3 ) + Πcm1 Df3 (xs3 , 0)(¯ xs3 , x ¯c3 ) . In order to show lim

(˜ xs0 ,˜ xc0 )→(xs0 ,xc0 )

kDf (˜ xs0 , x ˜c0 ) − Df (xs0 , xc0 )k = 0,

(8.38)

we let I1 = (Df (xs0 , xc0 ) − Df (˜ xs0 , x ˜c0 )) DG2 (0, 0)     s 0 Πm3 s c I2 = Df (˜ x0 , x − DG2 (0, 0) ˜0 ) DG3 (0, 0) 0 Πcm3   I3 = Πum1 Πsm2 − Πsm3 ⊕ Πum1 Πcm2 − Πcm3    s 0 Πm3 u s s I4 = Πm1 Df2 (x2 , 0) − Df3 (x3 , 0) . 0 Πcm3 It is easy to verify that I1 = I2 + I3 + I4 .

(8.39)

By Lemma 4.3, we have that as (˜ xs0 , x ˜c0 ) → (xs0 , xc0 ) m3 + xs3 + f3 (xs3 , 0) → m2 + xs2 + f2 (xs2 , 0) and m3 → m2 . Therefore, from the continuity of Πα m in m, we have that lim

(˜ xs0 ,˜ xc0 )→(xs0 ,xc0 )

I3 = 0.

Express I4 as I4 = +

Πum1



Df2 (xs2 , 0)

Πum1 (Πum2





Πum3 )

−1 u Πum3 |Xm Df3 (xs3 , 0) 2

−1 u Πum3 |Xm Df3 (xs3 , 0) 2





Πsm3 0

Πsm3 0

 0 Πcm3  0 . Πcm3

92

Using (8.32) and the continuity of Πum in m, we obtain lim

I4 = 0.

lim

I2 = 0

lim

I1 = 0.

(˜ xs0 ,˜ xc0 )→(xs0 ,xc0 )

Similarly, (˜ xs0 ,˜ xc0 )→(xs0 ,xc0 )

Thus, from (8.39) it follows (˜ xs0 ,˜ xc0 )→(xs0 ,xc0 )

s c To show (8.38), it is enough to show that DG2 (0, 0) is an isomorphism from Xm ×Xm 2 2 s c onto Xm × Xm . We must show that 1 1

xs = Πsm1 (¯ xs2 + x ¯c2 ) + Πsm1 Df2 (xs2 , 0)(¯ xs2 , x ¯c2 ) xc = Πcm1 (¯ xs2 + x ¯c2 ) + Πcm1 Df2 (xs2 , 0)(¯ xs2 , x ¯c2 ) is uniquely solvable for (¯ xs2 , x ¯c2 ). Note that from (8.33) |m1 − m2 | ≤ 5. From (4.5), by choosing ∗ sufficiently small such that m2 is in an η-neighborhood of m1 ,  −1  −1 c s c | Πsm1 |Xm and Π exist. Hence, X m1 m1 2 x ¯s2

=



s Πsm1 |Xm 2

−1

   c s −1 x − Πsm1 Xm Πsm1 − Πsm2 x ¯2 s

2

−1



 s s s − Πsm1 |Xm Π − Π Df2 (xs2 , 0)(¯ xs2 , x ¯c2 ) m m 1 2 2   −1 −1  s c c c c c c Πcm1 − Πcm2 x ¯2 x ¯2 = Πm1 |Xm2 x − Πm1 xm2 −



c Πcm1 |Xm 2

−1

 Πcm1 − Πcm2 Df2 (xs2 , 0)(¯ xs2 , x ¯c2 ).

We may choose ∗ sufficiently small so that the right hand side of the above is a linear s c s c contraction on Xm × Xm uniformly with respect to (xs , xc ) ∈ Xm × Xm . This 2 2 1 1 completes the proof. ˜ cs () locally From Lemma 7.1 one may also represent the center-stable manifold W in terms of functions g˜cs (m, x ˜s ). It is not hard to see, by using the implicit function cs 1 theorem, that g˜ is C . Summarizing all above propositions gives Theorem 8.1.

93

§9. Smoothness of Center-Unstable Manifolds. All results obtained in Section 8 also hold for center-unstable manifolds. In particular, √ Theorem 9.1. Let λ1 ∈ (λ, 1), ρ ∈ (1, 1/ λ), and µ ∈ (0, 1) such that µρ < 1/2. Then there exist positive constants ˜∗ , ∗ = ∗ (˜ ), δ ∗ = δ ∗ () <  and σ = σ(, δ) such ˜ cu () obtained that if ˜ < ˜∗ ,  < ∗ , δ < δ ∗ , and T˜ satisfies ||T˜ − T t ||1 < σ, then W 1 in Section 6 is a C manifold. Generally, a few modifications are needed to adapt the proofs presented in Section 8 to the case of the center unstable manifold. The most significant differences are the definition of the graph transform and the associated spaces. We shall outline the proofs and leave the details to the interested reader. The center-unstable manifold we obtained in section 6 is the graph of a Lipschitz section over X u (), which is denoted by W cu () and satisfies ˜ cu ()) ∩ Θ(X u () ⊕ X s ()) = W ˜ cu () T˜(W

(9.1)

˜ cu ()) ∩ W ˜ cu () → W ˜ cu () T˜ : T˜−1 (W

(9.2)

is a homeomorphism. Corresponding to Σcs,l θ , we define Σcu,l = {γ : W cu () → J θ (X u ⊕ X c , X s ; 0, 0), ||γ|| < ∞} θ where J θ (X u ⊕ X c , X s ; 0, 0) is the jet bundle over M with the jet fiber u c s J θ (Xm ⊕ Xm , Xm ; 0, 0) u c s = {j ∈ J b (Xm ⊕ Xm , Xm ; 0, 0) : j has a representative g satisfying Lip(g|U ) ≤ θ} u c , Remember that Lip(g|U ) is same as in Section 8, U is a neighborhood of 0 in Xm ×Xm cu,l and θ = µρ < 1/2. The norm in Σθ is given by

||γ|| = sup{||γ(m1 + xu1 + xs1 )|| : m1 + xu1 + xs1 ∈ W cu ()}. In order to show the smoothness of W cu (), we need find a candidate for the tangent bundle of W cu () then to show it indeed is the tangent bundle. The outline of the approach is as follows. The first step is to construct γ˜ ∈ Σcu,l for each γ ∈ Σcu,l such θ θ ˜ that γ˜ is the image of γ under DT in certain sense, defining a graph transform F(γ) = γ˜ .

94

In the case of the center-stable manifold, we constructed the preimage γ˜ of γ instead of the image. The next step is to show that F has a unique fixed point in Σcu,l θ . The difficulty cu,l here is that Σθ is not complete. To overcome this difficulty, we first show that F k u s is a contraction in Σcu,l θ , thus the iterations F (γ)(m1 + x1 + x1 ) define a Cauchy sequence in J c (m1 ). Since J c (m1 ) is complete, F k (γ)(m1 + xu1 + xs1 ) converges to some γ0 (m1 + xu1 + xs1 ) in J c (m1 ) as k → 0. On the other hand, the local coordinate representative of W cu (), (see Lemma 6.2,) m0 + xu + xs + f (xu , xc ) ˜ cu produces γ˜0 ∈ Σcu,l ˜0 is a fixed θ . It turns out from the invariance of W () that γ point of F. Hence, γ˜0 = γ0 . The differentiability of f follows from the fact that F(J a ) ⊂ J a and J a = J d is closed. The final step is to show Df is C 0 . The idea here is to prove there is a continuous section of a vector bundle, which is a fixed point of F. ˜ cu (), Let us first look at how to construct F. For each fixed x2 = m2 +xu2 +xs2 ∈ W ˜ cu () such that from (9.1) and (9.2) there is a unique point x1 = m1 + xu1 + xs1 ∈ W T˜(x1 ) = x2 . For fixed γ ∈ Σcu,l θ , let j1 = γ(x1 ). We choose a Lipschitz representative g1 from jet j1 satisfying ||g1 (xu , xc ) − g1 (¯ xu , x ¯c )|| ≤ θ(|xu − x ¯u | + |xc − x ¯c |), for (xu , xc ), (¯ xu , x ¯c ) ∈ B1u (0, r1 ) × B1c (0, r1 ) α where B1α (0, r1 ), α = u, c, are closed balls in Xm . 1 We first want to construct a Lipschitz map,

g˜2 : B2u (0, r2 ) × B2c (0, r2 ) → B2s (0, r2 ) with Lip g˜2 ≤ θ such that for (¯ xu , x ¯c ) ∈ B2u (0, r2 ) × B2c (0, r2 ), there exists a unique (xu , xc ) ∈ B1u (0, r1 ) × B1c (0, r1 ) such that DT˜(x1 )(xu + xc + g1 (xu , xc )) = x ¯u + x ¯c + g˜2 (¯ xu , x ¯c ).

(9.3)

95

To see this, for each (¯ xu + x ¯c ) ∈ B2u (0, r2 ) × B2c (0, r2 ), we define a map E from u c u c B1 (0, r1 ) × B1 (0, r1 ) to Xm1 × Xm by E = (E u , E c ), where 1 −1    α α u c u c ˜ ˜ α D T (x )| x ¯ D T (x )(x E α (xu , xc ) = Πα − Π + x + g (x , x )) + xα 1 X 1 1 m2 m2 m 1

for α = u, c. In the same fashion as for E in Lemma 8.3 and 8.4, one may show that there exists r1∗ > 0 such that if r1 < r1∗ then E is well-defined and is a contraction from B1u (0, r1 ) × B1c (0, r1 ) into itself. Thus E has a unique fixed point (xu , xs ) = (xu (¯ xu , x ¯c ), xc (¯ xu , x ¯c )) which yields a map x ¯s = g˜2 (¯ xu , x ¯c ) = Πsm2 DT˜(x1 )(xu + xc + g1 (xu + xs )). As in Lemma 8.4 and Lemma 8.5, one may also show that g˜2 is Lipschitz with Lip g˜2 ≤ θ and [˜ g2 ], the jet equivalent class, does not depend on the choice of g1 ∈ j1 . Hence one has γ˜ ∈ Σcu,l θ , determined uniquely by γ, where γ˜ (x2 ) = [˜ g2 ]. Consequently, one may define a graph transform F(γ) = γ˜ . We claim F is a contraction from Σcu,l to Σcu,l θ θ . There is no significant change in the proof of this claim, comparing with Lemma 8.8. As we mentioned before, since Σcu,l is not complete, we cannot directly claim F θ cu,l has a fixed point in Σθ . In order to show that F has a unique fixed point in Σcu,l θ one first observes that the iteration F k (x2 ) is a Cauchy sequence in J c (m2 ). Since J c (m2 ) is complete, there exists γ0 (x2 ) ∈ J c (m2 ) such that F k (x2 ) → γ0 (x2 ) as k → ∞. On the other hand, the local coordinate representative (see Lemma 6.2) m + xu + xc + f (xu , xc ) produces γ˜0 ∈ Σcu,l defined by θ γ˜0 (x2 ) = [f (xu2 + xu , xc ) − f (xu2 , 0)] ˜ ( cu)() at m2 . where f gives the local coordinate representation of W

