Period-multiplying cascades for diffeomorphisms of ...

Math. Proc. Camb. Phil. Soc. (1994), 116, 359 Printed in Great Britain

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Period-multiplying cascades for diffeomorphisms of the disc BY JEAN-MARC GAMBAUDO, JOHN GUASCHI AND TOBY HALL Institut Non-Line'aire de Nice, Universite de Nice Sophia Antipolis, Faculte des Sciences, 06108 Nice Cedex 2, France [Received 14 October 1993) Introduction It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2' for each integer i ^ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (Pi)i>0 with per(P{) = 2* which form a perioddoubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many 'zero-entropy' cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point). It seems probable that the requirement of differentiability is essential in the twodimensional context, since it is possible for surface homeomorphisms to have positive entropy in the absence of periodic orbits [19]. In this paper we require the diffeomorphism / to be C1+e: this enables us to make use of a result of Katok[13] guaranteeing t h a t / h a s a periodic orbit whose braid type has positive entropy. The idea of the proof is to show that the canonical (Nielsen—Thurston) representative of such a braid type has infinitely many zero-entropy cascades of periodic orbits; using isotopy persistence results, it follows that so does any homeomorphism having a periodic orbit of the given braid type. We shall assume familiarity with the main ideas and results of the Nielsen-Thurston classification of isotopy classes of surface homeomorphisms [8, 20], and of its application to two-dimensional topological dynamics [5, 7]. In Section 2 we give the definitions necessary to state precisely the onedimensional result: they are presented in such a way as to make explicit the analogy with the two-dimensional theory. The natural way to discuss cascades of periodic orbits for homeomorphisms of surfaces is in terms of the braid type of an infinite collection of orbits, which is defined in Section 3: the definition is a simple generalization of the one introduced by Boyland for finite collections [5]. Having defined the notions of extension and zero-entropy cascade for diffeomorphisms of the disc, we shall be able to state our main result, and indicate how it gives rise to the corollary that positive entropy diffeomorphisms of the disc have infinitely many zero-entropy cascades. In Section 4 we prove the main theorem under the assumption

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that a particular fixed point of a certain pseudo-Anosov map is a regular point of the invariant foliations; and finally, in Section 5, we indicate the modifications required to make the proof work in the case where this fixed point is a singular point of the foliations. 2. Extensions and cascades in one-dimensional dynamics In this section we shall give some definitions which will enable us to make a precise statement of the result that interval maps with positive topological entropy have a period-doubling cascade (Theorem 2-1). We adopt a rather convoluted approach so as to prepare the ground for the introduction of corresponding two-dimensional definitions which are less well known and less obvious. We begin by showing how the familiar notion of the permutation type [2, 16] of a finite collection of periodic orbits of a map of the interval can be extended to cover infinite collections (see also [18]). We can then define a period-doubling cascade for a m a p / : / - > / to be a collection of periodic orbits whose permutation type conforms to one of a set of simple standard models. Along the way, we shall make some definitions (for example, of forcing, extensions) which are not required for the statement of Theorem 2-1: again, this is in order to introduce some of the ideas of the next section in a more familiar context. L e t / : 7X ->/j and g: I2 -*-/2 be continuous maps of two oriented copies Ix and I2 of the unit interval / = [0,1], having countable (possibly finite) collections of periodic orbits P = {Pj}jej and Q = {Q^^^K respectively. Let/ S be a closed interval containing It in its interior, and let / a n d g be (arbitrary) extensions of/ and g to I1 and 72 respectively. We say that (P,/) and (Q,g) have the same permutation type if there is a bisection fi: J^K such that for each finite subset F ^ J, there is an orientationpreserving homeomorphism ^ : / j ^ / 2 sending P^ onto Q^ for each JEF, with the property that h~1ogoh is homotopic to / relative to U^ 6F i^: this definition is independent of the details of the extensions. Having the same permutation type is an equivalence relation on the set of all possible pairs (P,/), and we write n(P,f) for the equivalence class containing (P,/), which we call the permutation type of the collection P of periodic orbits /. We write PT^, for the set of all permutation types. If P is a finite collection with a period sum n, then TT(P,/) can be described by a permutation on n symbols, which indicates the way that the points of the orbit, ordered by the ordering on the interval, are permuted b y / (thus it has one cycle for each periodic orbit in the collection). We write PT £ PT^, for the set of all such finite permutation types: by the remark we have just made, PT bijects naturally with U B ^jiS n (where Sn is the group of permutations on n symbols), and we shall make this correspondence without further comment. We shall also write Cn for the subset of Sn consisting of cyclic permutations: that is, for the set of permutation types of single period n orbits. Given p ePT^, we say that a map / exhibits p if it has a collection of periodic orbits whose permutation type is p. If cr is another permutation type, then we shall say that p forces cr if every continuous map of the interval which exhibits p also exhibits cr. It can be shown that if p e P T , then there is a 'simplest' (piecewise linear) m a p / p exhibiting p. with the property that p forces cr if and only if fp exhibits a [2]. For pePTm, the entropy h(p) of p is defined to be the infimum of the set of topological

