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3. See Section 10.6.2 below. An extension of
3.2 Persistent tangencies, coexistence of attractors
33
this construction, in [68], also gives coexistence of infinitely many non-trivial compact invariant sets where the dynamics is uniquely ergodic (an adding machine). See Section 10.6.3. It was recently remarked by Bonatti, Moreira that adding machines are also present, as quasi-attractors 5 , in the unfolding of homoclinic tangencies in dimension 2. This is because the unfolding of a homoclinic tangency creates small disks strictly invariant by some iterate and containing new homoclinic tangencies. By iterating this procedure one obtains a nested sequence of disks, the intersection of which contains the adding machine. In addition, it is interesting to point out that the phenomenon of coexistence of periodic attractors has been extended by Colli [133] for strange attractors. Indeed, he constructs open sets of diffeomorphisms containing dense subsets for which there are infinitely many coexisting Henon-like attractors. Observe that the argument must be much more delicate than in the periodic case, because Henon-like attractors are persistent in a measure-theoretical sense only (see Theorem 4.4). Conjecture 3.9 ([426, 341]). In the unfolding of a homoclinic tangency associated to an area-dissipative (sectionally dissipative in higher dimensions, see Section 3.5), the set of parameter values corresponding to coexistence of infinitely many periodic or Henon-like attractors has Lebesgue measure zero. Similarly for quasi-attractors. Also very interesting, Theorem 3.6 has been extended to conservative maps by Duarte [171, 172]: any area preserving map with a homoclinic tangency is C2 approximated by an open domain in the space of area preserving maps exhibiting persistent tangencies; generic diffeomorphisms on such a domain have infinitely many "independent" elliptic islands (which take the role of attractors or repellers in the non-conservative case). Another striking property of Newhouse's domains with persistent tangencies has been discovered recently by Kaloshin: super-exponential growth of the number of periodic orbits. Indeed, he proves Theorem 3.10 (Kaloshin [228]). Let U C Diff2(M2) be an open set with persistent tangencies. Given any sequence (j]n) —> oo there exists a residual subset oflA for which the number of periodic points of period n is larger than 7]n for all large n.
In a few words, Kaloshin exploits the fact that diffeomorphisms with highly degenerate saddle-nodes (high contact with the identity along the central direction) are dense in the Newhouse domain hi (Gonchenko, Shil'nikov, Turaev [193]), to produce abnormal rates of growth of the number of periodic 5
A quasi- attract or is a decreasing intersection A = nnAn where each An is a topological attractor, that is, the maximal invariant set in an open neighborhood Un such that the closure of f(Un) is contained in Un .
34
3 Homoclinic Tangencies
points. Quite in contrast, Hunt, Kaloshin [221] announce that the number of periodic points of period n is bounded by const exp(n 1+5 ), for a "full probability" subset of diffeomorphisms.
3.3 Hyperbolicity and fractal dimensions We are going to see that a refinement of the previous analysis leads to a deep connection between frequency of hyperbolicity in the unfolding of a homoclinic tangency and fractal dimensions of invariant sets. Let {ffj,)fj, be a smooth parametrized family of diffeomorphisms such that f — f0 has a quadratic homoclinic tangency q associated to a periodic point p, and this tangency is generically unfolded as the parameter \i varies: the stable manifold and the unstable manifold of the continuation p^ of p move with respect to each other with non-zero velocity near the point q, so that the tangency gives rise to a pair of transverse intersections when /i > 0. As before, we suppose that p is part of a hyperbolic basic set iJ, which may or may not be reduced to the orbit of p. We are going to suppose that \i — 0 is a first bifurcation, meaning that /M is uniformly hyperbolic for every \x < 0 close to zero. This is for simplicity only: the conclusions hold in general, restricted to the dynamics of f^ related to the homoclinic tangency of / . More precisely, in the general case one must refer to the maximal invariant set
Z(UW,)
(3.2)
inside a neighborhood U of H union a C/i-neighborhood V^ of the orbit of tangency. See [345, Chapter V]. In the first bifurcation case the non-wandering set fi(fn) is contained in the union of E^ with a finite number of hyperbolic basic sets, so that the diffeomorphism is uniformly hyperbolic if E^ is. As we have seen, hyperbolicity is directly related to existence of intersections, or almost intersections, between the Cantor sets if* := if*. , * = s,u. Let us again look at the way these Cantor sets vary with the parameter /i. Refer to Figure 3.1. The stable foliation T^ and the unstable foliation T^ may be chosen such that their tangent bundles vary i n a C 1 fashion with \i. Then so does the curve of tangencies i^. Wefixparametrizations of these curves, jointly C1 in phase space and parameter space. Of course, we may choose the point of Wu(p^) D i^ close to the tangency to lie at the origin of the coordinates induced in t^ by this parametrization. The main effect on the variation of the two Cantor sets comes from the generic unfolding of the tangency: K^ is translated with respect to K™ when the parameter varies, so that their convex hulls become linked when /i > 0. With our choices of coordinates, and rescaling the parameter \i is necessary, K^KU
and K£ & Ks + fj.
3.3 Hyperbolicity and fractal dimensions
35
for all ji close to zero. Thus, K™ and K^ are close to intersecting each other if and only if Ku and Ks + \i are close to intersecting each other. In this way, one is led to studying the arithmetic difference Ku - Ks = {fi e R : /i = KU - KS for some KU G KU and K3 G KS}
= {fj, e R : Ku H (Ks + /i) is non-empty}. And Problem 3.1 directly points to: Problem 3.11. Let K1 and K2 be two Cantor sets in the real line. How big is their arithmetic difference K1 — K21 The question is of special interest in the case of dynamically defined Cantor sets, that is, arising in connection with hyperbolic sets of diffeomorphisms in the way described in Section 3.1 (a formal definition will appear later). A very precise answer to this problem may be given in terms of metric invariants of K1 and K2. To begin with, it follows from the gap lemma 3.2 that if r(Ki)r(K2) > 1 then the arithmetic difference has non-empty interior. More refined statements are in terms of fractal dimensions, like the Hausdorff dimension and the limit capacity (box dimension). Let X be a compact metric space and N(X, s) denote the minimum number of ball of radius e > 0 required to cover X. The limit capacity of X is c(X) = inf{d > 0 : N(X,e) < s~d for every small s > 0}. Now we define the Hausdorff dimension of X. Given any d > 0 the Hausdorff d-measure of X is
md(X) = lim (inf ] T diam(f/)d J where the infimum is taken over all coverings U of X by sets U with diameter less than e > 0. It is easy to see that there is a unique number 0 < HD(X) < oo, the Hausdorff dimension of X, such that md(X) = ooifd HD(X).
Detailed information about these notions can be found in [178, 353] and [345, Chapter IV]. It is easy to show that HD(X) < c(X), and 0 < HD(X) < c{X) < d for every compact subset of a d-dimensional manifold. Moreover, HD(y)
< Z/^HDPO
and
c(Y) < v~
if Y is the image of X by a z/-H61der continuous transformation. In particular, both dimensions are preserved by bi-Lipschitz homeomorphisms. Since stable and unstable holonomies are C1, in this 2-dimensional setting, it follows that
36
3 Homoclinic Tangencies
c(Kp = c(W?oc(Pll)nHti)
and HD(i^) = HD(W£c(pM) n Hj
(3.3)
and analogously for K™ and Wf^p^) n H^. In fact, our dynamical sets are much more regular than arbitrary Cantor subsets of R. In particular, c(K*) = HD(iT*) £ (0,1) for * = 5, u and c(# M ) = HD(F M ) = H D ( i ^ ) + H D ( i ^ ) e (0,2). The second equality follows from the fact that H^ is locally bi-Lipschitz homeomorphic to the product K^ x K™, and c(X xY)
HD(X) +
ED(Y)
(so that we have equality whenever c = HD). See [178, 345]. Moreover, the fractal dimension c(K^) = HD(i^*) varies continuously with the diffeomorphism / M , in the C1 topology. See [345, Chapter IV] and [285, 346]. The following simple lemma shows that the arithmetic difference of sets with small limit capacity is a zero Lebesgue measure set. The proof is an exercise. L e m m a 3.12. Let K1 ,K2 C R be compact sets. Then c(Kl - K2) < c{Kx x K2) < ciK1) + c(K2). In particular, if c(Kx) + c(K2) < 1 then m^1
- K2) = 0.
Starting from this observation, Palis, Takens show that if H has fractal dimension less than 1 then most values of /a close to /i = 0 correspond to hyperbolicity. The case when the set H reduces to the orbit of p had been treated in [324].
Theorem 3.13 (Palis, Takens [344]). Ifc(H) < 1 then n = 0 is a Lebesgue density point for the set H of parameter values for which / M is uniformly hyperbolic:
Conversely, Palis, Yoccoz prove that if the fractal dimension is bigger than 1 then non-hyperbolic parameters correspond to a sizable portion of parameter space near /i = 0. The statement is slightly more involved than in the case of Theorem 3.13. They consider smooth two-parameter families of surface diffeomorphisms (f^^)^^ such that (/^o)^ unfolds generically a homoclinic tangency associated to a hyperbolic set H of /o,o- For every v small the family (fij,,u)fjb unfolds generically a homoclinic tangency associated to the continuation of H; we may suppose that the tangency always takes place at /i = 0. Let £fj,}Js be the maximal invariant set of /M;I/ defined as in (3.2). Then
3.3 Hyperbolicity and fractal dimensions
37
Theorem 3.14 (Palis, Yoccoz [348]). 7/HD(#) > 1 then liminf
-
< 1, for Lebesgue-almost every v close to zero,
S
£->0
where Tiy is the set of values of /u for which /MjI/ is uniformly hyperbolic. We shall later see that the limsup is also smaller than 1. At the heart of the proof of Theorem 3.14 is the following deep result which provides a converse to Lemma 3.12. The need for the extra parameter v in Theorem 3.14 is related to the presence of the parameter A in Marstrand's theorem: Theorem 3.15 (Marstrand [286]). Let K1 and K2 be compact subsets of R with H D ( i f x ) + H.D(K2) > 1. Then for Lebesgue almost every A G I there exists a positive Lebesgue measure set T\ such that for every t £ T\ X
n (XK2 + t))>d>0,
d = HD(iT 1 ) 4- ED(K2)
- 1.