96

˜ cu (), one easily shows that γ˜0 is a fixed point of F, and From the invariance of W ˜ cu () therefore γ˜0 = γ0 . It is not hard to see that if γ(x2 ) ∈ J a (m2 ) for all x2 ∈ W then F(γ)(x2 ) ∈ J a (m2 ). From the Theorem on Lipschitz jets, J a (m2 ) = J d (m2 ) ⊂ J c (m2 ) are Banach spaces. ˜ cu (), we obtain Thus, choosing γ ∈ Σcu,l such that γ(x2 ) ∈ J a (m2 ) for all x2 ∈ W θ F k (γ) → γ˜0 and γ˜0 (x2 ) ∈ J a (m2 ) = J d (m2 ), which implies f is differentiable at (xu2 , 0). The proof of the differentiability of f at (xu , xs ) follows in the same fashion as in Lemma 8.10. Finally we want to show that Df is continuous. As we did in section 8, we shall consider the vector bundle L(X u ⊕ X c , X s ) set ˜ cu = {` : W ˜ cu () → L(X u ⊕ X c , X s ) is a C 0 section and ||`|| ≤ θ}. Λ θ ˜ cu with norm Endow Λ θ ˜ cu ()}. ||`|| = sup{||`(x1 )|| : x1 ∈ W ˜ cu is a complete metric space, which may be regarded as It is not hard to check Λ θ a closed subspace of Σcu,l . Here we identify ` with [`] ∈ Σcu,l θ θ . In the same fashion cu ˜ as in Lemma 8.11 one may show that F map Λθ into itself. Observe that F˜ is a ˜ cu and therefore has a unique fixed point in Λ ˜ cu , which yields contraction in Λ θ θ m2 + xu2 + fm2 (xu2 , 0) → Df (xu2 , 0) is continuous. Using the same approach as Lemma 8.13 one may show that Df (xu , xc ) is continuous in (xu , xc ). Using the Implicit Function Theorem, one may prove that g˜cu given in Lemma 6.1 is C 1 .

97

§10. Persistence of Invariant Manifold. The kep point here is that the center-unstable and center-stable manifolds constructed earlier are transversal. Actually, the Lipschitz nature of these manifolds provides the intersection, but we wish to also prove its smoothness. ˜ =W ˜ cu () ∩ W ˜ cs (). Then we have Let M Theorem 10.1. There exists ∗ > 0 such that for each  < ∗ there is σ > 0 such that if ||T˜ − T t ||1 < σ, then there exists C 1 diffeomorphism ˜ K = KT˜ : M → M which satisfies ||KT˜ − I||C 1 (M,X) → 0

(10.1)

˜ is a C 1 invariant manifold for T˜. as ||T˜ − T t ||1 → 0. Furthermore M ˜ is C 1 close to M . This theorem implies that M ˜ is invariant. Take m ˜ . By Theorem 7.1, we have Proof. We first show that M ˜ ∈M cs cs ˜ () is forward invariant, hence T˜(m) ˜ (). On the other hand, from that W ˜ ∈ W Theorem 6.3, we have ˜ cu ()) ∩ (X u () ⊕ X s ()) = W ˜ cu (). T˜(W ˜ cs (), which implies that T˜(m) Hence, by using T˜(m) ˜ ∈ W ˜ ∈ X u () ⊕ X s (), we conclude that ˜ cu (). T˜(m) ˜ ∈W ˜ , which means M ˜ is positively invariant. Therefore T˜(m) ˜ ∈M ˜ cu Next we construct K. Fix m1 ∈ M , let g˜cu and g˜cs be local representatives of h cs ˜ and h , respectively, which are given in Lemma 6.1 and Lemma 7.1. Define a map u u A : B(m1 , ρ) ∩ M × Xm (ρ−1 ) → Xm 1 1 u

cs

cu

by

u

A(m, x ) = g˜ (m, g˜ (m, x )). We claim that for each m ∈ B(m1 , ρ) ∩ M, A(m, xu ) has a unique fixed point xu = xu (m) ∈ X u (ρ−1 ), which is C 1 in m. Recall, from the proofs of Lemma 6.1 and Lemma 7.1, that |˜ g cs (m, 0)| < δ

(10.2)

|˜ g cu (m, 0)| < δ.

(10.3)

and

98

As in Section 8, we require that µρ < 1/2. By (10.2) and (10.3), if we choose δ < we have that max{|˜ g cs (m, 0)|, |˜ g cu (m, 0)|} + µ < ρ−1 .

 4

We first show that A is well-defined. Using Lemma 6.1, we obtain |˜ g cu (m, xu )| ≤ |˜ g cu (m, 0)| + ρµ|xu | ≤ |˜ g cu (m, 0)| + µ < ρ−1  which implies that A is well-defined. Furthermore, |A(m, xu )| < ρ−1 . u Hence for each m, A(m, ·) is a map from Xm (ρ−1 ) into itself. 1 Next we have |A(m, xu ) − A(m, x ¯u )| < (ρu)2 |xu − x ¯u |. u Thus, A(m, ·) is a contraction from Xm (ρ−1 ) into itself. From sections 8 and 9, g˜cs 1 and g˜cu are C 1 , hence A(m, xu ) is C 1 in (m, xu ). Hence by the uniform contraction mapping theorem, A(m, ·) has a unique fixed point xu = xu (m) which is C 1 in m. This completes the proof of the claim. Moreover,

|xu (m)| ≤

1 (|˜ g cs (m, 0)| + µρ|˜ g cu (m, 0)|). 1 − (ρu)2

Let xs (m) = g˜cu (m, xu (m)) and m ˜ = m + Πum xu (m) + Πsm xs (m).

(10.4)

Observe that n o n o u s ˜ cu (m, x ˜ cs (m, x {m} ˜ = m+x ˜u + h ˜u ) : x ˜u ∈ Xm () ∩ m + x ˜s + h ˜s ) : x ˜s ∈ Xm () provided that ∗ is sufficiently small. This implies that m ˜ is uniquely determined by ˜ m and independent of the choice of m1 . Hence, we have a bijection from M onto M m ˜ = K(m). 1 u s 1 1 Since Πα m is C in m and x (m) and x (m) are C , K is a C bijection. To show the convergence (10.1), from (10.4) we have

|K(m) − m| = |Πum xu (m)| + |Πsm xs (m)| ≤ (1 + η)(|xu (m)| + |xs (m)|) ≤ 2(1 + η).

99

Estimating Dxu (m) and Dxs (m), we find ||Dxu (m)|| ρµ ≤ 1 − ρµ and

(ρµ)2 . ||Dx (m)|| ≤ ρµ + 1 − ρµ s

Hence |DK(m) − I| ≤ ||DΠum || |xu (m)| + ||DΠsm || |xs (m)| + (1 + η)(||Dxu (m)|| + ||Dxs (m)||)   2(ρµ)2 . ≤ O() + (1 + η) ρµ + ( 1 − ρµ We can take  and µ as small as desired by choosing σ sufficiently small. This estab˜ is a C 1 manifold and that K is a diffeomorphism. lishes (10.1) and also implies that M This completes the proof. ˜ →M ˜ is a C 1 diffeomorphism. Proposition 10.2. T˜ : M ˜ is compact. Since T t is a diffeomorphism from Proof. Theorem 10.1 implies that M ˜ → M ˜ is a M onto M , again using Proposition 10.1, we have K ◦ T t ◦ K −1 : M diffeomorphism and ||K ◦ T t ◦ K −1 − T˜||1 → 0 as ||T˜ − T t ||1 → 0. Therefore, as long ˜ →M ˜ is a diffeomorphism . as ||T˜ − T t ||1 is small enough, T˜ : M ˜ cu () there is a From Proposition 6.12 it follows that for each m0 + xu0 + xs0 ∈ W ˜ cu () for k ≥ 1 such that for k = 1, 2, · · · , unique sequence mk + xuk + xsk ∈ W T˜(mk + xuk + xsk ) = mk−1 + xuk−1 + xsk−1 . ˜ cu () in the That is, T˜−k (m0 + xu0 + xs0 ) is well-defined for each m0 + xu0 + xs0 ∈ W u s tubular neighborhood Θ(X () ⊕ X ()). Theorem 10.3. ˜ cs (), (i) For each m + xu + xs ∈ W ˜ ) = 0, uniformly on W ˜ cs () lim d(T˜k (m + xu + xs ), M

k→∞

(10.5)

100

˜ cu (), (i) For each m + xu + xs ∈ W ˜ ) = 0, uniformly on W ˜ cu () lim d(T˜−k (m + xu + xs ), M

k→∞

(10.6)

˜ ) = inf where d(x, M ˜ ˜ |x − m|. m∈ ˜ M ˜ cs (). Let mk + xu + xs = Proof. We first consider (10.5). For each m + xu + xs ∈ W k k T˜k (m + xu + xs ). Observe that mk + xuk + xsk ∈ Ak where Ak is given in Proposition 6.10. (6.46) gives ˜ cu (mk , xu )| ≤ 2λk |xsk − h (10.7) k 1 We note that ˜ cu (mk , x ˜ cs (mk , x K(mk ) = mk + x ¯uk + x ¯sk = mk + x ¯uk + h ¯uk ) = mk + x ¯sk + h ¯sk ). ˜ cs (), using Lemma 6.1, we obtain Thus, from the fact mk + xuk + xsk ∈ W |xuk − x ¯uk | ≤ µρ|xsk − x ¯sk | ≤

1 s |x − x ¯sk |. 2 k

(10.8)

From (10.7), we find ˜ cu (mk , xu ) − h ˜ cu (mk , x |xsk − x ¯sk | − |h ¯uk )| ≤ 2λk1 k which with (10.8) yields |xsk − x ¯sk | ≤

8 k λ . 3 1

Therefore, |K(mk ) − (mk + xuk + xsk )| ≤ 4λk1 . This completes the proof of (10.5). Similarly, one may show that (10.6) holds. The proof is complete.