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Fig. 1. (1 9 4 2 8 5) (3 7 6) is a simple (1 2) (3)-extension of (1. 3 2).

entropies of maps exhibiting p. If p e P T , then this infimum is attained by the entropy of fp (see for example [4]). Let reCn, and let p, crePT^. We say that a is a (simple) p-extension of T [16] if there exists a m a p / : I->I which cyclically permutes n mutually disjoint subintervals 7j,... ,In, and which has a collection P of periodic orbits contained in U 1 ^ i < n / j with n(P,f) = a, such that the following properties are satisfied: first, that for each i the set of points of the orbits of P contained in Ii has permutation type p as a collection of periodic orbits of/"| 7 : /0 the intervals I\,... ,Pn( are contained in I n t U ^ ' / J r 1 We say that a sequence P = (Pt)i>0 of periodic orbits of a m a p / : / - > / is a perioddoubling cascade (for/) if n(P,f) is a period-doubling cascade permutation type, and if the orbits are labelled so that per (Pt+1) = 2 per (Pt) for each i. It is clear that in such a cascade n{Pi+1,f) is a doubling of n(Pt,f) for each i. The cascade is said to be complete if in addition Po is a fixed point. If a m a p / : /-»•/ has topological entropy zero, then the only permutation types of single periodic orbits which it can exhibit are those which arise in complete period-doubling cascades [3]. We can now state: THEOREM 2-1 (Misiurewicz[15]) Letf: 1^1 have positive topological entropy. Then j has a complete period-doubling cascade.

3. Extensions and cascades in two-dimensional dynamics Let/:Dj->Dj and g:D2^D2 be orientation-preserving homeomorphisms of two oriented copies Dx and D2 of the 2-disc D2 = {xeU2: \x\ ^ 1}, having countable

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(possibly finite) collections of periodic orbits P = {Pj}]eJ and Q = {Qk}keK respectively. Let I)i be a disc obtained from Dt by attaching an exterior collar, and l e t / a n d g be (arbitrary) extensions of/ and g to homeomorphisms of D1 and D2. We say that (P,/) and (Q,g) have the same braid type (via/i) if there is a bijection fi: J->K such that for each finite subset F £ J, there is an orientation-preserving homeomorphism h: Dl->D2, sending Pj onto Q/t^) for each jeF, such that h~logoh is isotopic to / relative to \JjeFPj\ this definition is independent of the details of the extensions. Having the same braid type is an equivalence relation on the set of all possible pairs (P,/), and we write b t ( P , / ) for the equivalence class containing (P,/), which we call the braid type of the collection P of periodic orbits of/. If J and K are finite, then this corresponds to the standard definition introduced by Boyland[5]. Notice that if b t ( P , / ) = bt(Q,-*-ft with fo=f and fx — g). Since braid type is invariant under isotopy connection, this gives the result. (The result is false if we take a collection P which contains more than one periodic orbit on 3D2 [6].) The set P is countable (since/is pseudo-Anosov), and we write P = {Pi)i>0, where Pt c A for 0 ^ i < j , and Pi $ A for i >j (some j ^ 0). Let Qt = Pt for i sj j . We construct periodic orbits Qf of g for i > j inductively as follows: suppose that we already have orbits Qo,... ,Qk such that {Qo,...,Qk} is isotopy connected to {Po,..., Pk} by an isotop}' {/J -f^g and periodic orbits {Pl0,.... Pk}. Let {st}: Id ~ s^ be an isotopy with the property that st(P\) = Pt for each t and each i (such an isotopy can easily be constructed by piecing together local segments). Then