In particular, T\ C K1 — XK2 and so miK1 — XK2) > 0 for almost every A e R. This admits the following geometric interpretation. Let n\ : R2 —» M be the projection along co-slope A: n\(x\,X2) — x\ — Xx2- For almost every A the image ix\(Kl x K2) = K1 — XK2 has positive Lebesgue measure. Actually, Marstrand proves that for almost every X the projection (7r\)*(mnD) of the Hausdorff measure mnD on Kl x K2 is absolutely continuous, with L2 density. Moreover, there even exists a positive Lebesgue measure subset T\ such that for t G Tx the fiber Tr-^t) n (K1 x K2) « K1 n (XK2 +1) has Hausdorff dimension at least d. Remark 3.16. The hypothesis of Theorem 3.14 is more general than that of Theorem 3.3: thick Cantor sets always have Hausdorff dimension close to 1, in particular, r(Ks)r(Ku)
>1
=>
HD(iT) = EB(KS) -I- RD(KU) > 1.
See Palis, Takens [345, Chapter 4] for a discussion of relations between thickness and fractal dimensions. Let us also mention that Diaz, Ures [161] show that saddle-node horseshoes with small Hausdorff dimension may lead to persistent homoclinic tangencies immediately after the bifurcation. Moreover, Rios [376] has extended a good part of the previous theory, including persistence of tangencies and control of hyperbolicity through fractal dimensions, to the more subtle situation of homoclinic tangencies inside the limit set (accumulated by periodic orbits).
38
3 Homoclinic Tangencies
3.4 Stable intersections of regular Cantor sets In the context of these results, Palis has proposed the following ambitious answer to Problem 3.11: Conjecture 3.17. The arithmetic difference K1 — K2 of generic dynamically defined Cantor sets either has zero Lebesgue measure or contains an interval. This conjecture was proved by Moreira, Yoccoz [316, 315], in a very strong form. From it they deduced a statement about homoclinic tangencies that improves Theorem 3.14 in more than one way. These outstanding results are the subject of the present section. Notice that the statement above can not be true for all pairs of dynamically defined Cantor sets, by [403]. A Cantor set K C M is called dynamically defined, or regular, if it is the limit set n=0
of a Markov family of contractions, that is, a family / = {fj : j = 1,..., N} of C r , r > 1, diffeomorphisms fj : Dj —» Rj between compact subintervals Dj and Rj of / = [0,1], with derivative bounded by some constant A < 1, and such that every domain Dj is the convex hull of some subset of ranges Ri. The union in (3.4) is over all admissible sequences j _ n , . . . , j _ i , that is, such that the composition is defined. The Cantor sets Ks, Ku, Wj^c(p), Wfoc(p) introduced above are regular with r > 1, that is, the contractions fj may be chosen so that their derivatives are at least Holder continuous. See [345, Chapter IV]. In what follows we fix r > 1. The space of regular Cantor sets inherits a natural topology from the Cr topology in the space of such families (with fixed number of elements N). This induces a natural Cr topology in the space /C of pairs of regular Cantor sets (K\K2). Definition 3.18 (Moreira [313]). K1 and K2 have stable intersection if for every pair (Ki,K2) G K close to \K\,KQ) we have K\ n K2 ^ 0. Translates of a regular Cantor set K are also regular, and they are Cr close to K if the translation is small. So, if K1 and K2 have stable intersection or, more generally, if K1 and K2 -f t have stable intersection for some ( G 1, then K1 — K2 contains an interval. Moreira, Yoccoz prove that an open dense subset of pairs of regular Cantor sets whose sum of the Hausdorff dimensions is larger than 1 have stable intersection after convenient translation. In view of Lemma 3.12, and the fact that Hausdorff dimension and limit capacity coincide in this setting, this result implies the Palis Conjecture 3.17. Theorem 3.19 (Moreira, Yoccoz [316]). Let Q be the set of'(K1 ,K2) G K such that HD(iC1) -\-YiD{K2) > 1. Then there exists an open and dense subset W of f2 such that for every (K1^2) <E W the set
3.4 Stable intersections of regular Cantor sets
39
\ if 2) = {t e R : K1 and K2 + t have stable intersection) is open and dense in K1 — K2 . In addition, the complement has zero Lebesgue measure: more than that, HD^if 1 — K2) \ S^if^if 2 )) is less than 1. Let (/M)M be a smooth family of surface diffeomorphisms generically unfolding a homoclinic tangency associated to a hyperbolic set H of / = /Q. Adapting the proof of Theorem 3.19 and using results from [313], Moreira, Yoccoz show that, if the Hausdorff dimension of H is larger than 1, then persistent tangencies correspond to a definite fraction of parameters close to zero and the union of such parameters with those corresponding to hyperbolicity has full Lebesgue density at the bifurcation. More formally, Theorem 3.20 (Moreira, Yoccoz [315]). Assume HD(iJ) > 1. Then there exists an open set J\f in parameter space corresponding to persistent tangencies such that liminf —
—> 0
and
hminf —
— = 1,
where H is the set of parameters for which /M is uniformly hyperbolic.
The key to proving Theorem 3.19 is the fact, remarkable in itself, that for most pairs of regular Cantor sets the same geometric pattern is repeated over and over again at arbitrarily small scales. More precisely, Moreira, Yoccoz introduce a family of renormalization (or scale refinement) operators 1la,6i a n d prove that, up to small perturbation of the Cantor sets, this family admits a compact invariant set. In the sequel we are going to describe the main steps in the proof in more detail. We begin by explaining the meaning and giving the precise definition of the renormalization operators. 3.4.1 Renormalization and pattern recurrence For simplicity we consider Cantor sets dynamically denned by contractions fj : / —> / defined on the whole / = [0,1]. Then, clearly, every sequence 0 = (..., #_n , . . . , 0-i) with values in { 1 , . . . , N} is admissible. Let £ be the space of such sequences. For every n > 1 denote fg = fe_n-o • • • o fg_1. This is defined also when 0 is a finite sequence with length at least n. Endow £ with the metric d(M') = length (/^(I)), (3.5) where (5 = (0-k, • • •, 0-i) if 0-i — 0'-i for 1 < i < k and k is maximum. Given regular Cantor sets (if, /) and (if, / ) , sequences 0 and 0, with values in { 1 , . . . , iV}, and given integers m and m, the interval f™(I) is a connected component of the ra:th stage of the construction of if, and analogously for if, /, (9, m. Let a and a be two finite sequences of length k and fc, respectively,
40
3 Homoclinic Tangencies
with values in { 1 , . . . , N}. Roughly, the renormalization operator is meant to assign
Ka,a : (f^(I)jf(I))
~ (/£ + V),/£ +£ (/)) after rescaling.
Here 6a and 6a represent the concatenations of the corresponding pairs of sequences. Observe that the intervals on the right hand side are connected components of deeper stages of the construction of the two Cantor sets. For the formal definition one takes a kind of limit when ra, rh go to oo. For each 6 G £ and n > 1 let Kg be the orientation preserving affine map from fo'(I) to the interval /. The limit diffeomorphism
exists for every 6 G £ and depends Holder continuously on 6, because our maps are differentiable and the derivatives are Holder continuous. The images KQ(K) of K under the maps KQ are the limit geometries of the Cantor set. A direct important consequence of the definition is that for any sequence a with length k > 1, the set ne(f*(K)) = KG(K) D «* (/*(/)) is the affine image of another limit geometry, namely Kecx{K). For our purposes it is important to keep track also of relative positions, which are forgotten by the quotient by the affine group involved in the definition of limit geometries. The relative position of two oriented intervals Ji, J2 is the pair (A, t) G M* x R defined by h(J2) = \h(J1)+t, where h is the orientation preserving affine map sending J\ onto [0,1], and A < 0 if the orientations of J\ and J2 are opposite. Given any finite sequences a, a, the associated renormalization operator is defined by naA
: £ x £ x E* x R -> £ x E x R* x E,
(9,6,\,t)^>
where (Ai ,ti) is the relative position of K>e(fa(I)) Figure 3.3.
an
(6a, 0a, Xi , £1)
d ^§(
X1k§a (K)
Fig. 3.3. Renormalization operator 7£ttja
3.4 Stable intersections of regular Cantor sets
41
L e m m a 3.21. Suppose there exists a compact subset £ of £ x £ x R* x R such that for every (0, 0, A, £) G C there exist a, a such that
Then Ke{K) and \K§(K)+t have Cr stable intersection 6 for all (0, 0, A, t) £ C. The idea of the proof is the following: by successive renormalizations one constructs a sequence {9^n\9^n\ \^n\ t^) inside the compact set C. In particular, the sequence t^ is bounded: the intervals involved are at bounded distance, in terms of relative positions. Since the diameters are going to zero as n increases, this really means that these intervals get closer and closer. In this way we get a common point of accumulation, and this point is in the intersection of the two Cantor sets. The problem is now how to find such a recurrent set C as in the hypothesis of Lemma 3.21, for every pair of Cantor sets (K,K) in some open and dense subset W of Q. Density is the main issue, of course. Fix e > 0. We are going to explain that a recurrent set does exist, after some ICte-perturbation of the original pair. The main difficulty comes from the fact that the behavior of the renormalization operators along the variable A is mostly neutral; in contrast, the operators are strongly expanding along the t variable, corresponding to the rescalings involved in the definition. To see this, observe that the renormalization operator involves a rescaling, where the size £ of the selected components (see the overlapping intervals in Figure 3.3) is normalized to approximately 1. Under this rescaling, the effect of translations is great amplified: dti/dt w e~x. On the other hand, A acts by homothetic transformation, which commutes with the rescaling, and so this variable is essentially unaffected by the operator: Ai w A. For this reason, the construction is carried out in two steps. 3.4.2 The scale recurrence lemma The first step is to find a compact recurrent set C* for the renormalization operators acting on £ x £ x R*, that is, taking only the scale variable A in consideration. This is handled by the following crucial scale recurrence lemma. Choose a set £(e) of finite sequences a, with variable lengths ka, such that length ( / ^ (/)) « e and the intervals f^a (I) cover K. Choose £{e) analogously for K. Taking £ small,
Assume at least one of the Cantor sets, K say, is essentially non-affine: the corresponding expanding transformation ip is C2 and there exist #i, 92 £ £ and x £ K such that (K^1 O KQ2)"(X) ^ 0. The scale recurrence lemma says 6
For any r > 1 such that the regular Cantor sets are Cr.
