Structure of accessibility classes

STRUCTURE OF ACCESSIBILITY CLASSES

arXiv:1706.01156v1 [math.DS] 4 Jun 2017

´ JANA RODRIGUEZ–HERTZ AND CARLOS H. VASQUEZ Abstract. In this work we deal with partially hyperbolic diffeomorphisms whose central direction is two dimensional. We prove that the accessibility classes are smooth immersed manifolds and we give a proof that the class of accessible diffeomophisms is open.

1. Introduction Let M be a Riemannian closed manifold and let f : M → M be a partial hyperbolic diffeomorphim with an invariant splitting T M = Es ⊕ Ec ⊕ Eu. Here E s and E u are uniformly hyperbolic bundles contracting and expanding, respectively, while vectors in E c are neither contracted as strongly as the vectors in E s nor expanded as the vectors in E u . See the precise definition in Section 2. It is well known that partial hyperbolicity is a C 1 -open property. Also It is known that there are unique invariant foliations F s and F u tangent to E s and E u respectively [2, 7] but in general, E c , E cu = E c ⊕ E u , and E cs = E c ⊕ E s do not integrate to foliations, not even when E c is onedimensional (see [15]). A partially hyperbolic diffeomorphism is dynamically coherent if there are two invariant foliations: F cu tangent to E c ⊕ E u , and F cs tangent to E c ⊕ E s . If f is dynamically coherent, then there is an invariant foliation F c tangent to E c (just take the intersection of F cs and F cu ). We say that a point y ∈ M is su-accessible from x ∈ M if there exist a path γ : I → M , from now on an su-path, piecewise contained in the leaves of the strong stable and strong unstable foliations. This defines an equivalence relation on M . We denote by AC(x) = {y ∈ M : y is su − accessible from x} the accessibility class of x. A diffeomorphism has the accessibility property if there is a unique accessibility class. We shall mainly be working in the universal cover of M . When no confusion arises, AC(x) will be denoting the accessibility class of x in the universal cover which is not necessary the same as the preimage of AC(x) by the covering projection from the universal cover. The aim of this work is to describe the geometry of the accessibility classes: Theorem A. If f : M → M is a partially hyperbolic diffeomorphism with a two-dimensional center bundle, then the accessibility classes are immersed manifolds. Date: June 6, 2017. 2000 Mathematics Subject Classification. 37D30 (37A25,37C05, 57R30). Key words and phrases. Accesibility, Partially hyperbolic diffeomorphisms, stable ergodicity. CV was supported by Proyecto Fondecyt 1171427. 1

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´ JANA RODRIGUEZ-HERTZ AND C. H. VASQUEZ

Moreover, if AC(x) is not a codimension-one accessibility class, then either: (i) AC(x) is open, or else (ii) AC(x) is a codimension-two C 1 -immersed manifold. With some additional hypotheses we can also obtain regularity of the codimension-one accessibility classes. Theorem B. Let f : M → M be a dynamically coherent diffeomorphism with twodimensional center bundle. Then all accessibility classes are injectively immersed C 1 submanifolds if any of the two following conditions hold: (i) f is C 2 and satisfies the center bunching condition (2.4) (ii) f is C 1+α and satisfies the strong center bunching condition (2.5) Theorem B together with Proposition 6.1 partially answer Problem 16 in [17] Conjecture 1.1 ([17], Problem 16). Prove that the accessibility classes are topological manifolds that vary semi-continuously, as well as their dimensions. Prove that, with bunching, they are indeed smooth manifolds. Theorem B also gives positive evidence of the following conjecture by Wilkinson: Conjecture 1.2 (Wilkinson [18], Conjecture 1.3). Let f : M → M be C r , partially hyperbolic and r-bunched. Then the accessibility classes are injectively immersed C r submanifolds of M . Accessibility is a key notion in the program of Pugh and Shub [10, 11] to prove that stable ergodicity is C r -dense among volume preserving partially hyperbolic diffeomorphismst, r ≥ 2. Pugh and Shub also conjectured the stable accessibility is dense among the C r -partially hyperbolic diffeomorphisms, volume preserving or not, r ≥ 2. In the case dim E c = 1, the accessibility property is always stable [5], and C ∞ -dense amog the volume preserving diffeomorphisms [14]. Even in the case of one-dimensional center bundle the same result was proved in [4] for non conservative diffeomorphisms. Without any hypothesis on the dimension of the central bundle, Dolgopyat and Wilkinson [6] proved that stable accessibility is C 1 - dense in the space of C r -diffeomorphisms (any r ≥ 1). The following theorem has also been announced some years ago by Avila and Viana [1]. We found it convenient to publish it here, since our proof is different from theirs. Theorem C. Among partially hyperbolic diffeomorphisms with two-dimensional center bundle, the accessibility property is C 1 -open. 2. Preliminaries Let M be a Riemannian closed manifold. f : M → M is a partially hyperbolic diffeomorphism if there is a non trivial Df -invariant splitting of the tangent bundle T M = Es ⊕ Ec ⊕ Eu and there are continuous positive functions defined on M satisfying for every x ∈ M : µ(x) < ν(x) < γ(x) < γˆ (x)−1 < νˆ(x)−1 < µ ˆ(x)−1