101

§11. Persistence of Normal Hyperbolicity. In Section 10, we showed that the intersection of the center-stable manifold and center-unstable manifold is a C 1 compact connected invariant manifold. The basic ˜ is to construct idea to obtain normal hyperbolicity for the perturbed manifold M stable and unstable bundles from the tangent bundles of the center-stable and centerunstable manifolds by finding projection operators. ˜ Theorem 11.1. There exists σ > 0 such that if T˜ satisfies kT˜ − T t k ≤ σ, them M is a normally hyperbolic invariant manifold. We first review some basic results from previous sections. We shall again denote the center-stable and center-unstable manifolds for T t and T˜, respectively, by ˜ cs () and W ˜ cu (). W cs (), W cu (), W In sections 8 and 9, we showed that these manifolds are C 1 and consequently their tangent spaces are given, respectively, by the graphs of `cs ∈ C 0 (W cs (), L(X s ⊕ X c , X u )),

(11.1)

`cu ∈ C 0 (W cu (), L(X u ⊕ X c , X s )),

(11.2)

˜ cs (), L(X s ⊕ X c , X u )), `˜cs ∈ C 0 (W

(11.3)

˜ cu (), L(X u ⊕ X c , X s )). `˜cu ∈ C 0 (W

(11.4)

˜ cs () at point x1 = m1 +xu +xs ∈ More precisely, for example, the tangent space of W 1 1 cs ˜ () is given by W ˜ cs () = {xs + xc + `˜cs (xs , xc ) : xs + xc ∈ X s ⊕ X c }. Tx1 W x1 m1 m1 With the norm as before we have that ||`cs ||, ||`cu ||, ||`˜cs ||, ||`˜cu || ≤ θ = µρ < 1/2. ˜ cu () ∩ T˜−1 (W ˜ cu ()) → W ˜ cu () is a homeomorphism. From Proposition 6.12, T˜ : W ˜ cu () is a C 1 manifold, one may easily show that T˜ is a C 1 Furthermore, since W diffeomorphism. We shall establish Theorem 11.1 by first constructing projection operators. Throughout this section, for each m ∈ M we set m ˜ = K(m), where K is the diffeomorphism obtained in section 10. From (10.4), we notice that m ˜ = m + xu + xs , where u u xu + xs ∈ Xm () ⊕ Xm (). Our main goal is to find the invariant stable and unstable subspaces. We first try to ˜ cs ˜ cu use the tangent spaces Tm ˜ W () and Tm ˜ W () as a coordinate system to represent

102 u s c u s c each point in X = Xm ⊕ Xm ⊕ Xm . For each xu + xs + xc ∈ Xm ⊕ Xm ⊕ Xm , we consider the equation

xu + xs + xc = x ¯s + x ¯c + `˜cs xs , x ¯c ) + x ¯u + `˜cu xu , 0) m ˜ (¯ m ˜ (¯

(11.5)

α where x ¯α ∈ Xm , α = u, s, c. We denote the right hand side of (11.5) by u ψ˜m (¯ xu + x ¯s + x ¯c ). u Thus we may define a map ψ˜u from M to L(X u ⊕ X s ⊕ X c ) by ψ˜u (m) = ψ˜m , where u s c u s c L(X ⊕ X ⊕ X ) is the vector bundle over M with fiber L(Xm ⊕ Xm ⊕ Xm ). We ˜ cs should point out that x ¯u + `˜cu xu , 0) is transversal to Tm ˜ W (). m (¯ The next result shows that (11.5) has a unique solution x ¯u + x ¯s + x ¯c . u u s c Lemma 11.2. ψ˜m is a bounded linear isomorphism from Xm ⊕ Xm ⊕ Xm onto itself u 0 u s c ˜ and ψ ∈ C (M, L(X ⊕ X ⊕ X ))

˜cu are bounded linear operators and ||`˜cα || ≤ ρµ < Proof. From the fact that `˜cs m ˜ and `m ˜ m ˜ u 1/2, α = u, s, we obtain that ψ˜m is a bounded linear operator and satisfies 3 2 u u s c u ˜ |(I − ψm )(¯ x +x ¯ +x ¯ )| ≤ µρ(|¯ x | + |¯ xs | + |¯ xc |) 1 u ≤ (|¯ x | + |¯ xs | + |¯ xc |) 2 u ||ψ˜m || ≤ (1 + µρ) ≤

u is a bounded linear isomorphism. From (11.3) and (11.4) it which implies that ψ˜m u 0 ˜ follows that ψ ∈ C (M, L(X u ⊕ X s ⊕ X c )). The proof is complete. s u s c u s c Similarly, we define a map ψ˜m from Xm ⊕ Xm ⊕ Xm to Xm ⊕ Xm ⊕ Xm by s ψ˜m (¯ xu + x ¯s + x ¯c ) = x ¯u + x ¯c + `˜cu xu , x ¯c ) + x ¯s + `˜cs xs , 0) m ˜ (¯ m ˜ (¯ s . and define a map ψ˜s from M to L(X u ⊕ X s ⊕ X c ) by ψ˜s (m) = ψ˜m Then, we may show s u s c Lemma 11.3. ψ˜m is a bounded linear isomorphism from Xm ⊕ Xm ⊕ Xm onto itself s 0 u s c ˜ and ψ ∈ C (M, L(X ⊕ X ⊕ X )).

Next we define the followings sets of projection operators ˜ cs Λu ={P˜ u ∈ C 0 (M, L(X)) for m ∈ M, ker(P˜ u (m)) = Tm ˜ W (), ˜ cu (P˜ u (m))2 = P˜ u (m) and R(P˜ u (m)) ⊂ Tm ˜ W ()}

103

and ˜ cu Λs ={P˜ s ∈ C 0 (M, L(X)) for m ∈ M, ker(P˜ s (m)) = Tm ˜ W (), ˜ cs (P˜ s (m))2 = P˜ s (m) and R(P˜ s (m)) ⊂ Tm ˜ W ()}. It is easy to see that Λu and Λs are closed subsets of C 0 (M, L(X)) under the usual u norm. Let L(X, X u ⊕ X s ⊕ X c ) be the vector bundle over M with fiber L(X, Xm ⊕ s c Xm ⊕ Xm ). We have Lemma 11.4. Λu and Λs are not empty. Proof. We first define a section ω of the bundle L(X, X u ⊕ X s ⊕ X c ) by ω(m)x = Πum x + Πsm x + Πcm x. The proof of lemma 4.2 implies that ω is C 1 . We also define a section N u of the bundle L(X u ⊕ X s ⊕ X c , X) by N u (m)(xu + xs + xc ) = x ¯u + `˜cu xu , 0) m ˜ (¯ u −1 u α , α = u, s, c, and x ¯u is determined by x ¯u + x ¯s + x ¯c = (ψ˜m ) (x +xs + where xα ∈ Xm xc ). From Lemma 11.2 it follows that N u is well-defined and N u ∈ C 0 (M, L(X u ⊕ u ˜ cu ˜u X s ⊕ X c , X)). Observe that N u (m)(xu + xs + xc ) ∈ Tm ˜ W (). Let P = N ω. It is easy to verify P˜ u ∈ Λu . Hence Λu 6= φ. Similarly, we define a section N s of L(X u ⊕ X s ⊕ X c , X) by

N s (m)(xu + xs + xc ) = x ¯s + `˜cs xs , 0) m ˜ (¯ s −1 u where x ¯s is determined from x ¯u + x ¯s + x ¯c = (ψ˜m ) (x + xs + xc ). Then N s ω ∈ Λs . This completes the proof. ˜ u ∈ Λu satisfying the invariance property Next we want to find Π

˜ u (m1 )DT˜(m ˜ u (m0 ) = DT˜(m ˜ u (m0 ) Π ˜ 0 )Π ˜ 0 )Π

(11.6)

where m1 = K −1 (m ˜ 1 ) and m ˜ 1 = T˜(m ˜ 0 ), which leads us to define the following transformation. u s c For each xu + xs + xc ∈ Xm ⊕ Xm ⊕ Xm , by Lemma 11.2, there exists a unique 1 1 1 u s c u s c x ¯ +x ¯ +x ¯ ∈ Xm1 ⊕ Xm1 ⊕ Xm1 such that xu + xs + xc = x ¯s + x ¯c + `˜cs xs , x ¯c ) + x ¯u + `˜cu xu , 0). m ˜ 1 (¯ m ˜ 1 (¯

(11.7)

Note that xc = x ¯c . For P˜ u ∈ Λu and m1 , we define a map P u (m1 ) from X to X by P u (m1 ) = DT˜(m ˜ 0 )P˜ u (m0 )(DT˜(m ˜ 0 )|Tm˜

0

−1 u N (m1 )ω(m1 ), ˜ cu () ) W

where m0 = K −1 ◦ T˜−1 ◦ K(m1 ). Then P u (m1 ) satisfies

104

Lemma 11.5. (i) P u (m1 ) is well-defined; ˜ cs (ii) ker(P u (m1 )) = Tm ˜ 1 W (); ˜ cu (iii) R(P u (m1 )) ⊂ Tm ˜ 1 W (); u 2 u (iv) (P (m1 )) = P (m1 ). ˜ cu () ∩ T˜−1 (W ˜ cu ()) → W ˜ cu () is a C 1 difProof. As mentioned before, T˜ : W ˜ cu ˜ cu feomorphism, DT˜(m ˜ 0 ) is an isomorphism from Tm ˜ 0 W () to Tm ˜ 1 W (), where m ˜ 1 = T˜(m ˜ 0 ). Hence P u (m1 ) is well-defined. Clearly, P u (m1 ) is a bounded linear ˜ cs operator from X to X. To show (ii), we first prove that ker(P u (m1 )) ⊂ Tm ˜ 1 W (). u u s c Let P (m1 )(x + x + x ) = 0. Namely,  u ˜ ˜ D T (m ˜ 0 )P (m0 ) DT˜(m ˜ 0 )|Tm˜

−1 ˜

0W

cu ()

(¯ xu + `˜cu xu , 0)) = 0. m ˜ 1 (¯

˜ cu ˜ ˜ 0 )| Because P˜ u ∈ Λu , R(P˜ u (m0 )) ⊂ Tm ˜ cu () is an ˜ 0 W (). Thus, because D T (m Tm ˜ 0W isomorphism, we have  P˜ u (m0 ) DT˜(m ˜ 0 )|Tm˜

−1

˜ cu () W 0

(¯ xu + `˜cu xu , 0)) = 0. m ˜ 1 (¯

˜ cs Note that ker(P˜ u (m0 )) = Tm ˜ 0 W (). Hence, 

DT˜(m ˜ 0 )|Tm˜ 0 W cu ()

−1

˜ cs ˜ cu ˜ (¯ xu + `˜cu xu , 0)) ∈ Tm ˜ 0 W () ∩ Tm ˜ 0 W () = Tm ˜ 0M, m ˜ 1 (¯

which yields ˜ (¯ xu + `˜cu xu , 0)) ∈ Tm ˜ 1M. m ˜ 1 (¯ ˜cu k, k`˜cs k < µρ, we have ˜ Observe that for any xu + xs + xc ∈ Tm ˜ 1 M , since k`m ˜1 m ˜1 |xu | + |xs | ≤

2ρµ |xc |. 1 − ρµ

Therefore, x ¯u = 0. From (11.7) ˜ cs xu + xs + xc = x ¯s + x ¯c + `˜cs xs , x ¯c ) ∈ Tm ˜ 1 W (). m ˜ 1 (¯ ˜ cs On the other hand, from the definition of P u (m1 ) it is easy to see that Tm ˜ 1 W () ⊂ ker(P u (m1 )). Therefore (ii) holds. It is quite straightforward to check (iii). To

105

prove (iv), let y = P u (m1 )x and z = (I − N u (m1 )ω(m1 ))y so that z ∈ R(I − u ˜ cu ˜ cu N u (m1 )ω(m1 )) ⊂ Tm ˜ 1 W (). But by (iii), y ∈ Tm ˜ 1 W () and N (m1 )ω(m1 ))y ∈ ˜ cu ˜ Tm ˜ 1 W (). So z ∈ Tm ˜ 1 M , which implies P˜ u (m0 )(DT˜(m ˜ 0 )|Tm˜

˜

0W

cu ()

)−1 z = 0.