42
3 Homoclinic Tangencies
that given a large A > 0 and a family {LQ $ : (0, 0) G E x Z1} o/ subsets of '—A, A] with
LebQ-A, A] \ L^~) < 1 /or all (0, 0) G T x S, there exists a family {CQ § : (0,6) G E x i7} o/ compact sets, where each CQ § is near LQ z , such that L e b ( [ - A A] \ Cg>§) < 1 for all and for any A G JCQ § there is a positive fraction for which nai&(9,9,\)€mt(CeaM).
(9,6)e£x£, 7
of pairs (a, a) G E(s) x Z(^) (3.6)
To use this lemma, one considers sets LQ Q as follows. Notice that (the are diffeomorphisms, and so preserve dimensions)
= ED(K) 4and so Marstrand's theorem applies to (KO^K), RQ(K)). Then, for almost every A, the TTA-projection down to R of the Hausdorff measure on the product of these Cantor sets has L 2 density. Consider B ^> A ^> 0 and let L0 g be the set of A G [—A, A] for which the L2 norm is less than B. Using the CauchySchwarz inequality, we get that B2Leb(7TX(Y)) > mnD(Y)2
for every measurable set Y
(3.7)
and any A G L0 Q . This relation will be important in the second step. Then £* = {(#, 0, A) : A G Ce0 and (0,0) G E x Z1} is a compact recurrent set for the renormalization operator in Ex E x ~R*. The conclusion (3.6) contains A e £e,e = ^ Ai G Ceaj§&. Observe that Ax « A because length (AC^(/£*(/))) w e « length (^~(/~ 5 (/))). This corresponds to the fact we already mentioned that the renormalization operator acts in a neutral fashion along the A variable. Remark 3.22. The scale recurrence lemma implies the following beautiful formula: 1 + K2) = min{l, whenever K1 and K2 are regular Cantor sets, with K1 being C2 and essentially non-amne. See Moreira [312]. 7
More precisely, at least
C£"
HD
(^)- H D (^) o f them.
3.4 Stable intersections of regular Cantor sets
43
3.4.3 T h e probabilistic argument The hypothesis of non-affinity above is the first condition in the definition of the open dense set W in Theorem 3.19. The remaining (much less explicit) conditions correspond to additional perturbations to ensure, from £*, the existence of an invariant set C for the full renormalization operator acting on E x E x R* x R. Let (6,0, A) G £*. By Marstrand's theorem 3.15, there exists a positive measure set TQ Q X of values of t for which the Hausdorff dimension of Ke(K)n (\k§(K)-\-t) is at least d = HD(iT) + ED(K) - 1. Since length ( / ^ ( / ^ ( / ) ) ) « e, this implies (fix any d < d) that there are at least ce~d pairs (a, a) G E(e) x E(e) for which the corresponding intervals intersect. We shall refer to these as distinguished pairs and distinguished intervals. To be more precise, we satisfy also the conclusion (3.6) of the scale recurrence lemma. The reason we can do this is, essentially, the following. The pairs given by the recurrence lemma cover a definite fraction of the product Cantor set. Using (3.7) we see that this covered subset projects to a positive Lebesgue measure subset and, in fact, Marstrand's argument may be carried out within this subset. So we may pick cs~d distinguished pairs from among those (a, a) given by the scale recurrence lemma. Very roughly, we now take C£ to be the set of (0,0, A, t) G E x E x W x R such that t is in the ^-neighborhood of Te § A. We want to prove that this is a recurrent set: given any (0,0, A, t) G C£ there is a distinguished pair (a, a) such that nat&(9,0,X,t)eCe/2 (3.8) at least after a lOs-perturbation of the Cantor sets. To this end, we embed the expanding transformation -0 of K into a parametrized family with p w e~dl2 parameters with values in [—10s, 10s], such that varying one parameter causes a corresponding interval ^e(f^a(I)) to move with respect to the associated K§(f&& (/)), while keeping the other p— 1 distinguished intervals unchanged 8 . This displacement causes the relative position of the pair of intervals to move across a whole interval, of uniform size, along the t-direction. Noting that the two intervals intersect for the initial Cantor sets (K, K), we conclude that the probability (that is, the Lebesgue measure in parameter space) that 7Za,a £ Ar/2 is bounded below by some uniform r\ > 0. Since we are dealing with p w e~dl2 distinguished pairs, and the behavior of each one is ruled by a different parameter in an independent fashion, the probability that (3.8) fail for all these pairs is bounded above by 8
Such a family exists with C 1 bounded norm. The argument extends to the Cr topology, any r > 1, considering only p « s" 1 /^* 1 ) distinguished intervals and parameters.
44
3 Homoclinic Tangencies
Summarizing these steps, given any (9,8, A,£) G E x Z1 x R* x R the majority of values of the parameter gives rise to Cantor sets (K^K) such that 7Za,& is in the interior of C£ for some distinguished pair (a, a). We still need to find some parameter value such that this holds for all (8, 9, A, t) simultaneously. This uses the following pair of observations. On the one hand, the previous estimates have some room for accommodating variation of the point. Using continuity of the renormalization operators 7^a? 0 this means that
for any z G l (when a = 0 the limit is the Dirac distribution). See Appendix E. The starting point in the strategy to prove Theorem 4.29 is to identify a set AQ C A and a return map fR : Ao —> AQ , x \-+ fR(x\x), such that •
Ao has hyperbolic product structure and its intersection with every unstable leaf is a positive Lebesgue measure subset of the leaf,
80
•
4 Henon-like Dynamics
fR preserves this product structure and admits a Markov partition with count ably many rectangles.
These notions will be explained below. From fR : AQ —* AQ one constructs a Markov tower extension F : A —> A for / over U n >o/ n (/\o) which is essentially a uniformly hyperbolic transformation (on a non-compact space). One constructs an SRB measure fip for F and proves that (F, up) has exponential decay of correlations. The corresponding statement for (/, /i) is a fairly easy consequence. Moreover, these arguments provide an L2 control on the correlation sequence as in (E.14), so that Gordin's Theorem E.9 may be applied to get the central limit property.
Tu
Fig. 4.7. Product structure and Markov property
Before giving a more detailed outline of the proof, let us mention that Young [460] has isolated a small set of properties under which these arguments go through, and observed that similar "horseshoes with positive measure" satisfying those properties can be found in other situations, so that exponential decay of correlations can then be proved along similar lines. See Appendix E. We also mention that a different type of Markov extension, closer in spirit to the inducing method of Jakobson [227], has been introduced by Palis, Yoccoz [349] in a related situation, to study geometric and ergodic properties (including geometric invariant measures and speed of mixing) of non-uniformly hyperbolic horseshoes formed in the unfolding of homoclinic tangencies. See Section 4.6. A "horseshoe" with positive measure: Hyperbolic product structure (see Figure 4.7) means that A$ coincides with the set of intersections between the leaves of a pair of transverse laminations, Ts andJ^ n , such that •
the leaves of Ts are exponentially contracted by all forward iterates; analogously for Tu and backward iterates.
4.4 Decay of correlations and central limit theorem
81
A subset of AQ is an s-subset if it coincides with the union of entire sets 7 n AQ over some subset of stable leaves 7 G P . There is a dual notion of u-subset. By Markov partition we mean a family of pairwise disjoint s-subsets Ai covering almost all of AQ in the sense that ra7(Z\o \ ^%Ai) = 0 inside every unstable leaf 7 G Tu (m1 stands for arc-length), and satisfying also: there exist Ri G N such that the return time R(x) = Ri on each Ai and the image fRi(Ai) is a u-subset; the map fRi sends the stable leaves through Ai inside stable leaves, and its inverse has a dual property for the unstable leaves through the image. See Figure 4.7. For the construction of AQ (see [57, Section 3] for precise statements) one takes Tu consisting of escaping leaves (free segments of unstable manifold at distance « 5 from their binding critical points), and obtains Ts through Lemma 4.23. That is, the orbits of points of AQ approach the critical set at some bounded exponential rate: dist(/ n (z), (n) > pn
at every free return,
where (n is the binding critical point and p ^> b. Notice that there is some resemblance to the construction of the X set in Section 4.3.1. One difference is that AQ does not consist of entire unstable leaves, just positive measure subsets. Perhaps even more relevant, the return map fR has a much more complicated definition presently and, in particular, it needs not be the first return map. However, it is important that it does share the fast returns property (iii) in Section 4.3.1: •
the return time R has exponential tail: there are C > 0 and A < 1 such that ra7({:r G 7 : R(x) > n}) < CXn
for every n > 1 and 7 G Tu.