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with max(ν(x), νˆ(x)) < 1, such that for every unit vector v ∈ Tx M (2.1)

µ(x) < kDf (x)vk < ν(x),

for every v ∈ E s (x),

(2.2)

γ(x) < kDf (x)vk < γˆ (x)−1

for every v ∈ E c (x),

(2.3)

νˆ(x)−1 < kDf (x)vk < µ ˆ(x)−1 ,

for every v ∈ E u (x).

The inequalities (2.1) and (2.3) above mean that E s and E u are uniformly hyperbolic bundles (contracting and expanding, respectively) while (2.2) means vectors in E c are not contracted as strongly as the vectors in E s nor expanded as strongly as the vectors in E u . It is well known that partial hyperbolicity is a C 1 -open property. We will say that f satisfies the center bunching condition if the following holds: (2.4)

max{ν(x), νˆ(x)} ≤ γ(x)ˆ γ (x)−1 .

When f is a C 2 partially hyperbolic diffeomorphism dynamically coherent, then the (stable/unstable) holonomy maps are smooth restricted to the center stable/unstable leaves [12]. Recently, Brown [3] proved that the same statement hold if f is a C 1+α partially hyperbolic diffeomorphism satisfying an stronger bunching condition. More precisely, take 0 < β < α such that ν(x)γ(x)−1 < µ(x)β

and

νˆ(x)ˆ γ (x)−1 < µ ˆ(x)β .

we will say that f satisfies the strong bunching condition if for some 0 < θ < β < α such that ν(x)α γ(x)−α ≤ ν(x)θ and νˆ(x)α γˆ (x)−α ≤ νˆ(x)θ we have (2.5)

max{ν(x), νˆ(x)}θ ≤ γ(x)ˆ γ (x)−1 . 3. Proof of Theorem A

A subset K ⊂ Rn is said to be topologically homogeneous if for every pair of points x, y ∈ K there exist neighborhoods Ux , Uy ⊂ Rn of x and y respectively and a homeomorphism ϕ : Ux → Uy , such that ϕ(Ux ∩ K) = Uy ∩ K, and ϕ(x) = y, respectively. When ϕ can be chosen a C r -diffeomorphim, r ≥ 1, we say that K ⊂ Rn is C r -homogeneous. Let K be a locally compact subset of Rn . The following result holds for partially hyperbolic diffeomorphisms with any center dimension. Lemma 3.1. Let x, y ∈ AC(x), and two c-dimensional discs Dx , Dy transverse, respectively, to E s (x) ⊕ E u (x) and E s (y) ⊕ E u (y) centered at x and y. Then there exist two c-discs x ∈ Ux ⊂ Dx and y ∈ Uy ⊂ Dy and a homeomorphism h : Ux → Uy such that h(x) = y and h(ξ) ∈ AC(ξ) for all ξ ∈ Ux .