Therefore, (P u (m1 ))2 x = DT˜(m ˜ 0 )P˜ u (m0 )(DT˜(m ˜ 0 )|

−1 (y ˜ cu () ) Tm ˜ 0W

= DT˜(m ˜ 0 )P˜ u (m0 )(DT˜(m ˜ 0 )|Tm˜

0

˜ cu () ) W

−1

− z)

DT˜(m ˜ 0 )P˜ u (m0 )(DT˜(m ˜ 0 )|Tm˜

0

−1 ˜ cu () ) W

N u (m1 )ω(m1 )x = P u (m1 )x The proof is complete. ˜ , m1 can be taken to be any point of Note that since T˜ is a diffeomorphism on M M . Thus, P u is a map from M to L(X). From continuity of DT˜, P˜ u , N u , and ω it follows that P u ∈ C 0 (M, L(X)). Summarizing above discussion gives next result. Proposition 11.6. P u ∈ Λu . Thus, we may define a map F u from Λu to Λu by F u (P˜ u ) = P u . In order to show F u has a unique fixed point in Λu , we shall introduce an equivalent metric in Λu under which F u is a contraction. c s u , by Lemma 11.2, there exists a ⊕ Xm ⊕ Xm Let m ∈ M . For xu + xs + xc ∈ Xm c s u s c u unique x ¯ +x ¯ +x ¯ ∈ Xm ⊕ Xm ⊕ Xm such that cs s xu + xs + xc = x ¯s + x ¯c + `˜m x ,x ¯c ) + x ¯u + `˜cu xu , 0). ˜ (¯ m ˜ (¯

˜ cu Let P˜1u , P˜2u ∈ Λu , since R(P˜iu (m)) ⊂ Tm ˆu + ˜ W (), i = 1, 2, there exists a unique x u c x ˆc ∈ Xm ⊕ Xm such that (P˜1u (m) − P˜2u (m))(xu + xs + xc ) = (P˜1u (m) − P˜2u (m))(¯ xu + `˜cu xu , 0)) = x ˆu + x ˆc + `˜cu xu , x ˆc ). m ˜ (¯ m ˜ (ˆ (11.8) Define δu (P˜1u (m), P˜2u (m)) ≡ sup (|ˆ xu | + |ˆ xc |). |¯ xu |=1

Then we have

(11.9)

106

Lemma 11.7. There exists a constant C > 0 such that 1 ˜u ||P1 (m) − P˜2u (m)|| ≤ δu (P˜1u (m), P˜2u (m)) ≤ C||P˜1u (m) − P˜2u (m)||, for m ∈ M. C Proof. From(11.8) and the fact that ||`˜cu m ˜ || ≤ µρ < 1/2, we obtain (1 − µρ)(|ˆ xu | + |ˆ xc |) ≤ ||P˜1u (m) − P˜2u (m)||(1 + µρ)|¯ xu | which yields δu (P˜1u (m), P˜2u (m)) ≤ 3||P˜1u (m) − P˜2u (m)||. On the other hand, for any x ∈ X, writing it as (11.7) and using the fact that u c P˜1u , P˜2u ∈ Λu , there exist x ˆu + x ˆc ∈ Xm ⊕ Xm such that (P˜1u (m) − P˜2u (m))x = (P˜1u (m) − P˜2u (m))(¯ xu + `cu xu , 0)) m ˜ (¯ =x ˆu + x ˆc + `ˆcu (ˆ xu , x ˆc ). m ˜

Thus, by definition (11.9), |(P˜1u (m) − P˜2u (m))x| ≤ (1 + µρ)(|ˆ xu | + |ˆ xc |) 3 u x |δu (P˜1u (m), P˜2u (m)). ≤ |¯ 2 Observe that x ¯u = (Πum N u ω)x. Hence |¯ xu | ≤ kΠum kkN u (m)kkω(m)k|x| ≤ C|x| for some positive constant C independent of m ∈ M . Choosing a possiblely larger constant C completes the proof. Set du (P˜1u , P˜2u ) ≡ sup δu (P˜1u (m), P˜2u (m)).

(11.10)

m∈M

Then Lemma 11.7 implies that (11.10) defines an equivalent metric in Λu . Proposition 11.8. There exists σ > 0 such that if ||T˜ − T t ||1 ≤ σ, then F u : Λu → Λu is a contraction under the metric (11.10). u Proof. For x ¯u ∈ Xm , let 1

 xu + xs + xc = DT˜(m ˜ 0 )|Tm˜

0

−1   u cu u ˜ x ¯ + ` (¯ x , 0) . ˜ cu () m ˜1 W

(11.11)

107 α Here m ˜ 1 = T˜(m ˜ 0 ) and xα ∈ Xm for α = u, s, c. Thus, 0

x ¯u + `˜cu xu , 0) = DT˜(m ˜ 0 )(xu + xs + xc ). m ˜ 1 (¯

(11.12)

We also observe, from the invariance, that u c xs = `˜cu m ˜ 0 (x , x ).

(11.13)

u s c As in (11.7), by Lemma 11.2, there exists x ˆu + x ˆs + x ˆc ∈ Xm ⊕ Xm ⊕ Xm such that 0 0 0

xu + xs + xc = x ˆs + x ˆc + `˜cs xs , x ˆc ) + x ˆu + `˜cu xu , 0) m ˜ 0 (ˆ m ˜ 0 (ˆ Note that xc = x ˆc . We claim   u |¯ xu | ≥ inf{|DT t (m0 )xu | : |xu | = 1, xu ∈ Xm } − O() − Cσ |ˆ xu | 0

(11.14)

(11.15)

˜ To show this claim, we first observe that x ˆs + xc + `˜cs xs , xc ) ∈ Tm ˜ 0 M . In fact, m ˜ 0 (ˆ ˜ cu (11.13) implies that x ˆs + xc + `˜cs xs , xc ) + x ˆu + `˜cu xu , 0) ∈ Tm ˜ 0 W (), thus m0 (ˆ m ˜ 0 (ˆ ˜ cu x ˆs + xc + `˜cs xs , xc ) ∈ Tm ˜ 0 W (). m ˜ 0 (ˆ ˜ Hence, x ˆs + xc + `˜cs xs , xc ) ∈ Tm ˜ 0M. m ˜ 0 (ˆ A simple calculation gives |ˆ xs | + |`˜cs xs , xc )| ≤ m ˜ 0 (ˆ

2µρ |xc | ≤ 2|xc |. 1 − µρ

(11.16)

From (10.4) we have for i = 0, 1, |mi − m ˜ i | ≤ 2.

(11.17)

Thus, One may obtain, for some positive constant C, |m1 − T t (m0 )| ≤ C( + σ).

(11.18)

Applying the projection Πcm1 to (11.12), 0 = Πcm1 DT˜(m ˜ 0 )(xu + xs + xc ). Using (11.14), (11.16), (11.17), (11.18), and an estimate similar to (8.8), we have |DT t (m0 )xc | = |DT t (m0 )xc − Πcm1 DT˜(m ˜ 0 )(xu + xs + xc )| ≤ |DT t (m0 )xc − Πc DT˜(m ˜ 0 )xc | + |Πc DT˜(m ˜ 0 )(xu + xs )| m1

c

m1

u

≤ (O() + Cσ)|x | + (O() + Cσ)|ˆ x |

108

which yields, by using (H3), |xc | ≤ (O() + Cσ)|ˆ xu |

(11.19)

Similarly, applying the projection Πum1 to (11.12), we obtain o n  u |ˆ xu | − O() + Cσ (|xc | + |ˆ xu |) |¯ xu | ≥ inf |DT t (m0 )xu | : |xu | = 1, xu ∈ Xm 0 which, with (11.19), gives (11.15). This completes the proof of the claim. For P˜1u , P˜2u ∈ Λu and m0 ∈ M , let Piu = F u (P˜iu )(m1 ) where m1 = K −1 ◦ T˜ ◦ K(m0 ). Observe that for i = 1, 2, P˜iu (m0 )(I − P˜iu (m0 )) = 0, and Piu (m1 )(I − Piu (m1 )) = 0, which implies ˜ cs R(I − P˜iu (m0 )) ⊂ Tm ˜ 0 W (), ˜ cs R(I − Piu (m1 )) ⊂ Tm ˜ W (). 1

Thus, ˜ R(P˜1u (m0 ) − P˜2u (m0 )) ⊂ Tm ˜ 0M,

(11.20)

˜ R(P1u (m1 ) − P2u (m1 )) ⊂ Tm ˜ 1M.

(11.21)

and u , let For x ¯u ∈ Xm 1

xu1 + xs1 + xc1 = (P1u (m1 ) − P2u (m1 ))(¯ xu + `˜cu xu , 0))). m ˜ 1 (¯ From the definition of F u , using (11.11) and (11.14), we find xu1 + xs1 + xc1  = DT˜(m0 )(P˜1u (m0 ) − P˜2u (m0 )) DT˜(m ˜ 0 )|Tm˜

˜ cu () W 0

−1

(¯ xu + `˜cu xu , 0)) m ˜ 1 (¯

= DT˜(m0 )(P˜1u (m0 ) − P˜2u (m0 ))(ˆ xu + `˜cu xu , 0)). m ˜ 0 (ˆ

(11.22)

Set x ˜u + x ˜s + x ˜c = (P˜1u (m0 ) − P˜2u (m0 ))(ˆ xu + `˜cu xu , 0)). m ˜ 0 (ˆ

(11.23)

109

˜ From (11.20), we have x ˜u + x ˜s + x ˜c ∈ Tm ˜ 0 M , hence, 2ρµ |˜ xu | + |˜ xs | ≤ |˜ xc | (11.24) 1 − ρµ and 1 |˜ xc |. (11.25) |˜ xu | + |˜ xc | ≤ 1 − ρµ Applying the projection Πcm1 to (11.22), using (11.17), (11.18), (11.24), and an estimate similar to (8.8), we obtain |xc1 | = |Πcm DT˜(m ˜ 0 )(˜ xu + x ˜s + x ˜c )| 1

c k |˜ ≤ kDT t (m0 )|Xm xc | + |(DT t (m0 ) − Πcm1 DT˜(m ˜ 0 ))˜ xc | + |Πcm1 DT˜(m ˜ 0 )(˜ xu + x ˜s )| 0   t c k + O() + Cσ |˜ ≤ kDT (m0 )|Xm xc | 0   ≤ kDT t (m0 )|X c k + O() + Cσ du (P˜1u , P˜2u )|ˆ xu |, m0

where (11.9) and (11.23) are used in the last estimate. Thus, by (11.15), we have   ˜ u , P˜ u ) c k + O() + Cσ) du (P kDT t (m0 )|Xm 1 2 0 c  |¯ xu |. |x1 | ≤  t u u u u inf{|DT (m0 )x | : |x | = 1, x ∈ Xm0 } − O() − Cσ ˜ Note that xu1 + xs1 + xc1 ∈ Tm ˜ 1 M . Thus, as in (11.25), we have 1 |xu1 | + |xc1 | ≤ |xc |, 1 − ρµ 1 hence,   t c kDT (m )| k + O() + Cσ) du (P˜1u , P˜2u ) u c 0 Xm0 1 |x1 + |x1 |  . ≤ |¯ xu | 1 − ρµ inf{|DT t (m )xu | : |xu | = 1, xu ∈ X u } − O() − Cσ 0

m0

Observe that (H3) implies that c k kDT t (m0 )|Xm

0 < λ. u } inf{|DT t (m0 )xu | : |xu | = 1, xu ∈ Xm 0 By choosing µ, ∗ and σ sufficiently small, we have for T˜ satisfying ||T˜ − T t ||1 ≤ σ δu (P u (m1 ), P u (m1 )) ≤ λ1 du (P˜ u , P˜ u ).