(4.12)
A few other properties follow from the construction and are needed for the sequel of the argument: backward iterates do not distort the Jacobian along unstable leaves 7 G Tu too much; • the stable lamination Ts is absolutely continuous relative to the Lebesgue measure (arc-length) on unstable leaves; • the greatest common divisor of the values taken by R(x) is equal to 1. The last property means that, if n denotes the projection along the stable lamination between two unstable leaves 71 and 72 , then 7r*ra7l is absolutely continuous with respect to ml2 . A technical, yet relevant point, is that, rather than Riemannian distance, it is convenient to consider on each unstable leaf a dynamical distance f3s^x^ where /3 < 1 and s(-, •) is the separation time: •
s(x,y) = min{n > 0 : (fR)n(x)
and (fR)n(y) are in different Markov rectangles}.
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4 Henon-like Dynamics
Markov extension and consequences: The next step is to construct a Markov extension for / restricted to the invariant set A* = Un>o/n(Z\o)- In general, by an extension of a given map f : A* -* A* one means another map F : Z\ —>• A together with a (surjective) projection IT : Z\ —> A* such that TTOF = f on. Benedicks, Young [57] propose a tower extension, that is, A is a disjoint union of copies of subsets of A* , such that the return map to the ground level of the tower corresponds to the return map fR. More precisely, they define A = {(x,£) <E Ao x N : R(x) > £} = ( J ({a; : #(x) > ^} the projection TT : Z\ —> A* by 7r(x,t) — f£(x), and the map F : Z\ —» Z\ by / fl (x),0) if iJ(z)
=i+l
The ground level is {0,0) : R(x) > 0} = Ao x {0}. Observe that if / R were the first return map to Ao then n would be bijective, and so A would be naturally identified with A* . This observation helps to understand why we need the extension in the first place: we want to be able to treat points fn(x) returning to Ao prior to time R{x) as "different points" (e.g. to consider different Riemannian structures on their tangent spaces and those of their iterates) even if they happen to coincide as points in A* C M. With this flexibility, one can introduce a Riemannian structure on A relative to which F is uniformly hyperbolic (with countably many Markov branches). The property of exponential decay of return times ensures that this structure has finite volume. Now classical arguments to prove exponential mixing (see Bowen [86]) may be adapted to F, as follows. In simplified terms (see [57, Section 6], [460, Section 3], and Appendix E for more information), one considers the map F + : Z\+ —> A+ induced by F in the quotient space A+ of A by the stable lamination. This F + is a uniformly expanding Markov map, and one proves that the corresponding transfer operator £, relative to Lebesgue measure 2 on Z\+, has a spectral gap. It follows that F+ has an SRB measure /i + absolutely continuous with respect to Lebesgue measure, and the system (F + ,/i + ) has exponential decay of correlations in the space of Holder functions. Then the same is true for F : A —> A, the corresponding SRB measure fip being absolutely continuous with respect to Lebesgue measure along unstable leaves. Then \i — ?r*/fp is an /-invariant probability in A* absolutely continuous with respect to Lebesgue measure along unstable manifolds. By uniqueness, \x must be the SRB measure of the Henon-like map. This also proves that A* is a full /i-measure subset of the 2 There is a well defined notion of Lebesgue measure in the quotient space, up to equivalence, because the stable lamination is absolutely continuous.
4.5 Stochastic stability
83
attractor A. Now, every Holder function cp : A* —> M. lifts to a Holder function on Z\, via (p — i > y? o TT. It follows that the system (/,/i) has exponential decay of correlations in the space of Holder functions, as claimed in Theorem 4.29.
4.5 Stochastic stability Despite being metrically persistent in parameter space, Henon-like attractors are very fragile under small perturbations of the map. A good measure of this is given by the result of Ures [434] stating that the parameter values exhibited in the proof of Theorem 4.3 are approximated by other parameters for which the map has a homoclinic tangency associated to the fixed point P. Consequently, under arbitrarily small perturbations one may cause the closure of WU(P) to contain infinitely many periodic attractors! Quite in contrast, Theorem 4.30 below asserts that these systems are remarkably stable under small random noise. Let us explain this first in intuitive terms. Consider a point z close to the attractor, and suppose as we iterate it a mistake is made at each step: one uses some nearby diffeomorphism fj in the place of / . Thus, instead of the orbit fn(z) one obtains a random orbit Z, fl(z),
(/2o/l)(4
. . . , (/nO.-.O/!)(*),
....
The theorem says that, assuming the mistakes are small and independent, the statistical behavior of typical random orbits (that is, for almost all choices of fj and almost every z) is essentially the same as that of typical genuine orbits: for any continuous function cp the time average 1 n ~1 lim — V^ 0} where each ve is supported in the ^-neighborhood of / relative to the C2 topology. This means that we consider iterates
3
This is sometimes seen as evidence that computer pictures of Henon attractors are accurate, despite calculation errors and the chaotic character of such systems (but round-off errors and the like are not strictly random).
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4 Henon-like Dynamics
where each fj is picked at random, independently of all previous choices, according to the probability distribution v£ . Throughout, e will be smaller than the distance from f(U) to the complement of U, so that every random orbit starting in U remains there for all times. The process is described by a Markov chain in [/, with transition probabilities p£(- \ z) defined by pe(E\z)
=
Ve{{g:g{z) c\ > 0 such that e-2
< dpe(',z) dm
^ £ _2
^
for
z and £
(in dimension d, replace e~2 by e~d). In fact, the next theorem is valid for much more general noise (see [53, Section 1.5]). Theorem 4.30 (Benedicks, Viana [53]). Let f be a Henon-like map, \i be its SRB measure, and {ye : e > 0} be a random perturbations scheme whose transition probabilities satisfy (4.14). Then there is a unique stationary measure \ie supported in the basin B(A), and it is ergodic. Moreover, as e —> 0 the measure \i£ converges to \i in the weak* topology: j cp d\i£ —> J (pd\± for every continuous cp : U —> M. Uniqueness also implies that, for any continuous function (p : U —* K. and almost every random orbit, 1 n~1 j=o
f = / J
So, the conclusion of the theorem does give (4.13) for small e. In what follows we outline the main steps in the proof of Theorem 4.30. An upper bound on random iterates of measures: The claim that the stationary measure fi£ is unique is relatively easy, relying on the fact that / | A is transitive and the transition probabilities p£(- \ z) are supported on a whole neighborhood of the corresponding f(z). In order to prove the main statement, \i£ —* \i as e —> 0, let us introduce the random transfer operator T£ , defined in the space of probabilities in U by
4.5 Stochastic stability
Te{r,){E) = JPe{E
85
| z)dr,{z) =
A measure is stationary if and only it is fixed by T£ . Moreover, every accumulation point of the sequence n~1 Yl^Zo 1~J (rj) is a stationary measure, for any initial measure 77. So, since the space of probabilities is compact for the weak* topology, uniqueness of the stationary measure proves that
U
j=0
converges to \i£ as n —» 00, regardless of the initial measure 77 one takes. Fix 77 = normalized arc-length on some segment of WU(P). The main technical step in the proof is the following upper bound: 00
/* (Vn I {e(0 > 4 ) + Me,n,N +
fle,Jv(0-
(4-16)
s=0
for every £ > 0 , n > l , i V > l , where e(-) denotes the escape time for the unperturbed map / (recall the explanations following (4.9) above, and see also [53, Section 4.3] for a formal definition) and •
the A£?n are measures on the X set (Section 4.3.1), absolutely continuous on unstable leaves and with density and total variation bounded by C; • the M£jriiN are measures on the attractor A, with total variation bounded by Ce~ck; • the R£JN a r e positive functional in the space of C1 functions which, for each fixed iV > 1, converge to zero pointwise when e —•> 0. Here c and C denote various constants, respectively small and large, independent of £, n, N. Given measures a and j3 and a positive functional r(-) in the space of C1 functions, we write a < /3 + r(-) to mean that there exists a third measure 7 such that j(E) < f3(E) for all measurable set E and I
cpda —
(pd^f < r((p)
for every cp G C1.
A simple example that illustrates the way we use this terminology: Let /3 be normalized arc-length on some line segment £, and a be normalized area on the rectangle whose larger axis is £ and with smaller axis of radius e. Then a < (3 + r£(-) where r£(-) converges pointwise to zero when £ —> 0. Shadowing of random orbits: Let us explain briefly where (4.16) comes from. The basic idea is to introduce a notion of escape time for random orbits (see [53, Section 5.7]) and to split the expression (4.15) into three parts:
86
4 Henon-like Dynamics
(i) The mass carried by random orbits (z, j \ , . . . , fj) at escape times. One key ingredient in the whole proof is to show that such orbits are shadowed by genuine orbits in the attractor which are also at escape times. Thus, one proves that this part of \i£^n is bounded by
for some A£jn absolutely continuous along escaping leaves and positive functional r£(-) that converges pointwise to zero as e —> 0. Notice that A£jn is supported inside A which, of course, is not the case for \i£^n . The functional is there, precisely, to take care of that: the fact that r£(cp) converges to zero, for every C1 function (/?, depends on the fact that the support of /i£;Tl converges to A (uniformly in n) when e —> 0. (ii) The mass carried by random iterates in between escape times, with a cut-off iV from the previous escape time. Since all the maps are £-close to / , from (i) and a continuity argument we get that the 5th iterate is bounded by f:(Xs,n | {e(-) >s} + re)+pe>a(-) < f!{X£,n I {e(-) > s}) + re,s where, for each fixed s > 1, the functional p£jS and r£^s converge pointwise to zero as s —* 0. (iii) The mass carried by random orbits at iterates > N from the last previous escape time. Another crucial fact is that the total mass of random orbits with escape time larger than iV decays exponentially. So this remainder term in JJL£^ has total mass less than Ce~cn. Putting this together with the previous estimates one gets (4.16). Proving stochastic stability: Now we explain how Theorem 4.30 can be deduced from (4.16). Firstly, making n —» oo along a suitable subsequence, we get that [i£^n accumulates on the unique stationary measure \i£ and •
X£i7l accumulates on some measure A£ on 1 , absolutely continuous along unstable leaves with density and total bounded by some C > 0 • M£ n JV accumulates on some measure M£ N with total mass bounded by Therefore, (4.16) leads to fi£ < ^2 f*Xe + M^N + RNA')
for
all £ > 0 and N > 1.