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Proof. Let x and y be on the same Wεs (x), the argument then extends by induction. Take Dx and Dy , for any ξ in a suitable Ux ⊂ Dx , there is a unique w ∈ Wεs (ξ) ∩ Wεu (Dy ). Let h(ξ) = Wεu (w) ∩ Dy . Clearly, h(ξ) ∈ AC(ξ). Also, it is easy to see that h is continuous and open. The inverse of h is defined analogously, taking Wεu instead of Wεs and viceversa. Therefore h is a local homeomorphism. For every ε > 0 sufficiently small, in a sufficiently small ball of x, the following is well defined π s : B(x) → Dxu , such that π s (y) = Wεs (y) ∩ Dxu . Analogously we define π u . It is easy to verify that π s and π u are open maps. Let us call π su = π s ◦ π u . Define: Fε (x) = {y ∈ Dx : (π us )−1 (y) ∩ (π su )−1 (x) 6= ∅} Fε (x) = π us ((π su )−1 (x)). Fε (x) is arc-connected (see Lemma 3.3 below). Theorem 3.2. If dim E c = 2, then for each x ∈ M and sufficiently small ε > 0, one of the following holds: (i) Fε (x) is a point (ii) Fε (x) is the injective image of a segment or a circle (iii) the accessibility class of x, AC(x), is open The proof follows essentially the same steps as in [16], with some suitable changes. We include the steps for completeness. Lemma 3.3. Given x ∈ M , for any y ∈ Fε (x), there exists an open disc D ′ ⊂ Dx containing x and a continuous γ : D ′ × I → Dx such that γ(x, 0) = x, γ(x, 1) = y and γ(z, I) ⊂ Fε (z). Proof. Take y ∈ Fε (x), hence there exists w ∈ (π us )−1 (y) ∩ (π su )−1 (x). This implies that π su (w) = x and π us (w) = y. Therefore, there exists a (two-legged) su-path in M from w to y, η : I → Bx . The projection π su ◦ η gives a path in Dx from x to y that is contained s ⊕ E u . If D ′ ⊂ D is sufficiently small, then for in Fε (x). Take a disc Dw transverse to Ew x w ′ ′ ′ each x ∈ D there is a unique w ∈ Dw such that π su (w′ ) = x′ . By continuity of the stable and unstable foliations, we get close paths η ′ for each w′ , so we can choose a γ as in the statement. Proof of Theorem 3.2. From Lemma 3.3, it follows that Fε (x) is path-connected. Let us show that if Fε (x) contains a simple triod, then AC(x) must be open. A simple triod is a continuum homeomorphic to letter Y . If Fε (x) contains a simple triod Y , then by Lemma 3.1, there is h and a small neighborhood of x, Ux ⊂ Dx , such that h(Y ) is as in Figure 1; that is, there is an arc η1 separating U into two connected components A and B, and η2 separating B into two connected components B1 and B2 , such that η1 ∪ η2 = h(Y ), and such that A ∩ B1 ∩ B2 = {x}. Notice that η1 ∪ η2 ⊂ AC(x). Let y ∈ B1 ∩ B2 . By Lemma 3.3, there exists ψ : D ′ × I → D ′′ such that ψ(z, t) = h(γ(h−1 (z), t)); hence ψ(x, 1) = y and ψ(z, t) ⊂ h(Fε (z)) ⊂ AC(z). By continuity of ψ, for all points z in A ∩ D ′ , we have ψ(z, I) ∩ η1 6= ∅. This implies that AC(z) ∩ AC(x) 6= ∅ for all z ∈ A ∩ D ′ , hence A ∩ D ′ ⊂ AC(x). By continuity of π su , (π su )−1 (A ∩ D ′ ) is an open set in AC(x). By the center homogeneity of AC(x) (Lemma 3.1), AC(x) ∩ Dy is open for every small disc