1

2

1

2

Hence, du (F u (P˜1u ), F u (P˜2u )) ≤ λ1 du (P˜1u , P˜2u )

(11.26)

and the proof is complete. Note that Λu is complete under the metric (11.10) from Lemma 11.7. Since F u is a contraction, by the contraction mapping theorem, we obtain

110

Proposition 11.9. There exists σ > 0 such that if ||T˜ − T t ||1 ≤ σ, there exists ˜ u ∈ Λu such that Π ˜ u (K −1 (m ˜ u (K −1 (m ˜ u (K −1 (m Π ˜ 1 ))DT˜(m ˜ 0 )Π ˜ 0 )) = DT˜(m ˜ 0 )Π ˜ 0 )). ˜ u ||0 → 0 as ||T˜ − T t ||1 → 0. Proposition 11.10. ||Πu − Π Proof. Note that N u ω ∈ Λu from the proof of Lemma 11.4. We first estimate u s c d( N u ω, F u (N u ω)). For each x ˜u + x ˜s + x ˜c ∈ Xm ⊕ Xm ⊕ Xm , by Lemma 11.2, 1 1 1 u s c u s c there exist a unique x ¯ +x ¯ +x ¯ ∈ Xm1 ⊕ Xm1 ⊕ Xm1 such that x ˜u + x ˜s + x ˜c = x ¯s + x ¯c + `˜cs xs , x ¯c ) + x ¯u + `˜cu xu , 0). m ˜ 1 (¯ m ˜ 1 (¯ Note x ˜c = x ¯c . As in (11.11), let s

c

−1



x + x + x = DT˜(m ˜ 0 ) Tm˜ u

˜ cu () W 0

(¯ xu + `˜cu xu , 0)), m ˜ 1 (¯

and write xu + xs + xc as in (11.14). Combining (11.16) with (11.15) and (11.19) gives |ˆ xs | + |ˆ xc | + |`˜cs xs , x ˆc )| ≤ (O() + Cσ)|¯ xu |. m ˜ 0 (ˆ

(11.27)

Thus,    u u u | N (m1 )ω(m1 ) − F (N ω) (m1 ) (˜ xu + x ˜s + x ˜c )|     u = | N u (m1 )ω(m1 ) − F u (N u ω) (m1 ) x ¯u + `˜cu (¯ x , 0) | m ˜1 = |DT˜(m ˜ 0 )(ˆ xs + x ˆc + `˜cs xs , x ˆc ))| m ˜ 0 (ˆ ≤ (O() + Cσ)|¯ xu |, which implies du (N u ω, F u (N u ω))) ≤ O() + Cσ. ˜ u is the fixed point of the contraction mapping F u , using (11.26) gives Since Π ˜ u) ≤ ( du (N ω, Π u

∞ X

λk1 )(O() + Cσ) = O() + Cσ

(11.28)

k=0

with a different constant C. Next, we estimate ||N u ω − Πu ||. From (11.5), we obtain ||(ψ˜u (m1 ))−1 − I|| ≤

ρµ 1 − ρµ

(11.29)

111

and x ¯u + x ¯s + x ¯c = (ψ˜u (m1 ))−1 (˜ xu + x ˜s + x ˜c ), which implies |¯ xu − x ˜u | ≤ and |¯ xu | ≤

µρ (|˜ xu | + |˜ xs | + |˜ xc |) 1 − µρ

1 (|˜ xu | + |˜ xs | + |˜ xc |). 1 − µρ

Thus, |(N u (m1 )ω(m1 ) − Πum1 )(˜ xu + x ˜s + x ˜c )| ≤ |¯ xu + `cu xu , 0) − x ˜u | m ˜ 1 (¯ ≤ |¯ xu − x ˜u | + µρ|¯ xu | 2µρ ≤ (|˜ xu | + |˜ xs | + |˜ xc |) 1 − µρ 2µρ |˜ xu + x ˜s + x ˜c |, ≤C 1 − ρµ which implies ||N u ω − Πu || ≤ C

2ρµ . 1 − ρµ

Hence, using (11.28) and Lemma 11.7, we find ˜ u − Πu || ≤ ||Π ˜ u − N u ω|| + ||N u ω − Πu || ||Π ˜ u , N u ω) + ||N u ω − Πu || ≤ Cdu (Π ≤ O() + Cσ +

2µρC . 1 − µρ

2µρC For any E > 0, we first choose µ small enough that 1−µρ < 12 E, then we may choose ∗ and σ sufficiently small such that that if ||T˜ − T t ||1 < σ, then

˜ u − Πu || ≤ E ||Π which completes the proof. ˜ s ∈ Λs satisfying the Next we shall show that there is a unique projection map Π invariance property ˜ s (m1 )DT˜(m ˜ s (m0 ). Π ˜ 0 ) = DT˜(m ˜ 0 )Π

112

˜ s by directly applying the contraction mapping approach to the We cannot find Π above equation since DT˜(m ˜ 0 ) Tm˜ W ˜ cs () may not be invertible. To overcome this 0 difficulty, we consider the following equivalent equation instead ˜ s (m0 ) − I) = (Π ˜ s (m1 ) − I)DT˜(m DT˜(m ˜ 0 )(Π ˜ 0 ). ˜ s ∈ Λs Note that for Π ˜ s (m0 ) − I) ⊂ ker(Π ˜ s (m0 )) = Tm ˜ cu R(Π ˜ 0 W ()

(11.30)

˜ s (m1 ) − I) ⊂ ker(Π ˜ s (m1 )) = Tm ˜ cu R(Π ˜ 1 W ().

(11.31)

and Thus, −1

 ˜ s (m0 ) = I + DT˜(m Π ˜ 0 ) T

˜

m ˜ 0W

cu ()

˜ s (m1 ) − I)DT˜(m (Π ˜ 0)

which leads us to define the following transformation: For P˜ s ∈ Λs , let  P s (m0 ) = I + DT˜(m ˜ 0 ) T

−1 ˜

m ˜ 0W

cu ()

(P˜ s (m1 ) − I)DT˜(m ˜ 0 ).

Lemma 11.11. (i) (ii) (iii) (iv)

P s (m0 ) is well-defined; ˜ cu ker(P s (m0 )) = Tm ˜ 0 W (); ˜ cs R(P s (m0 )) ⊂ Tm ˜ 0 W (); s 2 s (P (m0 )) = P (m0 ).

s ˜ cu ˜ cu Proof. . Since DT˜(m ˜ 0 ) : Tm ˜ 0 W () → Tm ˜ 1 W () is an isomorphism, P (m0 ) is well-defined. Clearly P s (m0 ) is a bounded linear operator. The proofs of (ii) and (iv) u s ⊕ Xm ⊕ are quite straightforward. Let us look at (iii). For each xu + xs + xc ∈ Xm 0 0 u s c u s c c ¯ +x ¯ +x ¯ ∈ Xm0 ⊕ Xm0 ⊕ Xm0 such Xm0 , by Lemma 11.3, there exits a unique x that xu + xs + xc = x ¯u + x ¯c + `˜cu xu , x ¯c ) + x ¯s + `˜cs xs , 0). m ˜ 0 (¯ m0 (¯

Note that xc = x ¯c . By (11.31) and (ii) in this lemma, we have P s (m0 )(xu + xs + xc ) = P s (m0 )(¯ xs + `˜cs xs , 0)) m ˜ 0 (¯  s =x ¯s + `˜cs (¯ x , 0) − DT˜(m ˜ 0 ) T m ˜0

−1 ˜

m ˜ 0W

cu ()

(P˜ s (m1 ) − I)DT˜(m ˜ 0 )(¯ xs + `˜cs xs , 0)). m ˜ 0 (¯

113

˜ Observe that (P˜ s (m1 ) − I)DT˜(m ˜ 0 )(¯ xs + `˜cs xs , 0)) ∈ Tm ˜ 1 M . Hence, m0 (¯ ˜ P s (m0 )(¯ xs + `˜cs xs , 0)) − (¯ xs + `˜cs xs , 0)) ∈ Tm ˜ 0M m ˜ 0 (¯ m ˜ 0 (¯ which implies ˜ cs P s (m0 )(¯ xs + `˜cs xs , 0)) ∈ Tm ˜ 0 W (). m ˜ 0 (¯ This completes the proof. Lemma 11.11 defines a map P s from M to L(X). The continuity of DT˜ and P˜ s implies that P s ∈ C 0 (M, L(X)). So we have proved Lemma 11.12. P s ∈ Λs Define F s : Λs → Λs by F s (P˜ s ) = P s . As in Proposition 11.8, we shall show that F s is a contraction under an equivalent metric. Let m ∈ M . For xu + xs + xc ∈ c u s s c u ⊕ Xm ⊕ Xm ⊕ Xm , by Lemma 11.3, there exists a unique x ¯u + x ¯s + x ¯c ∈ Xm Xm ⊕ Xm such that xu + xs + xc = x ¯u + x ¯c + `˜cu xu , x ¯c ) + x ¯s + `˜cs xs , 0). m ˜ (¯ m ˜ (¯ ˜ cs Note xc = x ¯c . For P˜1s and P˜2s ∈ Λs , since R(P˜is (m)) ⊂ Tm ˜ W (), there exists c s such that x ˆs + x ˆc ∈ Xm ⊕ Xm (P˜1s (m) − P˜2s (m))(¯ xs + `˜cs xs , 0)) = x ˆs + x ˆc + `˜cs xs , x ˆc ). m ˜ (¯ m ˜ (ˆ Define δs (P˜1s (m), P˜2s (m)) = sup (|ˆ xs | + |ˆ xc |).

(11.32)

|¯ xs |=1

The next result is the analog to Lemma 11.7. Lemma 11.13. There exists a positive constant C such that 1 ˜s ||P1 (m) − P˜2s (m)|| ≤ δs (P˜1s (m), P˜2s (m)) ≤ C||P˜1s (m) − P˜2s (m)||. C One may prove this lemma in the same fashion as Lemma 11.7. Set ds (P˜1s , P˜2s ) = sup δs (P˜1s (m), P˜2s (m)). m∈M

Then (11.33) defines an equivalent metric on Λs .