Keeping N fixed and making e —> 0 along a suitable subsequence of any given sequence, we get that \i£ accumulates on some measure /io and
4.6 Chaotic dynamics near homoclinic tangencies
87
•
X£ accumulates on some measure A on X, absolutely continuous along unstable leaves with density and total mass bounded by C; • ME JV accumulates on some measure M^ whose total mass is less than Ce'cN. We also have i2/v,e(-) —> 0, pointwise. Therefore, the previous inequality gives oo
Mo < ] T /* (A I {e(-) > s}) + MN for all iV > 1. s=0
The limit measure /XQ is necessarily invariant, see [242, Theorem 1.1]. Finally, making N go to infinity, we obtain that CO
^o s}).
The measure Ao needs not be invariant, of course. On the other hand, it is absolutely continuous along unstable manifolds in the attractor A (because A is). It follows that the /-invariant measure /io is absolutely continuous along unstable manifolds of A. Since there exists a unique such probability measure [57], this /io must coincide with the SRB measure \i of the attractor. This finishes our outline of the proof of Theorem 4.30. As in most other situations where stochastic stability could be established, the previous arguments require much more detailed information on the dynamics of the unperturbed system than seems reasonable in view of the conclusion. At this point one may hope to obtain much more abstract statements of the following form: Problem 4.31. Is every Henon-like attractor supporting a unique SRB measure, hyperbolic and with the no-holes property, always stochastic stable?
4.6 Chaotic dynamics near homoclinic tangencies We conclude this chapter with a brief discussion on the occurrence of chaotic behavior and Henon-like attractors within certain bifurcation mechanisms. 4.6.1 Tangencies and strange attractors The behavior of Henon-like maps is very much characterized by the presence of critical regions hosting homoclinic phenomena, as we have seen. The next result means that a kind of converse is also true: the unfolding of homoclinic tangencies is also accompanied by the creation of Henon-like attractors. A periodic point is called sectionally dissipative if the product of any pair of eigenvalues is smaller than 1 in absolute value. This implies that the unstable manifold is 1-dimensional.
88
4 Henon-like Dynamics
Theorem 4.32 (Mora, Viana [297, 438]). For an open dense subset of C3 families (g^)^ of surface diffeomorphisms going through a homoclinic tangency at ILL = 0, there exists a positive Lebesgue measure set of parameters fi close to zero for which g^ has some Henon-like attractor or repeller. The same is true in arbitrary dimension, assuming the periodic saddle involved in the tangency is sectionally dissipative. This is a corollary of Theorem 4.4 together with the observation that any parametrized family of diffeomorphisms going through a homoclinic tangency associated to a dissipative saddle point "contains" a sequence of Henon-like families. More precisely, return maps of g^ to appropriate domains close to the tangency, and for certain parameter intervals close to ii = 0, are conjugate to Henon-like maps as in (4.2). Let us explain this in some detail. Let g^ : M — > M , / i G M , b e a smooth family of diffeomorphisms such that go has a homoclinic tangency associated to some fixed (or periodic) saddle point p. We assume the family to be generic, in the sense that the tangency is quadratic and is generically unfolded as the parameter /i varies. For the time being we take M to be a surface and go to be area-dissipative at p, meaning that | detDgo(p)\ is strictly smaller than 1, but we shall comment on the general case near the end of this section. Theorem 4.33. There is I > 1 and for each large n there exist • •
an affine reparametrization /i n : [1/4,4] —» R and Cr systems of local coordinates # n?a : [—4,4]2 —> M , 1/4 < a < 4
such that /i n ([l/4,4]) —> 0 and #n,o([—4,4]2) converges to the homoclinic tangency, and the sequence fn^a(x,y) = 6~*a o gn~^!a\ o 6na{x,y) converges to the 2 map ipa : (x, y) *—» (1 — ax ,0) in the Cr topology as n —> +oo. Here r > 2 is a fixed integer and convergence is meant in the strongest sense: as functions of (a, x, y), uniformly on [1/4,4] x [-4,4] 2 . The choice of this domain is rather arbitrary: uniform Cr convergence holds on any compact subset of M+ x M2. Figure 4.8 helps understand the contents of this result: the image of 0njO is a small domain close to the point of tangency, that returns (close) to itself after / -f n iterates, at least for the parameter values in the image of \in. Up to n-dependent blowing ups for the domain and the parameter range, given by 6n^a and /i n , the family of return maps converges to the quadratic map as n —-> oo. Let us describe the definition of nn and 0 n>a . For technical simplicity we assume that there exist Cr //-dependent coordinates (£, rf) on a neighborhood U of p linearizing g^ for /i small (this is a generic condition, and [387] shows how it can be removed): where a = a^ and A = AM satisfy |cr| > 1 > |A|. The dissipativeness assumption means that 0 < |<JA| < 1 for \i close to zero. Up to a convenient rescaling
4.6 Chaotic dynamics near homoclinic tangencies
89
Fig. 4.8. Renormalization in homoclinic tangencies
of the coordinates, we may take the set {(£, 77) : ||(£, 77) || < 2} to be contained in U and q — (1,0) to be a point in the orbit of tangency. Fix I > 1 such that 9o(o) = (0,770) G U and write #^(£,77) as (a(£ - I) 2 4- /?(£ - l)/i + 7/x2 -f 677 + v/i + r, 770 4- c(f - 1) + ^77 4- w\i 4- s) where a, /3,7, 6, c,d,v,w
G l and r = r(/x, ^,77), 5 = s(//, £, 77) are such that
r, s, Dr, Ds, % r , 9 ^ r , aMMr
all vanish at (0,1,0).
(4.17)
The hypotheses of nondegeneracy and generic unfolding of the tangency amount to having a / 0 and v ^ 0 and, up to reparametrizing (gM)M, we may suppose v = 1. First we consider the n-dependent reparametrization (4.18)
v =
It is easy to check that, given any large constant A, for n sufficiently large vn maps a small interval In close to \i = 0 diffeomorphically onto [—A, A]. We let p,n — (z^nl-fn)"1; in what follows we always take \i = \in{y). Then we introduce (n, i/)-dependent coordinates (x, y) given by (4.19)
= 0ni1J(x,y) = with p — ^\G\\. calculation gives
NOW
we denote fn^
fn,v(x, y) = (ax2 + j3x(ann) 4-
= 6n^ o g7^1 o ^n?zy. A straightforward 2n
r 2nrr, V ) 2 + ^ 4- bpny 4- cr ^ ± wpn(anfi) ± pncrns)
where ± is the sign of aX and r, s are taken at (//,£, 77) = Note that p n -> 0 and
(£Ln(is),0nitJb(x,y)).
90
4 Henon-like Dynamics anfi = (1 + (J~nv - ban\nf]o) -> 1.
as n —>• +oo. On the other hand, |<j2nr|, \pnans\ also converge to zero, as a consequence of (4.17) and, recall (4.18) and (4.19), \lM\(a) = - a / a - /?/(2a) + /? 2 /( 4< ^) ~ 7Thus, we may take nn(a) — \±noi/ and 0n^a = 0n^^ oha. Clearly, the domain of definition contains [1/4,4] x [—4,4]2, if A is taken large enough. Let us mention some extensions of this procedure. If | det Dgo(p)\ > 1 then, obviously, g^1 is area-dissipative at p and so we get the same conclusions as before, just by iterating backwards. The area-conservative case | det Dg^\ = 1 is particularly interesting: a reasoning similar to the one above may still be applied, and the limit is the conservative Henon family i/ja(x, y) = (1 — ax2 -fy, ±x), see [171, 387]. Moreover, the arguments above also extend in a natural way to higher dimensions when go is sectionally dissipative at p. In this case we get ipa(xi1X2,. • . , # m ) = (1 —ax 2 ,0,... ,0) as limit model, see [347, 438]. In fact, a more sophisticated restatement of this scheme is valid in any dimension for generic families unfolding a homoclinic tangency (with no dissipativeness conditions), see [387]. Using Theorem 4.32, Fornaess, Gavosto [184] concluded that the Henon family (4.1) contains Henon-like attractors for parameters close to the values a = 1.4 and b = 0.3 initially considered in [210]. These attractors have a local character, in the sense that they are fixed by some large iterate of the map. Henon's original assertion remains open: Problem 4.34. Is there a global strange attractor (containing the unstable manifold of the fixed point P) in the Henon family, for parameter values close to a = 1.4 and b = 0.3?
4.6.2 Saddle-node cycles and strange attractors Critical saddle-node cycles [325] are another main global bifurcation mechanism, closely related to homoclinic tangencies. By a saddle-node cycle of
4.6 Chaotic dynamics near homoclinic tangencies
91
a diffeomorphism we mean a finite family of periodic points p0 , p\ , ..., Pi-i , pi = po such that po is a saddle-node, pi is a hyperbolic saddle for i = 1, ...,£— 1, and every Wn(p£_i) has a transverse intersection with Ws(pi). It is assumed that the saddle-node is sectionally dissipative, that is, all hyperbolic eigenvalues are less than 1 in absolute value. Then, its stable set is a manifold with boundary (the strong-stable manifold of the saddle-node), with the same dimension as the ambient space, and it admits a unique strongstable foliation of codimension-1 having the strong-stable manifold as one of the leaves. The cycle is called critical if the unstable manifold of Wu{pi^\) is tangent to the strong-stable foliation of po at some point (then the same is true for Wu(po)). Figure 4.9 describes the formation of a 1-cycle, through the collision of a saddle and a sink.