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η2 D ′′ B1

y B2 x

η1

D′ A

Figure 1. A simple triod in AC(x) ∩ D transverse to Eys ⊕ Eyu , and hence AC(x) is open. For another proof of the openness of AC(x), see [HHU2008]. Remark 3.4. If W s and W u are not jointly integrable at x, then Fε (x) always contains a segment that separates a small neighborhood of x. Indeed α = π u (Wεs (x)) ⊂ Fε (x), and since x is not an endpoint of Wεs (x) and π u is open, then x is not an endpoint of the segment α. For any disc Dx transverse to Exs ⊕ Exu , let AC D (x) = cc(AC(x) ∩ Dx , x) We have the following : Theorem 3.5. For each x ∈ M and sufficiently small Dx , one of the following holds: (i) AC D (x) is a point (ii) AC D (x) is the injective image of a segment or a circle (iii) the accessibility class of x, AC(x), is open Proof. If Fε (ξ) is a point for all ξ ∈ AC(x), then W s and W u are jointly integrable at ξ for all ξ ∈ AC(x), and therefore AC D (x) is a point. (Moreover, in this case AC(x) is a C 1 -manifold by Journée’s argument, see [Didier], [Burns, Hertz, Hertz, Talitskaya, Ures]). On the other hand, if Fε (ξ) is open for any ξ ∈ AC(x), then AC(x) is open. Let us assume that Fε (ξ) is the injective continuous image of a segment for some ξ ∈ AC(x). By Lemma 3.1, there is a small disc Ux transverse to Exs ⊕Exu and a homeomorphism h such that α = h(Fε (ξ)) ⊂ AC D (x) ∩ Ux , and α separates Ux .


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K A B1 α x

y

D′

D ′′ B2

Figure 2. A continuum K not included in α If AC D (x) contains a continuum K that is not contained in α, then we may assume that α ∪ K separates a small ball around x into three connected components A, B1 and B2 as in Figure 2, so that K = A ∩ B1 , A ∩ B2 ⊂ α and B1 ∩ B2 ⊂ α. Take y ∈ B1 ∩ B2 . Then, due to Lemmas 3.3 and 3.1 there exists ψ : D ′ × I → D ′′ continuous such that ψ(x, 0) = 0, ψ(x, 1) = y, and ψ(z, I) ⊂ AC(z) for all z ∈ D ′ . Now, arguing as in Theorem 3.2, we obtain that for all z ∈ A ∩ D ′ , AC(z) ∩ AC(x) 6= ∅. Hence, AC(x) is open. Corollary 3.6. AC(x) is a topological manifold for all x. Moreover, if, AC D (ξ) is a point for all ξ ∈ AC(x), then AC(x) is a C 1 -manifold. Proof. The first statement follows from Theorem 3.5. If for all ξ ∈ AC(x), AC D (ξ) is a point, then W s and W u are jointly integrable at ξ for all ξ ∈ AC(x). This means that W s (η) and W u (η) are, restricted to AC(x), continuous transverse foliations with uniformly smooth leaves. Journé’s argument [8] then implies that AC(x) is a C 1 -immersed manifold. In [13] the authors shown that K is C 1 -homogeneous if and only if K is a C 1 -submanifold of Rn . This fact, allows us to conclude the following statement. 4. Proof of Theorem B Throughout this section we shall assume that f is a dynamically coherent partially hyperbolic diffeomorphism. 4.1. AC(x) is C 1 -homogeneous when restricted to center discs. The following three results hold for any center dimension.