(11.33)

114

Proposition 11.14. There exists σ > 0 such that if ||T˜ − T t ||1 ≤ σ, then F s : Λs → Λs is a contraction under the metric (11.33). s Proof. For x ¯s ∈ Xm let 0

xu + xs + xc = DT˜(m0 )(¯ xs + `˜cs xs , 0)). m ˜ 0 (¯

(11.34)

˜ cs () implies that xu = `˜cs (xs , xc ). By Lemma 11.3, there exists The invariance of W m ˜1 u s c a unique x ˆu + x ˆs + x ˆc ∈ Xm ⊕ X ⊕ X such that m1 m1 1 xu + xs + xc = x ˆu + x ˆc + `˜cu xu , x ˆc ) + x ˆs + `˜cs xs , 0). m ˜ 1 (ˆ m ˜ 1 (ˆ Note that xc = x ˆc . We claim that   s t xs |. |ˆ x | ≤ ||DT (m0 ) X s || + O() + Cσ |¯

(11.35)

(11.36)

m0

To prove this claim, we first observe that ˜ x ˆu + x ˆc + `˜cu xu , x ˆc ) ∈ Tm ˜ 1M. m ˜ 1 (ˆ Thus, |ˆ xu | + |`˜cu xu , x ˆc )| ≤ m ˜ 1 (ˆ

2ρµ |ˆ xc |. 1 − ρµ

(11.37)

Applying the projection Πcm1 in (11.34), we find |xc | = |Πcm1 DT˜(m ˜ 0 )(¯ xs + `˜cs xs , 0))| m ˜ 0 (¯ ≤ |Πcm1 (DT˜(m ˜ 0 ) − DT t (m0 ))(¯ xs + `˜cs xs , 0))| m ˜ 0 (¯ + ||Πc − Πc t |||DT t (m0 )(¯ xs + `˜cs (¯ xs , 0))| m1

T (m0 ) s

m ˜0

≤ (O() + Cσ)|¯ x |,

where (11.17) and (11.18) are used. Similarly, applying the projection Πsm1 to (11.34) and using (11.35) yields x ˆs + `˜cu xu , x ˆc ) = Πsm1 DT˜(m ˜ 0 )(¯ xs + `˜cs xs , 0)). m ˜ 1 (ˆ m ˜ 0 (¯ Hence, by (11.37) |ˆ xs | ≤ |`˜cu xs , x ˆc )| + (O() + Cσ)|¯ xs | + |DT t (m0 )¯ xs | m ˜ 1 (ˆ 2ρµ |xc | + (||DT t (m0 ) X s || + O() + Cσ)|¯ xs |, ≤ m1 1 − ρµ

(11.38)

115

which with (11.38) yields (11.36). This completes the proof of the claim. Let P˜1s , P˜2s ∈ Λs . For i = 1, 2, let Pis (m0 ) = (F s (P˜is ))(m0 ). Note that P˜1s (m1 )(I − P˜is (m1 )) = 0 and Pis (m0 )(I − Pis (m0 )) = 0. Thus, ˜ R(P˜1s (m1 ) − P˜2s (m1 )) ⊂ Tm ˜ 1M

(11.39)

˜ R(P1s (m0 ) − P2s (m0 )) ⊂ Tm ˜ 0M.

(11.40)

and s For x ¯s ∈ Xm set 0

xu0 + xs0 + xc0 = (P1s (m0 ) − P2s (m0 ))(¯ xs + `˜cs xs , 0)). m ˜ 0 (¯ By the definition of F s , we have xu0 + xs0 + xs0 = (P1s (m0 ) − P2s (m0 ))(¯ xs + `˜cs xs , 0)) m ˜ 0 (¯ −1  ˜ (P˜1s (m1 ) − P˜2s (m1 ))DT˜(m ˜ 0 )(¯ xs + `˜cs xs , 0)) = DT (m ˜ 0 ) Tm˜ W ˜ cu () m ˜ 0 (¯ 0

˜ Then, by the invariance, (11.40) yields that xu0 + xs0 + xc0 ∈ Tm ˜ 0M. Let xu1 + xs1 + xc1 = DT˜(m0 )(xu0 + xs0 + xc0 ).

(11.41)

˜ Then we have that xu1 + xs1 + xc1 ∈ Tm ˜ 1 M . Thus, 2ρµ |xc |, 1 − ρµ 0 1 |xs0 | + |xc0 | ≤ |xc0 |, 1 − ρµ |xu0 | + |xs0 | ≤

and |xu1 | + |xs1 | ≤

2ρµ |xc |. 1 − ρµ 1

(11.42) (11.43)

(11.44)

Applying the projection Πcm1 to (11.41) and using (11.17), (11.18) and (11.42), we obtain |xc1 | = |Πcm1 DT˜(m ˜ 0 )(xu0 + xs0 + xc0 )|   c ≥ inf{|DT t (m0 )xc | : |xc | = 1, xc ∈ Xm } − O() − Cσ |xc0 | 0

(11.45)

116

Hence, from (11.43) |xs1 | + |xc1 |  c } ≥ inf{|DT t (m0 )xc | : |xc | = 1, xc ∈ Xm 0  − O() − Cσ (1 − ρµ)(|xs0 | + |xc0 |).

(11.46)

Observe from (11.34) xu1 +xs1 +xc1 = (P˜1s (m1 )− P˜2s (m1 ))(xu +xs +xc ) = (P˜1s (m1 )− P˜2s (m1 ))(ˆ xs + `˜cs xs , 0)), m ˜ 1 (ˆ which yields |xs1 | + |xc1 | ≤ ds (P˜1s , P˜2s )|ˆ xs |. Thus, using (11.36) and (11.46), we have 

||DT (m0 ) X s || + O() + Cσ t



m0 |xs0 | + |xc0 |  ≤ ds (P˜1s , P˜2s ), s |¯ x | t c c c c inf{|DT (m0 )x | : |x | = 1, x ∈ Xm0 } − O() − Cσ (1 − ρµ)

which implies

ds (P1s , P2s ) ≤

  t ||DT (m0 ) X s || + O() + Cσ m0

inf{|DT t (m0 )xc |

:

|xc |

=

1, xc



c } Xm 0



− O() − Cσ (1 − ρµ)

ds (P˜1s , P˜2s )

Note that (H3) implies ||DT t (m0 ) X s || m0

inf{|DT t (m0 )xc |

:

|xc |

c } = 1, xc ∈ Xm 0

< λ.

Thus, by choosing µ, , and σ sufficiently small we have ds (F s (P˜1s ), F1s (P˜2s )) ≤ λ1 ds (P˜1s , P˜2s ). This completes the proof. Note that Λs is complete under the metric (11.33) from Lemma 11.7. Since F s is a contraction, by the contraction mapping theorem, we obtain

117

Proposition 11.15. There exists σ > 0 such that if ||T˜ − T t ||1 ≤ σ, there exists ˜ s ∈ Λs such that Π ˜ s (K −1 (m ˜ s (K −1 (m Π ˜ 1 ))DT˜(m ˜ 0 ) = DT˜(m ˜ 0 )Π ˜ 0 )). ˜ s ||0 → 0 as ||T˜ − T t ||1 → 0. Proposition 11.16. ||Πs − Π Proof. The idea of the proof is the same as that for Proposition 11.10. We first notice that N s ω ∈ Λs from the proof of Lemma 11.3. The first step is to estimate ds (N s ω, F s (N s ω)). u s c For each x ˜u + x ˜s + x ˜c ∈ Xm ⊕ Xm ⊕ Xm , by Lemma 11.3, there exists a unique 0 0 0 u s c u s c x ¯ +x ¯ +x ¯ ∈ Xm0 ⊕ Xm0 ⊕ Xm0 such that

x ˜u + x ˜s + x ˜c = x ¯u + x ¯c + `˜cu xu , x ¯c ) + x ¯s + `˜cs xs , 0). m ˜ 0 (¯ m ˜ 1 (¯ Note that x ˜c = x ¯c . Write DT˜(m0 )(¯ xs + `˜cs xs , 0))) = x ˆu + x ˆc + `˜cu xu , x ˆc ) + x ˆs + `˜cs xs , 0). m ˜ 0 (¯ m ˜ 1 (ˆ m ˜ 1 (ˆ

(11.47)

Let xu0

+

xs0

+

xc0



 = N (m0 )ω(m0 ) − (F (N ω))(m0 ) (¯ xs + `˜cs xs , 0)). m ˜ 0 (¯ s

s

s

s c s Clearly, xu0 = `˜cs m ˜ 0 (x0 , x0 ). On the other hand, by the definition of F



 N s (m0 )ω(m0 ) − (F s (N s ω))(m0 ) (¯ xs + `˜cs xs , 0)) m ˜ 0 (¯  −1 = DT˜(m ˜ 0 ) Tm˜ W (I − N s (m1 )ω(m1 ))DT˜(m ˜ 0 )(¯ xs + `˜cs xs , 0)), ˜ cu () m ˜ 1 (¯ 0

˜ cu which is in Tm ˜ 0 W (). Hence, ˜ xu0 + xs0 + xc0 ∈ Tm ˜ 0M. Thus, |xu0 | + |xs0 | ≤

2ρµ |xc0 | 1 − ρµ

(11.48)

|xs0 | + |xc0 | ≤

1 |xc |. 1 − ρµ 0

(11.49)

and

118

Computing DT˜(m ˜ 0 )(xu0 + xs0 + xc0 ), we find DT˜(m ˜ 0 )(xu0 + xs0 + xc0 ) = x ˆu + x ˆc + `˜cu xu , x ˆc ). m ˜ 1 (ˆ

(11.50)

˜ cu xu , x ˆc ) ∈ Tm From the invariance, we have that x ˆu + x ˆc + `˜cu ˜ 1 W (), and hence m ˜ 1 (ˆ ˜ xu , x x ˆu + x ˆc + `˜cu ˆc ) ∈ Tm ˜ 1M. m ˜ 1 (ˆ Again, |ˆ xu | + |`˜cu xu , x ˆc )| ≤ m ˜ 1 (ˆ

2ρµ |ˆ xc |. 1 − ρµ

Applying the projection Πcm1 to (11.50) and using (11.17), (11.18) and (11.48), we obtain |ˆ xc | = |Πcm1 DT˜(m ˜ 0 )(xu0 + xs0 + xc0 )|   t c c c c ≥ inf{|DT (m0 )x | : |x | = 1, x ∈ Xm0 } − O() − Cσ |xc0 | (11.51) Combining (11.47) with (11.34) and (11.35) and using (11.38), (11.49) and (11.51), we have |xs0 | + |xc0 | ≤ (O() + Cσ)|¯ xs | which implies that δs (N s (m0 )ω(m0 ), (F s (N s ω))(m0 )) ≤ O() + Cσ ˜ s is the fixed point of the mapping F s which has Lipschitz constant λ1 < 1, Since Π a simple computation yields ˜ s ||s ≤ ( ||N ω − Π s

∞ X

λk1 )(O() + Cσ) = O() + Cσ

(11.52)

k=0

with a different constant C. Next, we estimate ||N s ω − Πs ||. From the definition of ψ˜s (m), it is easy to see ρµ ||(ψ˜s (m0 ))−1 − I|| ≤ , (11.53) 1 − ρµ which implies |xs | ≤

1 (|˜ xs | + |˜ xu | + |˜ xc |). 1 − ρµ

119

Thus, xu + x |(N s (m0 )ω(m0 ) − Πsm0 )(˜ ˜s + x ˜c )| = |¯ xs + `˜cs (¯ xs , 0) − x ˜s | s

m ˜0 s

≤ |¯ x −x ˜ | + ρµ|ˆ xs | 2ρµ (|˜ xu | + |˜ xs | + |˜ xc |) ≤ 1 − ρµ 2ρµ ≤C |˜ xu + x ˜c + x ˜s |, 1 − ρµ which implies ||N s ω − Πs || ≤ C