Fig. 4.9. A saddle-node cycle at the boundary of Morse-Smale diffeomorphisms
Building on Theorem 4.32, Diaz, Rocha, Viana [159] proved that Henonlike strange attractors are always a prevalent phenomenon in the generic unfolding of critical saddle-node cycles: this bifurcation always includes the unfolding of homoclinic tangencies and, hence, the creation of Henon-like attractors, the key novelty being that these attractors occur for a set <S of parameters with positive Lebesgue density at the saddle-node cycle bifurcation. We shall see in the next section that this is not true for homoclinic tangencies. In the case of 1-cycles the results of [159] take a more global form: for an open set of cases, one can give an explicit characterization of the strange attractor, that ensures that its basin contains a fixed neighborhood of the cycle: Theorem 4.35 (Diaz, Rocha, Viana [159]). There exists an open set of 1-parameter families (g^)^ unfolding a critical saddle-node 1-cycle at \i — 0 such that there exists a set S of parameters with
and a neighborhood U ofWu(po) such that the maximal invariant set s a Henon-like attractor for every /i £ S.
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4 Henon-like Dynamics
We just make a few informal comments on the proof, to explain how prevalence comes about. As part of the definition of the open set in the statement, one requires that Wu(po) C Ws(po). The starting point is to define a partition Ik = (/ifc+i , Hk] of some interval (0, s] in parameter space, a domain C in the ambient manifold, and a return map R^ : C —> C, given by some iterate g1^ with I « fc for all fi G Ik- This domain is bounded by two strong-stable leaves such that one is mapped to the other; by identifying points in the same orbit inside these leaves, the domain C becomes a cylinder and R^ becomes a smooth map. Most iterates involved in R^ take place near the saddle-node, and so this return map is strongly volume dissipative. In fact, up to a convenient /c-dependent reparametrization \i = ijjk{a), a G [0,1), the families R^ , jji G Ik , converge to a family of maps of the cylinder C ~ S1 x B, of the form
Va-.S1 xB^S'xB,
&a(0,y) = {${9) + a,0),
where B is a ball in Euclidean space and (/> : S1 —> S1. The definition of the open set in the statement involves, mostly, conditions on the circle map <j> to ensure that these R&k(a) are Henon-like families in the sense of Section 4.1. Now prevalence follows from two main observations. Firstly, there is a uniform lower bound for the Lebesgue measure of the sets of parameters a corresponding to Henon-like attractors in each of these families: roughly, it corresponds to the measure of the set of parameters a for which 0 — t > 1. However, Costa [136, 137] has been able to prove an analogue of Theorem 4.35 for saddle-node horseshoes, that is, saddle-node cycles where the saddle-node is part of the homoclinic class of some hyperbolic saddle. We refer the reader to [154] for a survey of this and other recent results on this topic.
4.6.3 Tangencies and non-uniform hyperbolicity Concerning prevalence of strange attractors, the situation should be very different for homoclinic tangencies: Problem 4.36. Let ( ^ ) M be a generic one-parameter family of diffeomorphisms unfolding a homoclinic tangency, as in Theorem 4.32. Does the set <S of parameter values corresponding to Henon-like attractors or repellers have zero Lebesgue density at \i — 0? The density does vanish in the context of Theorem 3.13 (Hausdorff dimension less than 1): the set H of parameters corresponding to hyperbolicity has full Lebesgue density at /i = 0, and the two sets S and H are clearly disjoint.
4.6 Chaotic dynamics near homoclinic tangencies
93
A recent extension due to Palis, Yoccoz [349, 350] suggests that the answer to Problem 4.36 might be always positive. The setting is as in Section 3.3. Let (9fj,)(j, unfold a homoclinic tangency associated to a periodic point p contained in a hyperbolic basic set H of go (see Figure 1.2). Let ds(H) = HD(Wi«c(p) n H)
and du{H) = HD(W£c(p) n H)
be the stable and unstable dimensions of iJ, as defined in (3.3). Palis, Yoccoz show that if the Hausdorff dimension HD(i7) = ds(H) + du(H) is not much larger than 1 then, for most parameter values close to ji — 0, the maximal invariant set E^ in a neighborhood of H union the orbit of tangency is a "nonuniformly hyperbolic horseshoe". We just quote the following consequence of their, much more detailed, statement: Theorem 4.37 (Palis, Yoccoz [349, 350]). There exists a neighborhood V oftheset{(dSldu) : ds+du < 1} inside (0,1)2 such that if (d3(H),du(H)) eV then there exists a set W of parameters with full Lebesgue density at \i = 0 such that for every fi E W the stable set and the unstable set of E^ have zero Lebesgue measure in M and, consequently, E^ contains neither attractors nor repellers. The strategy, inspired from Benedicks, Carleson [52], is to show that for most parameters \i close to zero the set E^ is hyperbolic, in a delicate nonuniform sense. Essentially, although E^ may contain tangencies, these correspond to very special points: at the majority of points there are transverse directions which are asymptotically contracted by forward and backward iterates, respectively. As in the Henon-like case, the proof requires a careful analysis of how trajectories return close to the tangencies and, most importantly, a precise definition of what a "tangency" ("critical point") is. To ensure hyperbolicity, returns should not be too frequent nor too close. This is achieved through parameter exclusions. These exclusions are less and less significant near the original tangency parameter, and that is how full Lebesgue density of non-excluded parameters arises. The rate of formation of tangencies as one iterates is a crucial ingredient in the exclusions estimates, and is closely related to the Hausdorff dimension of the original horseshoe. The assumption that this dimension is not much bigger than 1 ensures that the number of tangencies that must be considered at each stage grows fairly slowly (exponential growth with exponent not too big), so that a fairly small fraction of parameters needs to be excluded each time. Returns close to the tangencies yield quadratic type folds. The condition on the frequency and depth of returns is used to ensure that the folds always are "ironed-out" before a new very close return occurs. In this way, one never has to deal with folds of order larger than 2. One can certainly expect to relax the hypothesis on the Hausdorff dimension, at the price of having to deal with higher order folds, and it may even be that the conclusion remain true in all cases:
94
4 Henon-like Dynamics
Conjecture 4.38. The conclusion of Theorem 4.37 remains true for generic unfoldings of homoclinic tangencies by surface diffeomorphisms (with no assumption on the Hausdorff dimension). On the horizon lies the case of area-preserving maps, like the conservative Henon maps (6 = ±1), or the family of standard maps on T 2
for which the limit set (the whole ambient space) has dimension 2. The main dynamical issue in these maps is, once more, recurrence of criticalities. In fact, in this case one will probably have to deal with folds of all orders simultaneously. Notice that for K = 0 the standard map is an integrable twist map: the circles in the direction (1,1) are invariant and rotated by the map, with rotation angles varying monotonically with the circle. At this point the topological entropy vanishes and, consequently, so does the entropy relative to Lebesgue measure. Moreover, gK has elliptic islands for all small K. The most important problem in this domain is Problem 4.39. Is there a positive Lebesgue measure set of values of K for which the standard map gK (i) has positive entropy 4 relative to Lebesgue measure? (ii) has no Kolmogorov-Arnold-Moser elliptic islands? (iii) is ergodic and non-uniformly hyperbolic (non-zero Lyapunov exponents)? In the direction of this problem it should be useful to handle first the following simplified model, inspired from [52] and [440], that we are going to describe. In order to motivate this model, let us informally recall a bit of the previous discussion. We have seen that control of the recurrence of tangencies (critical orbits) through parameter exclusions very much depends on the growth of the number of critical points one has to consider at each scale. The latter is related to fractal dimensions of the limit set, which is, itself related to the conservative /dissipative character of the dynamics: for strongly dissipative Henon-like maps the dimension is 1 + £, whereas for the conservative standard map it is 2. Our model lives on the product of an interval by a Cantor set. Geometric aspects of the dynamics correspond to the interval direction. The fractal dimension of the Cantor set is a free parameter, that allow us to interpolate between strongly dissipative and conservative cases. More precisely, let M — [—1,1] xK(b) where K{b) is the mid-(l — b) Cantor set in [0,1]. Let CL(K) = a -f <j){n) where a is a parameter and <j> is some small Lipschitz function (e.g. / = 0). Consider the transformation / : M —> M given by . _ , J{X,K) 4
-
(1
, . a{K)x
2
x ,«ij,
f bn/2 Avi -
<j
1
j_
if x < 0 / O i f
x
> Q
The topological entropy is known to grow as CK,1^3. See [171], for instance.
4.6 Chaotic dynamics near homoclinic tangencies
95
That is, / sends every half horizontal [—1,0) x {^} and [0,1] x {/^} inside some horizontal [—1,1] x {fri} in a quadratic fashion, with a critical point at x — 0. The points (l,ft), K G K(b) play the role of critical values. Let f denote the derivative along the horizontal. Let 6^ be the uniformly distributed probability on K(b). Problem 4.40. For any fixed b G (0,1), is there a positive Lebesgue measure set of values of a such that the derivative (/ n y (1, n) grows exponentially with n for ^-almost every K G K^? IS the growth rate uniform? The answer is positive when b w 0, by a much simplified version of the arguments in [52].