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Theorem 4.1. [12] Suppose that f : M → M is a C 2 partially hyperbolic diffeomorphism that is dynamically coherent and satisfies the center bunching condition (2.4). Then the local unstable and local stable holonomy maps are uniformly C 1 when restricted to each center unstable and each center stable leaves respectively. Theorem 4.2. [3] Let f : M → M be a C 1+α partially hyperbolic diffeomorphism that is dynamically coherent that satisfies the strong bunching condition (2.5) for some 0 < θ < α. Then the local unstable and local stable holonomy maps are uniformly C 1+θ when restricted to the center unstable and center stable leaves respectively. Let Wεc (x) denote the c-disc of radius ε > 0 and center x inside the leaf W c (x) tangent to E c . Proposition 4.3. For each x ∈ M , the connected component of AC(x)∩Wεc (x) containing x is C 1 -homogeneous if f is in the hypothesis of Theorem B. Proof. Let x, y ∈ AC(x). In order to prove the proposition it suffices to show that the homeomorphisms h defined in Lemma 3.1 are C 1 when Ux and Uy are taken inside Wεc (x) and Wεc (y) respectively. Let us assume that y ∈ Wεs (x), then the argument extends by induction. If f is in the hypothesis of Theorem B then either f is in the hypothesis of Theorem 4.1, in which case the local stable holonomy map is C 1 when restricted to Wεsc (x), or f is in the hypothesis of Theorem 4.2, in which case the local stable holonomy map is C 1+Hölder when restricted to Wεsc (x). In either case, the corresponding h, which is exactly the stable holonomy map restricted to Wεsc (x) is C 1 . Proposition 4.4. If f is in the hypothesis of Theorem B, then for each x ∈ M , the connected component of AC(x) ∩ Wεc (x) is a C 1 -manifold. The proof of this proposition follows immediately from Proposition 4.3 above and the following theorem: Theorem 4.5. [13] Let K be a locally compact (possibly non-closed) subset of Rn . Then K is C 1 -homogeneous if and only if K is a C 1 -submanifold of Rn . In order to prove Theorem B, the idea is to apply the following theorem by Journé: Theorem 4.6. [8] Let E and F be two continuous transverse foliations with uniformly smooth leaves, of some manifold. If φ is uniformly smooth along the leaves of E and F, then φ is smooth. We want to see that the immersions of the sets BC(x), the connected component of AC(x) ∩ Bε (x) containing x, are smooth along transverse foliations. However, in our case we do not have smoothness along two transverse foliations a priori, so we proceed as Wilkinson in [18]: first we prove that the immersions of BC(x) are uniformly smooth along unstable and center leaves in a neighborhood of x. Since F u and F c are transverse foliations in each center-unstable leaf W cu (y), applying Journé to each of this leaf we

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obtain smoothness along the foliation F cu . Note that this smoothness is uniform in a neighborhood of each x (Proposition 4.7). In this way we obtain two continuous transverse foliations, F s and F cu , with uniformly smooth leaves, along whose leaves the immersion of BC(x) is uniformly smooth. Journé’s Theorem 4.6 then implies that BC(x) is smooth. This, in turn, implies that AC(x) is a C 1 -immersed manifold, and Theorem B follows. Proposition 4.7. Let f : M → M be a dynamically coherent partially hyperbolic diffeomorphism. Let φ be uniformly smooth along the leaves of F c and F u in a neighborhood of x ∈ M , then φ is uniformly smooth along the leaves of F cu in a neighborhood of x Proof. This follows from Journé’s Theorem proof. A more detailed and reader friendly proof may be found in [9], Section 3, Preparatory results from analysis, and in particular, Section 3.3. Journé’s Theorem. So, in order to finish the proof, the only thing left is to show that the immersions of the sets BC(x) are uniformly smooth along the leaves of F c in a neighborhood of each x ∈ M . This is proved in the following proposition: Proposition 4.8. The immersions of BC(x) are uniformly smooth along the center leaves of F c . Proof. For each y ∈ BC(x), call C(y) = BC(x) ∩ Wεc (y). Now C(y) is the image of C(x) by the composition of the local stable holonomy restricted to W cs (x) and the local unstable holonomy restricted to W cu (y). Since the local stable and unstable holonomies are uniformly C 1 when restricted to each center stable and center unstable leaves respectively, then y 7→ C(y) is uniformly C 1 in a neighborhood of x. 5. Proof of Theorem C The results in this section do not use Theorems 3.2 and 3.5. To prove Theorem C we proceed as in [4], that is, we use a semi-continuity argument. However, the persistence of the openness has to be different, since in center dimension two, there are more cases. The proof of the key Proposition 5.1 is therefore completely different. Call Γ(f ) = {x ∈ M : AC(x) is not open}. Γ(f ) is a compact su-saturated set. Let K(M ) be the family of compact subsets of M endowed with the Hausdorff metric. The proof consists in showing that the function Γ : Diff r (M ) → K(M ) is upper semi-continuous with respect to the Hausdorff metric. Indeed, if f 7→ Γ(f ) is upper semi-continuous and f has the accessibility property, then Γ(f ) = ∅, and this implies there must be a C 1 -open neighborhood U such that Γ(g) = ∅ for all g ∈ U . This gives the result. The upper semi-continuity of f 7→ Γ(f ) is immediate if we prove the following: Proposition 5.1. If ACf (x) is open, there exists ε > 0 and U (f ) such that Bε (x) ⊂ ACg (x) for all g ∈ U (f ). Indeed, let xn ∈ Γ(fn ) with fn → f in the C 1 -topology and xn → x. If x ∈ Γ(f )c , then there would be a ball Bε (x) ⊂ ACg (x) for all g ∈ U , a contradiction.