Period-multiplying cascades 1

is an isotopy relative to Po U ... U Pk, so that s1ogos^ which is isotopy connected to Pk+1 by some isotopy

363 has a periodic orbit Q'k+i

... U Pt) -> (D2, Po U . . .

iD^P^

[)Pk).

Thus setting Qk+1 = ^(Qt+i)' the collection {Qo,.... Qk+1} of periodic orbits of g is isotopy connected to {Po,...,Pk+1} by the isotopy {sj1 ogtost}:f~g. Now it is easy to see that bt(P,/) = bt(Q,^) via fi = Id. For if F is a finite set of non-negative integers, then let k = m a x # : we have bt({P0,...,Pk},f)

=

bt({Q0,...,Qk},g),

and hence bt(U ieF P,-,/) = bt(UisFQi,g) as required. I For /?eBTOT, the entropy h(fi) of fi is defined to be the infimum of the set of topological entropies of homeomorphisms exhibiting fi. If/?eBT then this infimum is attained by the entropy of the map/^ [7]. For each rational r/se[0,1) (in lowest terms) we write ar/s for the braid type of one of the period s orbits of a rotation i?r/s of the disc through 2nr/s: clearly h(ar/s) = 0, since h(Rr/s) = 0. Recall that in the one-dimensional theory, the only permutation types of single periodic orbits which can be exhibited by maps with zero entropy are those which arise in complete period-doubling cascades. Thus there are two elementary zeroentropy permutation types, (1) and (12): all of the others can be obtained from these by iterated extension (always by (12), since an extension by (1) is trivial). In the two-dimensional theory, the elementary zero-entropy braid types are precisely {Z) 2 with per (P() = nt, satisfying the following properties: 1- ni+1 = si+1 .nt for each i ^ 0. 2. For each i ^ 0 there exist ni mutually disjoint discs D\,... ,Dln , cyclically permuted by /, each of which contains nilni points of P} for each j ^ i. For each we have bt({P, C\Di,Pi+1 0Dk},fn 1 be the pseudo-Anosov expansion factor of/. If g:D2^-D2 is a map and yeD2, then we write o(y,g) for the forward (/-orbit of y. Since x has index + 1, it must lie in IntZ) 2 . For each a > 0, let I(a) be the segment of stable leaf centred on x of length 2a (as measured using fiu). Let JS£ and J2J be the two half-leaves of the unstable foliation which emanate from x; and let pt(a) be the first intersection of .2? with I(a) away from x (so that (x,pt(a))u 0l(a) = 0). Write Dt(a) for the disc bounded by [x,pt(a)]s U [x, Pi(a)]u. We claim that i and a can be chosen in such a way that I(a + e) fl IntDt(a) = 0 for some e > 0; that is, in such a way that we have the configuration shown in Figure 2. To see that this is possible, start with some given value a0 of a. If I(ao + e) 0 IntD0(a0) #= 0 for all small e, then we have one of the three configurations shown in Figure 3: cases (ii) and (iii) reduce to case (i) (or else to the configuration of Figure 2) on exchanging the labels of the half-leaves =S^ and =S^. In case (i), let a > a0 be as small as possible so that po(a) 4= po(aQ)- If I(a + e) f] IntDQ(a) 4= 0 for all small e, then we must have the configuration of Figure 4: thus I(a + e) (1 IntD^a) = 0 for some e > 0. Having chosen i and a in this way, we write p for the first intersection point pt(a) and D for the disc Dt(a); and we write / for the segment [f(p),p]s of stable leaf, and J for the segment [x,p]u of unstable leaf. We shall show that, given the configuration of Figure 2, then for all n sufficiently large,/has a collection of periodic orbits whose

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(ii)

Fig. 3. Other possible configurations.