Non-Critical Dynamics and Hyperbolicity
As we have seen in the previous chapters, much of the richness of the dynamics of surface diffeomorphisms is due to phenomena associated to the unfolding of homoclinic tangencies, as in the Henon family. Moreover, these phenomena have many analogies with critical endomorphisms of the interval or the circle. For instance, homoclinic tangencies correspond in the one-dimensional setting to pre-periodic critical points. For non-critical maps in dimension 1 which are sufficiently regular, one has good control of the distortion and, thus, a lot of rigidity: Theorem 2.5 asserts that non-critical maps are almost hyperbolic. Here we discuss some remarkable recent results of Pujals, Sambarino [371, 372, 370] extending this conclusion to non-critical surface diffeomorphisms, that is, such that one cannot create homoclinic tangencies by small C1 perturbations of the diffeomorphism. In this two-dimensional setting, the main tool involved in avoiding tangencies is the concept of dominated splitting of the tangent bundle. One key result states that an invariant compact set admitting a dominated splitting and whose periodic orbits are all hyperbolic coincides with the union of a hyperbolic set with finitely many periodic normally hyperbolic circles on which the dynamics is an irrational rotation. See Section 5.1 for the precise statement and Section 5.2 for an outline of the proof. The main result is deduced in Section 5.3, and Section 5.4 presents another very nice consequence: variation of entropy is always accompanied by homoclinic tangencies. Finally, in Section 5.5 we discuss a few results and conjectures concerning non-critical behavior in higher dimensions.
5.1 Non-critical surface dynamics Let M be a closed Riemannian surface, / : M —> M be a diffeomorphism, and A be an /-invariant set. A Df-invariant splitting TXM — E{x) 0 F(x), x £ A of the tangent bundle over A is called dominated if there is m G N such that
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5 Non-Critical Dynamics and Hyperbolicity
\Dfn | £ ( » | • \{Dfn | Fix))'1] < 1/2 for all x e A and n > m. See Appendix B for much more information about this notion. In general terms, the results in this section show that surface diffeomorphisms far from homoclinic tangencies have dominated dynamics: the limit set admits a dominated splitting. Moreover, dominated diffeomorphisms on surfaces behave very much like non-critical one-dimensional maps. Theorem 5.1 (Pujals, Sambarino [371]). Let f be a C2 surface diffeomorphism and A C fl(f) be a compact f-invariant set admitting a dominated splitting T^M — E ® F. Assume all periodic points in A are hyperbolic saddles. Then A = A\ U A Ci is an irrational rotation, for some rrii > 1.
Here is a useful global reformulation of this theorem: Theorem 5.2 (Pujals, Sambarino [372]). Let f be a C2 surface diffeomorphism whose periodic points are all hyperbolic and which can not be C1 approximated by diffeomorphisms exhibiting homoclinic tangencies. Then the non-wandering set may be decomposed into compact f -invariant sets, i?(/) = i?i(/) U i?2(/), such that i?i(/) is hyperbolic and i?2(/) ^s the union of finitely many pairwise disjoint normally hyperbolic circles C i , . . . , C& such that / m i (C^) = Q and / m ? : : Ci —> Ci is an irrational rotation, for some rrii > 1. In particular, Lebesgue almost every point of M is in the basin of attraction of either a hyperbolic attractor or a normally attracting periodic circle. More recently, Pujals, Sambarino [370] extended these results by removing the assumption that all periodic points are hyperbolic: / / the limit set L(f) admits a dominated splitting then the periods of the non-hyperbolic periodic orbits of f are bounded above. Moreover, L(f) = L\ U L2 U £3, where L\ is the union of finitely many pairwise disjoint homoclinic classes, each class containing at most finitely many non-hyperbolic periodic points; L2 is the union of finitely many normally hyperbolic circles on which a power of f is a rotation; L3 consists of periodic points contained in a finite union of normally hyperbolic periodic intervals. Compare Mane's Theorem 2.5. A very interesting by-product of the arguments leading to the proof of this last result is the construction in [370] of surface diffeomorphisms having coexisting hyperbolic periodic orbits with normalized eigenvalues arbitrarily close to one, and such that there is no C2 perturbations breaking the hyperbolicity of any of these points. By Frank's lemma (Theorem A.8), this phenomenon cannot occur in the C1 topology.
5.2 Domination implies almost hyperbolicity
99
5.2 Domination implies almost hyperbolicity To prove Theorem 5.1 it is enough to see that, under the hypotheses, either A is hyperbolic or it contains a periodic circle where the dynamics of a power of / is conjugate to an irrational rotation. In fact, this implies the theorem, after noting that the number of periodic circles of / must be finite. The main step of the proof is Proposition 5.3. Consider a C2 surface diffeomorphism f and a nontrivial transitive f-invariant compact set £ admitting a dominated splitting TJJM — E 0 F. Suppose that every proper compact invariant subset of £ is hyperbolic, and £ does not contain normally hyperbolic periodic circles C such that fm(C) = C and the restriction of / m to C is conjugate to an irrational rotation, for some m > 1. Then £ itself is hyperbolic. Assuming this proposition we now sketch the proof of Theorem 5.1. Suppose A is not hyperbolic. Consider the family JC of compact /-invariant subsets of A which are not hyperbolic. The intersection of any decreasing sequence of non-hyperbolic invariant compact sets is also a non-hyperbolic invariant compact set. So, by Zorn's lemma, the family /C has some minimal element £. By hypothesis, every periodic point of / in A D £ is hyperbolic. Hence, £ cannot be trivial (a union of periodic orbits). We claim that £ is a transitive set. Thus, since it is also minimal, £ verifies all the hypotheses of Proposition 5.3. It follows that it is hyperbolic, which is a contradiction. To see that £ is a transitive set one uses the following useful alternative characterization of hyperbolicity, which is an analogue of Lemma 2.6 in higher dimensions: Lemma 5.4. Let £ be an f -invariant compact set having a dominated splitting TEM = E®F such that \Dfn(x) \ E(x)\ -» 0 and \Df~n(x) \ F(x)\ -> 0 as n —> oo for all x G £'. Then £ is a hyperbolic set: E = Es is uniformly contracting and F — Eu is uniformly expanding. We are going to show that there exists x G £ such that either £ = UJ(X) or £ = a(x): which implies transitivity. Indeed, suppose the cj-limit set of a point x G £ is properly contained in £. Then, by minimality, UJ{X) is a hyperbolic set. It follows that Dfn(x)(v) —> 0 for every vector v of E(x). In the same way, arguing with a-limit sets, Df~n(x)(v) —» 0 for every vector v of F(x). If this were valid for every x G £, Lemma 5.4 would imply that £ is hyperbolic, which is a contradiction. So, there is some x G £ such that either UJ(X) or a{x) is the whole £. Let us now present some main ingredients in the proof of Proposition 5.3. A first one is a Denjoy type property about intervals transverse to the indirection. Existence of a dominated splitting T^M = E © F is equivalent to existence of forward invariant and backward invariant cone fields containing F and E1, respectively, and defined on some neighborhood V of the set £ (see
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Appendix B). Then we say that a curve / contained in V is transverse to the E-direction if the tangent direction at every point x G / is contained in the corresponding forward invariant cone. Fix an open neighborhood U of E whose closure is contained in V. Let E+ be the maximal forward invariant set of / in the closure of U. By construction, E+ has a dominated splitting and it contains E. We call a curve / a (J, E)interval if it is contained in i7 + , all its forward iterates are transverse to the indirection, and length (/ n (/)) < S for every n > 0. Proposition 5.5. There is S > 0 such that the LJ-limit set CJ(I) of every (5, E)-interval I satisfies one of the following two possibilities: • •
either LJ (I) is a normally hyperbolic periodic circle C where the restriction °f fm f771 is the period of C) is conjugate to an irrational rotation, or LO(I) is contained in the set of periodic points f in the neighborhood V of A.
By definition, CJ(I) is the union of the sets uo{x) for x G / . The Denjoy type statement implies that if the set A in Theorem 5.1 does not contain normally hyperbolic periodic circles then w(I) is contained in the set of periodic points, for every (5, £")-interval / . Another key ingredient is to control center stable and center unstable manifolds. Existence of invariant families of curves W£s(x) and W£s{x) tangent to the subbundles E and F follows from general normal hyperbolicity theory [216]. In the present setting these curves are of class C2. Most important, they are dynamically defined: for every x £ E there exists e(x) > 0 there and fn(W^x)(x)) go to zero as is such that the lengths of f~n(W^x)(x)) n -» oo. In a second stage one proves that
E-length (f- (W^ (x))) n
x)
n=0
< oo and
] T length (fn(W*{x)(x))) < oo n=0
for every x in an open subset of U. One deduces that \Df n(x) \ F(x)\ —» 0 and \Dfn(x) \ E(x)\ —> 0 as n —> oo, for every x £ E. In view of Lemma 5.4 this gives the hyperbolicity of E. The proof of these facts uses ideas from the proof of Theorem 2.5, together the Denjoy-like Proposition 5.5 and the following property of the unstable, or stable, separatrices of hyperbolic points p of E: there is K such that if a separatrix L of Wu(p) \ {p} has length less than ft then the end-point of L other than p is either a sink or a non-hyperbolic saddle.