2ρµ . 1 − ρµ

By Lemma 11.13, we obtain ˜ s − Πs || ≤ ||Π ˜ s − N s ω|| + ||N s ω − Πs || ||Π ˜ s , N s ω) + ||N s ω − Πs || ≤ Cds (Π ≤ O() + Cσ +

2ρµC . 1 − ρµ

For any E > 0, choosing µ, ∗ and σ sufficiently small, we have that if ||T˜ − T t ||1 < σ, then ˜ s − Πs || < E. ||Π This completes the proof. Let ˜c = I − Π ˜u − Π ˜ s. Π It is easy to see Proposition 11.17. ˜ c ∈ C 0 (M, L(X, X)) (i) Π c 2 c ˜ (ii) R(Πc (m)) = Tm ˜ M and (Π (m)) = Π (m). ˜ c − Πc ||0 → 0 as ||T˜ − T t ||1 → 0. (iii) ||Π ˜α = Π ˜ α (K −1 (m)) ˜ α = R(Π ˜ α ). Proof of Theorem 11.1. For α = u, s, c, let Π ˜ and X m ˜ m ˜ m ˜ ˜ we have a decomposition Clearly, for each m ˜ ∈M u s c ˜m X=X ˜ ⊕ Xm ˜ ⊕ Xm ˜

120

˜ c the tangent space to M ˜ at m. of closed subspaces with X ˜ Propositions 11.9, 11.15, m ˜ and 11.17 yield α ˜m ˜α DT˜(m) ˜ :X for α = c, u, s. ˜ → XT˜ (m) ˜ ˜ u onto X ˜ u . Next we claim there Furthermore, DT˜(m) ˜ is an isomorphism from X m ˜ T˜ (m) ˜ t exists σ > 0 such that if ||T˜ − T ||1 ≤ σ then o o n n ˜ u u u u ˜ ˜ ˜ ∈ Xm x | = 1 > max 1, kDT (m) ˜ X˜ c k ˜ x :x λ1 inf DT (m)˜ ˜ , |˜ m ˜ oo n n ˜ c c c c ˜ ˜ ˜ ∈ Xm x |=1 > kDT (m) ˜ X˜ s k ˜ x :x λ1 min 1, inf DT (m)˜ ˜ , |˜ m ˜

Let us first look at o n u u u ˜m ˜ ∈X , |˜ x | = 1 > 1. ˜ xu : x λ1 inf DT˜(m)˜ ˜

(11.54)

From (H3), we have ζ = inf |DT t (m)xu | > u |x |=1

1 , λ

and so |DT t (m)xu | ≥ ζ|xu |. Thus, |DT˜(m)˜ ˜ xu | = |DT t (m)Πm x ˜u | − |DT˜(m)˜ ˜ xu − DT t (m)Πm x ˜u | ≥ ζ|Πum x ˜u | − ||DT˜(m) ˜ − DT t (m)|| |˜ xu | u − ||DT t (m)|| ||Πum xu | ˜ − Πm || |˜ u ˜ um ≥ (ζ − C||Π xu |, ˜ − Πm || − O() − Cσ)|˜

where (11.17) and (11.18) are used. Using Proposition 11.10, we may choose ∗ and σ sufficiently small so that 1 u ˜ um ζ − C||Π , ˜ − Πm || − O() − Cσ > λ1 giving (11.54). One may prove the other estimates in the same fashion by using (H3) and Propositions 11.10, 11.16 and 11.17. The proof is complete Another direct consequence of Propositions 11.9, 11.15 and 11.17 is

121

˜ Proposition 11.18. For each m ˜ ∈M c ˜ cs ˜ s˜ ⊕ X ˜m Tm ˜ W () = Xm ˜, ˜ cu ˜u ˜c Tm ˜ W () = X ⊕ X . m ˜

m ˜

We now point out how to obtain Theorems A–D, stated in Section 3. Theorem A is extracted from Theorem 6.3, Proposition 6.10, Proposition 6.12, and Theorem 9.1; Theorem B is extracted from Theorem 7.3, Proposition 7.10, and Theorem 8.1; Theorem C is extracted from Theorem 10.1, Proposition 10.2, Theorem 11.1, and Proposition 11.18; and finally, Theorem D follows from Proposition 6.10, Proposition 7.10, and Theorem 10.3.

122

12. Invariant Manifolds for Perturbed Semiflow. In the previous sections, we obtained the existence of a compact, normally hyper˜ , for the map T˜, a C 1 perturbation of the time t-map bolic invariant manifold, M T t , where t > t0 . In doing so, we also obtained the stable and unstable manifolds ˜ under the map T˜. We now consider the perturbed semiflow T˜t of T t . The of M argument we use is different from that typically used in the finite dimensional case since we do not have a vector field, nor do we cut-off the semiflow outside the tubular neighborhood. Assume that the perturbed semiflow T˜t is continuous on [0, ∞) × X into X, and that for each t ≥ 0, T˜t : X → X is C 1 . Let us first recall some notation from Section 3. We use B to denote a fixed neighborhood of M in X containing the tubular neighborhood Θ(X u (0 )) ⊕ X s (0 )). We defined kT˜t − T t k0 ≡ sup |T˜t (x) − T t (x)| x∈B

and kT˜t − T t k1 ≡ kT˜t − T t k0 + sup kDT˜t (x) − DT t (x)k. x∈B

Let t1 > t0 be fixed. By Theorem 9.1 and Proposition 6.10, there exists ∗ > 0 such that for each  < ∗ there is a σ > 0 such that if ||T˜t1 −T t1 ||1 < σ then T˜t1 has a unique ˜ cu () in the tubular neighborhood Θ(X u () ⊕ X s ()) C 1 center-unstable manifold W which satisfies ˜ cu ()) ∩ Θ(X u () ⊕ X s ()) = W ˜ cu (). T˜t1 (W Proposition 12.1. There exists ∗ > 0 such that for each  < ∗ there are σ > 0 and 0 <  such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 , then for all t ∈ [0, t1 ] ˜ cu (0 )) ⊂ W ˜ cu () T˜t (W Proof. For the fixed t1 > t0 , by Theorem 9.1, there exist positive constants ˜∗ , ∗ = ∗ (˜ ), δ ∗ = δ ∗ () <  and σ = σ(, δ) such that if ˜ < ˜∗ ,  < ∗ , δ < δ ∗ , and T˜t satisfies ||T˜t1 − T t1 ||1 < σ, then T˜t1 has a C 1 center-unstable manifold with Lipschitz constant ρµ, ˜ cu )), ˜ cu () = Θ(gr(h W ˜ cu ∈ Γcu (, µ, δ, ˜). Let  < ∗ be fixed. Then, by Lemma 5.1, there exists where h 0 ≤  such that for t ∈ [0, t1 ] 1 1 T t (Θ(X u (0 ) ⊕ X s (0 ))) ⊂ Θ(X u ( ) ⊕ X s ( )). 2 2

123

By choosing σ sufficiently small, we have that if kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 , then for each t ∈ [0, t1 ] T˜t (Θ(X u (0 ) ⊕ X s (0 ))) ⊂ Θ(X u () ⊕ X s ()).

(12.1)

˜ cu (0 ) and W ˜ cu () exist. We We also choose σ sufficiently small such that both W cu 0 cu cu ˜ for  and  by h ˜ 0 and h ˜ respectively. Proposition denote the corresponding h   cu cu u 0 ˜ 0 =h ˜ on X ( ). We want to show that T˜t (W ˜ cu (0 )) ⊂ W ˜ cu () 6.10 implies that h   for t ∈ [0, t1 ]. ˜ cu (0 )) ∩ (X u (0 ) ⊕ X s (0 )) = For each (m0 , xu0 ) ∈ X u (0 ), from the fact that T˜t1 (W ˜ cu (0 ), there is a sequence of points (mk , xu ) ∈ X u (0 ) such that for k ≥ 0 W k ˜ cu (mk+1 , xu )) = mk + xu + h ˜ cu (mk , xu ). T˜t1 (mk+1 + xuk+1 + h  k+1 k  k ˜ cu (mk , xu ) and mk (t) + xu (t) + xs (t) = T˜t (mk + xu + xs ) for k ≥ 0. From Let xsk = h  k k k k k (12.1), (mk (t), xuk (t) + xsk (t)) ∈ X u () ⊕ X s (). Observe that T˜t1 (mk+1 (t) + xuk+1 (t) + xuk+1 (t)) = mk (t) + xuk (t) + xsk (t) Thus, applying Lemma 6.8, we obtain for all k ≥ 0 ˜ cu (mk (t), xu (t))| ≤ λ1 |xs (t) − h ˜ cu (mk+1 (t), xu (t))| |xsk (t) − h  k k+1  k+1

(12.2)

Note that λ1 < 1 and |xα k (t)| ≤  for α = u, s. Hence, (12.2) yields ˜ cu (m0 (t), xu (t)), xs0 (t) = h  0 ˜ cu0 (m0 , xu )) ∈ W ˜ cu (), for 0 ≤ t ≤ t1 , which completes In other words, T˜t (m0 + xu0 + h 0  the proof. Theorem 12.2. There exits 0 and σ such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 , then ˜ cu (0 )) ∩ Θ(X u (0 ) ⊕ X s (0 )) ⊂ W ˜ cu (0 ) for 0 ≤ t ≤ t1 ; (i) T˜t (W u u 0 t u ˜ cu (m0 , xu )) ∈ X u (0 ) ⊕ X s (0 ) for (ii) For (m0 , x0 ) ∈ X ( ), if T˜ (m0 + x0 + h 0 ˜ cu (m0 , xu )) ∈ W ˜ cu (0 ) for t ∈ [0, t2 ]; 0 ≤ t ≤ t2 , then T˜t (m0 + xu0 + h 0 ˜ cu (0 ) ⊃ ∩∞ A˜t , where (iii) W t=0  A˜t = (m0 , xu0 , xs0 ) ∈ X u (0 ) ⊕ X s (0 ) : ∃ (m1 , xu1 , xs1 ) ∈ X u (0 ) ⊕ X s (0 ) such that T˜t (m1 + xu1 + xs1 ) = m0 + xu0 + xs0 , and T˜τ (m1 + xu1 + xs1 ) ∈ X u (0 ) ⊕ X s (0 ), for all 0 ≤ τ ≤ t

124

Proof. (i)–(ii) follows directly from Proposition 12.1. Note that A˜kt1 ⊂ Ak , where Ak is given in Proposition 6.10. Hence (iii) holds. From Theorem 8.1 and Proposition 7.10, there exists ∗ > 0 such that for each  < ∗ there is a σ > 0 such that if ||T˜t1 − T t1 ||1 < σ then T t1 has a unique C 1 ˜ cs () in the tubular neighborhood Θ(X u () ⊕ X s ()) which center-stable manifold W satisfies ˜ cs ()) ⊂ W ˜ cs (). T˜t1 (W Proposition 12.3. There exists ∗ > 0 such that for each  < ∗ there are σ > 0 and 0 <  such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 , then ˜ cs (0 )) ⊂ W ˜ cs (), for 0 ≤ t ≤ t1 . T˜t (W Proof. One may establish this proposition in the same fashion as Proposition 12.1. The only difference is that we define the sequence (mk , xuk ) by ˜ cs0 (mk−1 , xu )) = mk + xu + h ˜ cs0 (mk , xu ). T˜t1 (mk−1 + xuk−1 + h  k−1 k  k We omit the details. Theorem 12.4. There exist 0 and σ such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 , then ˜ cs (0 )) ∩ Θ(X u (0 ) ⊕ X s (0 )) ⊂ W cs (0 ) (i) For all t ≥ 0, T˜t (W  and t ˜ cs (0 ). (ii) x : T˜ (x) ∈ Θ(X u (0 ) ⊕ X s (0 )), t ≥ 0 ⊂ W Proof. (i) is a direct consequence of Proposition 12.3. (ii) follows from Proposition 7.10. ˜ =W ˜ cs () ∩ W ˜ cu () is a C 1 compact invariant From Theorem 10.1, we have that M t ˜ is invariant for the manifold for the time-t1 map T˜ 1 . The next result gives that M t ˜ semiflow T .