Fig. 4. The case I(a + e) n Int D0(a) 4= 0 .

braid type is a /?hs-extension of an ]s traces out an anticlockwise loop, we can construct similarly a /?hs-extension of an a+1/TC-extension of a1/2. Notice that the fact that x has index + 1 means that the two half-leaves of the stable foliation emanating from x are exchanged under the action of/, as are the two halfleaves of the unstable foliation. Because J doesn't intersect (f(p),p)s, it follows that

f(J) = [x,f(p)]u and/ 2 (J) = [x,f(p)]u don't intersect (x,f(p))s.

Since/ 2 is orientation-preserving, points of/ 2 (J) sufficiently close to f2(p) lie in 2), and therefore/ 2 (J) intersects (f2(p),p)s- Further, if q is the intersection point which lies closest to p along I, then q is closer to p along I than any intersection of /(J) with (x,p)s (see Figure 5). For if/(J) meets (x,p)s, let r be the point of intersection which lies closest to p along I. Then the arcs [r,p]u and [r,p]s meet only at r and p, and therefore bound a disc A. Because/(J) doesn't intersect (x,f2(p))s, there is an arc a. (lying very close to (x,f2(p))s U J) from f2(p) to p in D which doesn't meet dA except at p; it follows that/ 2 (p)e A if and only if all of the points of/ 2 (J)\J sufficiently close to p are not in A (see Figure 6 for examples in the two cases/2(ja)eA and/ 2 (p)^A). T h u s / 2 ( J ) \ J = (p,f2(p)]u must cross dA: that is, it must intersect (r,p)s. Let S > 0 be small enough that there are no singularities of the invariant foliations within a ^-neighbourhood Jf of l\Jf(J) U/ 2 (J). Let k be a positive integer large enough that/ fc (p) is within distance 8 of a;, as measured along / using (iu: then for all e < 8/Xk, a foliation adapted rectangle R with base [q,p]s and height e satisfies

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f(p)

Fig. 5./ 2 (J) intersects (x,p)t closer to p than does/(n which is isotopic to the identity relative to the points of P n , such that kn(fZn(Rn)) = Rn. Then the homeomorphism gn = knof has the required properties. Step 2. bt(o(2 n ,/),/) is an a_1/n-extension of a1/2.

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Fig. 9. The disc

/(r

"y \U

J \U

Fig. 10. The subdisc Jfn with /(.FJ (]Fn = 0.

To see this, observe that if £ > 0 is sufficiently small, and f „ is the outer boundary of a ^-neighbourhood of Fn bounding a disc En, then En contains o(zn,f2) and is disjoint from o(f(zn),f2) by Fact 1; and/ 2 (f n ) is isotopic to fn relative to o(zn,f) by Fact 2. Now [x,f(p)]u, and hence /(F n ), can only intersect F n along yn. Thus for sufficiently small £ there is a subdisc Fn of En, obtained by pushing the boundary of En away from intersections with/(F re ), such that 8Fn is isotopic to dEn rel. o(zn,f), and f(Fn) is disjoint from Fn (see Figure 10). Thus there is a homeomorphism hn: D2->D2 isotopic to/relative to o(zn,f), for which hn(Fn) is disjoint from Fn, and h?n(Fn) = Fn. It follows that bt (o(zn,f),f) is an extension of some period 2 braid type: but the only such braid type is oc1/2. It therefore only remains to show that bt (o(zn,f2),f2) = oc._1/n. Let St be the arc 2i f (yn) for 0 ^ i ^ n - 2 . Then ^ joins (/2)'(zB.) to (/ 2 ) m (z n ); and these arcs intersect only at their endpoints. The isotopy class of/ 2 relative to o(zn,f2) is determined by the isotopy classes of the arcs f2{St) for 0 ^ % ^ n — 2 [11]: however /2(5;) =i fi(^), where i? is the rotation through -2n/n which takes f2i(zn) to fi+2(zn) for each i. I 5. Modifications for the case of a singular fixed point The proof of Theorem 3-4 in the case of a fixed point which lies at a singularity of the invariant foliations involves no new ideas beyond those employed in the proof of Theorem 4-1. In this section we shall outline the main modifications which need to be made to that proof in order to treat this special case, leaving the details to the reader.