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Proof of Proposition 5.1. For each sufficiently small ε > 0 and each n ∈ N, let us define: Eε(n) (x) = {y ∈ AC(x) : ∃x0 = x, x1 , . . . , xn = y such that xi ∈ Wεs (xi−1 ) ∪ Wεu (xi−1 )} (n)

It is easy to verify that Eε (x) is locally arc-connected. Let us assume that Wεσ (x) = {y ∈ (n) W σ (x) : there exists a segment α ⊂ W σ (x) with length(α) ≤ ε}, σ = s, u. Then Eε (x) is a compact set for all n ∈ N. Since [ AC(x) = Eε(n) (x), n (n)

if AC(x) is open, then there exists n such that Eε (x) has non-empty interior. By Lemma (n+k) 3.1, there exists k ∈ N and an open set Ux such that x ∈ Ux ⊂ Eε (x). Consider a disc s u su Dx transverse to Ex ⊕ Ex , and take Bρ (x) ⊂ π (Ux ). N

E x W

S Figure 3. An X in AC(x). Consider four neighborhoods N, S, E and W in ∂Bρ (x) so that any arc from N to S (n) intersects any arc from E to W , as in Figure 3. Since Eε (x) is locally arc-connected, and (n) Bρ (x) ⊂ π su (Eε (x)), there are four paths in Bρ (x) joining x with points, respectively, in N, S, E or W . Each of these paths is the π su -image of a finite su-path α; that is, α contains s (x u x0 = x, . . . , xn = L with π su (L) ∈ N, S, E or W , such that xi ∈ Wf,ε i−1 ) ∪ Wf,ε (xi−1 ), s (x u and such that α|[xi−1 ,xi ] ⊂ Wf,ε i−1 ) ∪ Wf,ε (xi−1 ); x and L are the endpoints of α.


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Since Wfs , Wfu and πfsu vary continuously with respect to the C 1 -topology, there exists a C 1 -neighborhood U (f ) of f , such that for all g ∈ U , there are four paths in Bρ (x) joining x with, respectively, N, S, E and W , such that each of them is the πgsu -image of a finite (n)

su-path in Eg,ε (x) ∩ Ux . For each g ∈ U , call αg the path joining W with E and passing through x, and call βg the map joining N with S passing through x, obtained as described above. g 7→ αg , βg are continuous at f . There is a point y ∈ βf and δ > 0 such that Bδ (y) ∩ αg = ∅ and βg ∩ Bδ (y) 6= ∅ for all g ∈ U . Let us assume y belongs to the connected component of Bρ (x) \ αf that contains S. s (x), W u (x), π su ), for each ξ ∈ B (y) and for each Due to the continuity of (x, g) 7→ (Wε,g δ ε,g x,g (n)