125

Proposition 12.5. There exists ∗ > 0 such that for each  < ∗ there is σ > 0 such that if ||T˜t1 − T t1 ||1 < σ and kT˜t − T t k0 < σ, for 0 ≤ t ≤ t1 , ˜ in the tubular neighborhood then T˜t has a unique C 1 compact invariant manifold M u s t 1 ˜ ˜ onto M ˜. Θ(X () ⊕ X ()) and for each t ≥ 0, T is a C diffeomorphism from M Proof. Let σ and 0 satisfy the requirements of Proposition 12.1 and Proposition 12.3. Since T˜t1 has a unique compact connected invariant manifold in Θ(X u () ⊕ X s ()), we have ˜ cs (0 ) ∩ W ˜ cu (0 ) = W ˜ cs () ∩ W ˜ cu (). W Propositions 12.1 and 12.3 imply that, for 0 ≤ t ≤ t1 , ˜) ⊂ M ˜. T˜t (M Note that each t > t1 may be written as t = kt1 + t˜ for a positive integer k and ˜) ⊂ M ˜ . For each 0 ≤ t˜ ≤ t1 . Hence, T˜t = T˜kt1 +t˜ = T˜kt1 T˜t˜ which implies that T t (M 0 ≤ t ≤ t1 , write T˜t1 = T˜t1 −t T t . Then, it follows that T˜t is a C 1 diffeomorphism ˜ onto M ˜ , which yields that T˜t is from the fact that T˜t1 is a diffeomorphism from M a C 1 diffeomorphism for all t ≥ 0. This completes the proof. As a corollary of Theorem 10.3, by using Propositions 12.1 and 12.3, we obtain Theorem 12.6. ˜ ) = 0, uniformly for x ∈ W ˜ cs (0 ). (i) limt→∞ d(T˜t (x), M and ˜ ) = 0, uniformly for x ∈ W ˜ cu (0 ). (ii) limt→∞ d(T˜−t (x), M ˜ is a normally hyperbolic manifold. Theorem 12.7. M ˜ α for α = u, s, c, defined in Section 11, are invariant for Proof. We first show that X the semiflow T˜ provided that σ is sufficiently small. Proposition 12.5 and Proposition ˜ c is invariant. We consider X ˜ s . For m ˜ and x ˜ s , it is 11.17 imply that X ˜0 ∈ M ˜s ∈ X m ˜0 ˜st enough to show that DT˜t (m ˜ 0 )˜ xs ∈ X for 0 ≤ t ≤ t . Write 1 T˜ (m ˜ ) 0

DT˜t (m ˜ 0 )˜ xs = x ˜u (t) + x ˜s (t) + x ˜c (t),

(12.3)

˜ cs () implies that ˜ αt . Proposition 11.15 and the invariance of W where x ˜α (t) ∈ X T˜ (m ˜ 0) x ˜u (t) = 0. We claim that x ˜c (t) = 0, for 0 ≤ t ≤ t1 . Suppose that for some t¯ ∈ [0, t1 ], c ¯ t x ˜ (t) 6= 0. Let m(t) ˜ = T˜ (m ˜ 0 ) and m(t) = K −1 m(t). ˜ For simplicity, we set x ˜c (kt1 + t¯) = DT˜kt1 (m( ˜ t¯))˜ xc (t¯), x ˜s (kt1 + t¯) = DT˜kt1 (m( ˜ t¯))˜ xs (t¯)

(12.4)

126

Note that for α = s, c x ˜α ((k + 1)t1 + t¯) = DT˜t1 (m(kt ˜ 1 + t¯))˜ xα (kt1 + t¯). Thus, from Theorem 11.1 it follows that |˜ xc ((k − 1)t1 + t¯)||˜ xs (kt1 + t¯)| ≤ λ1 |˜ xs ((k − 1)t1 + t¯)||˜ xc (kt1 + t¯)|, which implies

|˜ xs (t¯)| c |˜ x (kt1 + t¯)|, |˜ xs (kt1 + t¯)| ≤ λk1 c |˜ x (t¯)|

(12.5)

where the assumption that x ˜c (t¯) 6= 0 is used. Thus, using (12.3) and (12.4), we have DT˜(k+1)t1 (m ˜ 0 )˜ xs ¯ ¯ = DT˜t1 −t (m(kt ˜ 1 + t¯))DT˜kt1 +t (m ˜ 0 )˜ xs ¯ = DT˜t1 −t (m(kt ˜ 1 + t¯))(˜ xs (kt1 + t¯) + x ˜c (kt1 + t¯)).

(12.6)

˜c Applying the projection Π to (12.6), we obtain m((k+1)t ˜ 1)   s c ˜c ˜t1 −t¯(m(kt ¯ ¯ ¯ 0=Π D T ˜ + t ))(˜ x (kt + t ) + x ˜ (kt + t ) . 1 1 1 m((k+1)t ˜ 1) Using (12.5), we obtain ¯ 0 ≥ |DT˜t1 −t (m(kt ˜ 1 + t¯))˜ xc (kt1 + t¯)| |˜ xs (t¯)| c ¯ − C||DT˜t1 −t ||M˜ λk1 c |˜ x (kt1 + t¯)|, |˜ x (t¯)|

(12.7)

˜ of the norms of the operators. Since T˜t1 −t¯ where || · ||M˜ is the supremum over M ˜ and M ˜ is compact, this inequality is impossible for k is a diffeomorphism on M s ˜ is invariant for T˜t . Similarly, one may prove that X ˜ u is large enough. Therefore X invariant. ˜ is a normally hyperbolic invariant manifold for T˜t , first we notice that To prove M ˜ without any modification. We also need the Lemma 2.2 can be applied to T˜t and M following lemma. Lemma 12.8. There exists t¯0 > 0 such that for t4 > t3 > t¯0 there is a constant ˜ and t ∈ [t3 , t4 ]. C > 0 such that ||DT˜t (m)|| ˜ ≤ C for all m ˜ ∈M The proof of this lemma will be given after we complete the proof Theorem 12.7.

127

We want to show that for sufficiently large t, n o n o u ˜ u , |˜ ˜t (m)| λ1 inf |DT˜t (m)˜ ˜ xu | : x ˜u ∈ X x | = 1 > max 1, ||D T ˜ || c ˜ m ˜ X m ˜

(12.8)

and n n oo t c c c c ˜ ˜ λ1 min 1, inf |DT (m)˜ ˜ x | : x ˜ ∈ Xm x |=1 > ||DT˜t (m)| ˜ X˜ s || ˜ , |˜ m ˜

(12.9)

From Section 11, (12.8), and (12.9) hold for t = t1 . It follows from Lemma 12.8 that there exist a positive integer k0 and a constant C > 0 such that ||DT˜t (m)|| ˜ < C for ˜ ˜ ˜ all t ∈ [k0 t1 , (k0 + 1)t1 ] and m ˜ ∈ M . Hence, for t = kt1 + t, where t ∈ [k0 t1 , (k0 + 1)t1 ] ˜, and k is a positive integer, we have for all m ˜ ∈M ||DT˜t (m)| ˜ X˜ s || < Cλk1 . m ˜

(12.10)

Note that λ1 ∈ (λ, 1), so for large k ||DT˜t (m)| ˜ X˜ s || < λ1 . m ˜

With the help of Lemma 2.2, we can prove the second part of (12.9) in the same ˜ cu and considering the inverse of T˜t on it, fashion. Restricting our attention to W (12.8) follows. Proof of lemma 12.8. ˜  = {x ∈ X | d(x, M ˜ ) < }. Let M First, we claim that for any fixed t > 0, there exists  > 0 such that LipT˜t |M˜  < ∞. ˜ is compact and DT˜t (·) In fact, there exists a < ∞ such that ||DT˜t ||M˜ = a, since M ˜t is continuous. For each m, ˜ there exists m < a + 1. ˜ > 0 such that ||D T ||B(m, ˜ m ˜) i k ˜ ˜ ¯ Choose a finite number of m ˜ i ∈ M such that M ⊂ ∪i=1 B(m ˜ i , 5(1+C) ¯ ), where C is a constant satisfying ¯m |m ˜ −m ˜ 0 | ≤ d(m, ˜ m ˜ 0 ) ≤ C| ˜ −m ˜ 0| ˜ , where d( · ) is defined in the proof of Lemma 2.2 (but on M ˜ ), and for all m, ˜ m ˜0 ∈ M i = m ˜i  i For x1 ∈ B(m ˜ i , 5(1+ ˜ j , 5(1+j C) ¯ ), x2 ∈ B(m ¯ ), we have C) i + j ¯m |T˜t (x1 ) − T˜t (x2 )| ≤ (a + 1)(C| ˜i −m ˜ j| + ¯ ). 5(1 + C) If

i +j ¯ 5(1+C)

(12.11)

¯m ¯ 1 − x2 | then we have + C| ˜i −m ˜ j | ≤ (1 + C)|x ¯ 1 − x2 |. |T˜t (x1 ) − T˜t (x2 )| ≤ (a + 1)(1 + C)|x

(12.12)

128

Otherwise, without loss of generality, we may assume i ≥ j and we have ¯ 1 − x2 | < i + j + C| ¯m ¯ 1 − x 2 | + i + j , (1 + C)|x ˜i −m ˜ j | ≤ C|x ¯ 5 5(1 + C) which yields |x1 − x2 | ≤

2i . 5

Thus, x1 , x2 ∈ B(m ˜ i , i ). Hence, |T˜t (x1 ) − T˜t (x2 )| ≤ (a + 1)|x1 − x2 |.

(12.13)

i ˜t ˜ ≤ (a + 1)(1 + ˜  ⊂ ∪k B(m Let  > 0 be so small that M ˜ i , 5(1+ ¯ ), therefore LipT |M i=1 C)  ¯ This completes the proof of the claim. C). Let Ea, = {t ∈ [0, 1] : LipT˜t | ˜ ≤ a}, where a is a positive integer and  ∈ M

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