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F i g . 11. T h e rectangles f2'(Rn)

for t h e case of figure 3(i).

Let us begin by considering the first step in the proof. In our treatment of the case of a regular fixed point we were able to choose, for each a > 0, one of the unstable half-leaves emanating from x in such a way that it first intersected I(a) in the configuration of Figure 2 or Figure 3(i): that is, in such a way that/(a) 0 IntDt(a) = 0. By increasing a if necessary, we were then able to reduce the case of Figure 3(i) to the case of Figure 2, in which I(a + e) f] IntDt(a) = 0 for some e > 0. However, another approach which we could have taken would have been to construct the cascades of periodic orbits separately for the configurations of Figures 2 and 3(i). Since this is the approach which we shall need to take in dealing with the special case, let us begin by outlining the changes which have to be made to the proof of Theorem 4-1 in order to treat the configuration of Figure 3(i). As for the case treated in Section 4, we write p for the first intersection point p{(a) and D for the disc Dt(a); and we write / = [f(p),p]s and J = [x, p]u. We again define q to be the intersection point of/ 2 (J) with (f2(p),p)s which lies closest to p along / : unlike the earlier case it is possible that/(J) ma}' intersect (q,p)s, but because of the different configuration this will not be important. The picture corresponding to Figure 7 is given in Figure 11. Notice that in this case we have to take the rectangles Rn to have base slightly wider than [q,p]s in order to create a horseshoe. The fixed point zn of/2" 1^ with index + 1 now lies on the side of Rn closer to q. In the proof that b t ( P n , / ) is a /?hs-extension of bt (o(zn,f),f), observe that it is possible now to ha,vefin~1(Rn) fl Rn 4= 0 for all n (iif(J) intersects (q,p)s). However, such an intersection is non-essential: we can find a homeomorphism /„ isotopic to / relative to the points of P n such that fln(Rn) =j'{Rn) for 0 ^ i ^ 2n — 2, but / 2 n - 1 (i? n ) C\Rn = 0. The rest of the proof goes through as before. Now let us consider the special case in which x lies at a ^-pronged singularity of the invariant foliations of/. For each a > 0, let S(a) be the star formed by the length a segments of stable leaf emanating from x. Let J^,..., ££t_x be the t half-leaves of the

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unstable foliation which emanate from z\ and let pt(a) be the first intersection of J% with S(a) away from x. Write Dt{a) for the disc bounded by [x,p((a)]s U [x,pt(a)]u. Then it is easy to show that i and a can be chosen in such a way that S(a) n IntDf(a) = 0. Having done this, we write p for the first intersection point pt{a), 0 such that S(a + e) n lx\tDt(a) = 0: the case in which no such e exists can be treated similarly. The proof now runs through in a manner very similar to that of Theorem 4*1. It is straightforward to show that/ s (J) intersects (fs{p),p)s, and that the intersection point q which is closest to p along [x, p]s lies closer to p along [x, p]s than any of the intersections of /*( J) with {fs{p), p)s for 1 ^ t ^ s— 1. As in the proof of Theorem 4-1, we construct a sequence of rectangles Rn with base [q,p]s and height e n -»0 such that Rn,fs(Rn),... ,/s("~1>(/?n) are mutually disjoint, while/ 5 " \R is a horseshoe (in choosing the neighbourhood Jf of S U/(