g ∈ U there exists a path contained in πgsu (Eε,g (ξ)) ∩ Bρ (x) that reaches N , hence cutting αg . Therefore the 2-dimensional disc Bδ (y) ⊂ ACg (x) for all g ∈ U . Due to the continuity of (x, g) 7→ πgsu (x), there is an n-dimensional ball Bδ′ (y) such that Bδ′ (y) ⊂ (πgsu )−1 (Bδ (y)) for all g ∈ U . Finally, y is connected to x by a finite su-path x0 = x, x1 , . . . , xn , such that xi ∈ σi s (x), W u (x)), Wf,ε (xi−1 ), with σi = s or u, i = 1, . . . , n. Then, by continuity of (x, g) 7→ (Wg,ε g,ε we have that σ1 σ2 (Wg,ε (. . . (W σn (Bδ′ (y))))) ⊃ Bε (x) Wg,ε contains a ball Bε (x) for all g ∈ U if ε > 0 and U are sufficiently small. This implies that ACg (x) ⊃ Bε (x) 6. Further description of accessibility classes In this section, we give further description of accessibility classes. Let us recall that φ : M → 2M is lower semicontinuous if for any open set W ⊆ M the set of x ∈ M such that φ(x) ∩ W 6= ∅ is open in M . The following is a direct consequence of Lemma 3.1: Proposition 6.1. The function x 7→ AC(x) is lower semicontinuous. Proof. Let W be an open set, and let x be such that y ∈ AC(x) ∩ W 6= ∅. Take Ux and h as in Lemma 3.1, such that h(Ux ) ⊂ W . Then for all w ∈ Ux , h(w) ∈ AC(w) ∩ W 6= ∅. Now V = (π su )−1 (Ux ) is an open neighborhood of x, and for all z ∈ V , AC(z) = AC(π su (z)), and AC(π su (z)) ∩ W 6= ∅. Proposition 6.2. Let K ⊂ Γ(f ) be a minimal set. Then there exists fn → f in the C r -topology, such that fn |K = f |K and ACfn (x) is either a codimension-one immersed manifold or an open set for all x ∈ K. If AC(x) were open for some x ∈ K, then all AC(ξ) with ξ ∈ K would be open, since every orbit is dense, and the orbit of any ξ would eventually fall in AC(x). Proposition 6.2 makes use of the following: Lemma 6.3 ([14, 4]). The set of points that are non-recurrent in the future {z : z ∈ / ω(z)} is dense in every leaf of F u . Lemma 6.4. Let K ⊂ Γ(f ) be a minimal set. Then there exists fn → f in the C r -topology, such that fn |K = f |K and Ffn ,ε (x) 6= {x} for some x ∈ K.


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The proof of Lemma 6.4 is Lemma 6.1. in [4]. Indeed, what it is proved there is that the joint integrability at any x ∈ K can be broken by a small C r push outside K. This yields openness of the accessibility class of x of the perturbed system in the case dim E c = 1, but only implies Fε,fn (x) 6= {x} in the case dim E c = 2. Lemma 6.5. If Fε,f (x) 6= {x}, then there exist an open neighborhood U (x)of x and a C 1 -neighborhood U (f ) of f such that Fε,g (y) 6= {y} for all g ∈ U . Also, there exists µ > 0 such that diam (Fε,g (y)) > µ for all g ∈ U . Proof. If Fε,f (x) 6= ∅, then there exists z ∈ Dx such that z ∈ πfsu ((π su )−1 (x)). Let Bρ (z) ⊂ Dx such that x ∈ / Bρ (z). Due to the continuity of (x, f ) 7→ (Wfs (x), Wfu (x)), there exist an open neighborhood U (x) of x and a C 1 -neighborhood U (f ) of f , such that Fε,g (y) = πgus ((π su )−1 (y)) ∩ Bρ (z) 6= ∅ for all g ∈ U and all y ∈ U ; therefore, Fε,g (y) 6= ∅ for all g ∈ U and all y ∈ U . Also diamFε,g (y) > µ > 0 for all g ∈ U and y ∈ U , due to continuity. References [1] A. Avila and M. Viana. Stable accessibility with 2-dimensional center. preprint. [2] M. Brin and Y. Pesin. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat., 38:170–212, 1974. [3] Aaron Brown. Smoothness of stable holonomies inside center-stable manifolds and the C 2 hypothesis in Pugh-Shub and Ledrappier-Young theory. arXiv preprint arXiv:1608.05886, 2016. [4] K. Burns, F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Talitskaya, and R. Ures. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete Contin. Dyn. Syst., 22(1-2):75–88, 2008. [5] Ph. Didier. Stability of accessibility. Ergodic Theory Dynam. Systems, 23(6):1717–1731, 2003. [6] Dmitry Dolgopyat and Amie Wilkinson. Stable accessibility is C 1 dense. Astérisque, (287):xvii, 33–60, 2003. Geometric methods in dynamics. II. [7] M. Hirsch, C. Pugh, and M. Shub. Invariant manifolds. Springer-Verlag, Berlin, 1977. Lecture Notes in Mathematics, Vol. 583. [8] J.-L. Journé. A regularity lemma for functions of several variables. Rev. Mat. Iberoamericana, 4(2):187– 193, 1988. [9] Anatole Katok and Viorel Nitica. Rigidity in Higher Rank Abelian Group Actions. Cambridge University Press, 2009. [10] Charles Pugh and Michael Shub. Stable ergodicity and partial hyperbolicity. In International Conference on Dynamical Systems (Montevideo, 1995), volume 362 of Pitman Res. Notes Math. Ser., pages 182–187. Longman, Harlow, 1996. [11] Charles Pugh and Michael Shub. Stable ergodicity and julienne quasi-conformality. J. Eur. Math. Soc. (JEMS), 2(1):1–52, 2000. [12] Charles Pugh, Michael Shub, and Amie Wilkinson. Holder foliations. Duke Mathematical Journal, 86(3):517–546, 1997. ˇcepin. C 1 -homogeneous compacta in Rn are [13] Duˇsan Repovˇs, Arkadij B. Skopenkov, and Evgenij V. Sˇ 1 n C -submanifolds of R . Proc. Amer. Math. Soc., 124(4):1219–1226, 1996. [14] F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures. Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle. Invent. Math., 172(2):353–381, 2008. [15] F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures. A non-dynamically coherent example on T3 . Ann. Inst. H. Poincaré Anal. Non Linéaire, 33(4):1023–1032, 2016.

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[16] Federico Rodriguez Hertz. Stable ergodicity of certain linear automorphisms of the torus. Ann. of Math. (2), 162(1):65–107, 2005. [17] Federico Rodriguez Hertz, Maria Alejandra Rodriguez Hertz, and Raul Ures. A survey of partially hyperbolic dynamics. partially hyperbolic dynamics, laminations, and teichm¨ uller flow, 35–87. Fields Inst. Commun, 51, 2007. [18] Amie Wilkinson. The cohomological equation for partially hyperbolic diffeomorphisms. Astérisque, (358):75–165, 2013. Jana Rodriguez–Hertz, Department of Mathematics, Southern University of Science and Technology of China. No 1088, xueyuan Rd., Xili, Nanshan District, Shenzhen,Guangdong, China 518055. E-mail address: [email protected] ´ squez, Instituto de Matema ´ tica, Pontificia Universidad Cato ´ lica de ValCarlos H. Va ´ n, Valpara´ıso-Chile. para´ıso. Blanco Viel 596, Cerro Baro E-mail address: [email protected]