5.3 Homoclinic tangencies vs. Axiom A An important consequence of Theorem 5.1 is that every diffeomorphism / that cannot be approximated by diffeomorphisms exhibiting homoclinic tangencies can be approximated by an Axiom A diffeomorphism:
5.3 Homoclinic tangencies vs. Axiom A
101
Theorem 5.6 (Pujals-Sambarino [371]). Let M be a closed surface. Then there is a dense subset V o/Diff1(M) of diffeomorphisms f such that either f has a homoclinic tangency or f satisfies the Axiom A. The proof has two main parts. The first one is the construction of a dominated splitting defined on the non-wandering set of every diffeomorphism g in an open and dense subset of a neighborhood of / . The second step is to prove that such a splitting is hyperbolic. The latter is done by approximating the diffeomorphism by a C2 one, in the C1 topology. This allows us to use Theorem 5.2 to decompose the dynamics into two parts, one corresponding to irrational rotations and another which is hyperbolic. Let us now comment both parts in more detail. Consider the set Vi defined as the complement in Diff1(M) of the closure of the diffeomorphisms exhibiting homoclinic tangencies. Theorem 5.6 just claims the density of the Axiom A diffeomorphisms in hi. Let H be the dense subset of Kupka-Smale diffeomorphisms / £ U. In particular, every periodic point of / G H is hyperbolic. For / G W, we let E{f) be the set of (possibly infinitely many) sinks and sources, and 4?o(/) be the complement of S(f) inside the non-wandering set f2(f). Clearly, i?o(/) is /-invariant and compact. The next step is to see that i?o(/) n a s a dominated splitting. To prove this, one first observes that every point x G 4?o(/) is approximated by sequences (pn)n such that each pn is a hyperbolic periodic point of a diffeomorphism gn arbitrarily C1 close to / in the C1 topology. This follows from the closing lemma, Theorem A.I. The key point here is that the natural splitting over the set of periodic points, given by the hyperbolic directions, is dominated uniformly in gn, and so extends to the closure. The proof of this key fact is based on Franks' lemma (Theorem A.8) and arguments from the proof of the stability conjecture in [283]. In a few words, there are two cases. First, if the angles between the stable and unstable are of the periodic points pn may be taken arbitrarily small, then we can obtain a homoclinic tangency after a small perturbation. So, this case is forbidden by the fact that / G hi. Second, if the splitting is not dominated, a small perturbation allows us to recline the unstable direction against to the stable direction. Together with lack of domination, this leads to small angles between the two subspaces, and again we obtain a contradiction. See Section 7.2.1 for more details. Now we move to the second and last part of the proof of Theorem 5.6. We only have to show that the diffeomorphisms in a dense subset of H verify the Axiom A. From the first part we know that, for every / G 7Y, the set i?o(/) has a dominated splitting and so it verifies the hypotheses of Theorem 5.1. Since existence of normally hyperbolic curves supporting irrational rotations is a non-generic, Theorem 5.1 implies that i?o(/) is a hyperbolic set, for every / in a dense subset V of H. Thus, the non-wandering set i?(/) of / G V is hyperbolic if, and only if, the set S(f) of sinks and sources is hyperbolic. The hyperbolicity of S(f) is evident if this set is finite, and so we only have to consider the infinite case. Since J?o(/) is hyperbolic, it admit a neigh-
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borhood V such every /-invariant set in V is hyperbolic. Observe that every accumulation point of sinks or sources must be in J?o(/). Therefore, all but finitely many elements of S(f) are contained in V, which is a contradiction. It follows that S(f) is hyperbolic, and this concludes the proof of the hyperbolicity of fi(g).
5.4 Entropy and homoclinic points on surfaces The topological entropy of a transformation / : M —> M is a number that, roughly speaking, measures how many distinct trajectories the transformation has. In precise terms, it is defined by (see [86]) htop(f) = lim limsup —logs(n,£), where s(n, e) is the maximum number of points with e-distinct orbits of length n. We say that x and y have ^-distinct orbits of length n if d(ji (x), p (y)) > e for some 0 < j < n. Existence of transverse homoclinic points implies the existence of horseshoes (nontrivial hyperbolic sets), and so diffeomorphisms with transverse homoclinic points have strictly positive topological entropy. Conversely, the following deep result asserts that surface diffeomorphisms with positive topological entropy exhibit horseshoes x: Theorem 5.7 (Katok [231]). Let f be a C 1 + a diffeomorphism, a > 0, on a closed surface. If f has positive topological entropy then it has hyperbolic periodic saddles with transverse homoclinic points.
In view of these results, it is a natural important problem to try and characterize the diffeomorphisms having zero topological entropy, as well as to understand the dynamical consequences of the variation of the topological entropy. It is clear that Morse-Smale diffeomorphisms have zero entropy. The following consequence of Theorem 5.6 states that these are the only systems having zero topological entropy robustly. See also Gambaudo, Rocha [189]. Theorem 5.8 (Pujals, Sambarino [371]). Let M be a closed surface and MS denote the C1-closure of the set of Morse-Smale diffeomorphisms of Diffx(M). Then there is an open and dense subset o/Diff1(M)\MS consisting of diffeomorphisms with transverse homoclinic orbits. Thus, MS coincides with the closure of the interior of the set of diffeomorphisms with zero topological entropy. generally, for diffeomorphisms in any dimension, horseshoes are dense in the support of any hyperbolic invariant measure. Positive topological entropy implies the existence of ergodic measures with positive metric entropy, by the variational principle [445]. In dimension 2 such measures are necessarily hyperbolic.
5.4 Entropy and homoclinic points on surfaces
103
Since the set of diffeomorphisms with transverse homoclinic orbits is open, we only have to show that it is dense in Diff1(M) \ MS. This is an easy consequence of Theorem 5.6. Indeed, the theorem gives that close to any / ^ MS there is a diffeomorphism g either having a homoclinic tangency or verifying the Axiom A. In the first case, we may perturb g to create a transverse homoclinic orbit, thence proving the claim. In the second one, we may assume that the non-wander ing set of g is infinite: otherwise g would be Morse-Smale, contradicting that / ^ MS. Then, since g is Axiom A, the nonwandering set of g contains nontrivial hyperbolic sets and, thus, transverse homoclinic orbits. As an immediate consequence of Theorem 5.2 one gets the following result, that generalizes a theorem for diffeomorphisms of the 2-sphere in [189]: Theorem 5.9. Let f be a C2 surface diffeomorphism with infinitely many periodic points, all of them hyperbolic. Then f is C1 approximated by a diffeomorphism with a homoclinic point. This follows observing that if / can not be approximated by diffeomorphisms with homoclinic tangencies then it verifies Theorem 5.2. Since the non-wandering set is infinite, it must contain some nontrivial hyperbolic set and, hence, transverse homoclinic points. We mention yet another important consequence of Theorem 5.2, concerning variation of the topological entropy. A Ck diffeomorphism / is a point of entropy variation if every Ck neighborhood hi of / contains diffeomorphisms g such that the topological entropies of / and g are different. In what follows we consider only the case k = oo, to benefit from the fact that the topological entropy depends continuously on the dynamics relative to the C°° topology [322, 455]. Theorem 5.10 (Pujals, Sambarino [372]). Let f be a point of entropy variation. Then f is C1 accumulated by diffeomorphisms exhibiting homoclinic tangencies. Suppose / is far from homoclinic tangencies. Let g be any diffeomorphism C°° close to / . Consider an arc (ft)te[o,i] of C°° diffeomorphisms with fo = f and /i = g, contained in a small neighborhood of / . By [421] we may take this arc such that the set S of parameter values t such that ft is not Kupka-Smale is countable and [0,1] \ S is a countable union of intervals. The assumption that / is far from homoclinic tangencies ensures that ft satisfies the hypotheses of Theorem 5.2 for every t fi S. Since invariant circles do not contribute for the entropy, one deduces that the entropy is constant on each interval in the complement of S. On the other hand, htop(ft) is a continuous function of the parameter, because the topological entropy varies continuously with the dynamics in the C°° topology [322, 455]. As S is only countable, these two facts imply that the entropy is constant on the whole parameter interval [0,1]. This proves that htop(g) = htop(f) for every g close to / , and so / is not a point of entropy variation.
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5.5 Non-critical behavior in higher dimensions Not much is known about the dynamics of invariant sets far from tangencies in higher dimensions. Recently, Wen has shown that such non-critical behavior implies the existence of a dominated splitting, thus extending the first step of the proof of Theorem 5.6 to any dimension. Let us call a point x i-pseudoperiodic for / if there exists y arbitrarily close to x and g arbitrarily close to / in the C1 sense, such that y is a hyperbolic periodic point for g with stable manifold of dimension i. Then he proves that Theorem 5.11 (Wen [449]). If f : M —+ M is a diffeomorphism on a closed manifold of dimension d then f is C1 -far from diffeomorphism with homoclinic tangencies if and only if for every 0 < i < d the set ofi-pseudo-periodic points admits a dominated splitting E 0 F with dimE = i. This, and much of the progress in this direction is motivated by the following Conjecture 5.12 (Palis [341]). In any dimension, the diffeomorphisms exhibiting either a homoclinic tangency or heterodimensional cycles are Cr dense in the complement of the closure of the hyperbolic ones, for any r > 1. Of course, surface diffeomorphisms can not have heterodimensional cycles. So, in the 2-dimensional situation the case r = 1 of the conjecture reduces to Theorem 5.6, which had also been conjectured by Palis. A solution of Conjecture 5.12 has been announced very recently by Hayashi [206]. The following weaker version of Conjecture 5.12 has also been proposed: Conjecture 5.13 (Palis [341]). The subset of dynamical systems that either have their limit set consisting of finitely many hyperbolic periodic orbits or else they have transverse homoclinic orbits, are Cr dense in the set of all dynamical systems, r > 1. In this regard, Martin-Ribas, Mora [296] prove that every C2 surface diffeomorphism that admits an almost homoclinic sequence, that is, a sequence of points that have iterates arbitrarily close to the stable manifold and to the unstable manifold of some saddle, either has transverse homoclinic points or can be C1 approximated by diffeomorphisms with homoclinic tangencies. The proof goes as follows. Under the hypotheses, Hayashi's connecting lemma, Theorem A.4, gives C1 perturbations of the original map that have transverse homoclinic orbits. The closure of these homoclinic orbits contains a non-trivial invariant set with a dominated splitting. If the domination becomes weaker when the size of the perturbation goes to zero, methods of Marie [278] permit to find a homoclinic tangency for a nearby system. Otherwise, the unperturbed map also has a non-trivial invariant set with dominated splitting. From Theorem 5.1 it follows
5.5 Non-critical behavior in higher dimensions
105
that the original diffeomorphism has a nontrivial hyperbolic set, and so it has transverse homoclinic points. Recently, Arroyo, Rodriguez-Hertz [33] extended some of the previous results to vector fields on 3-dimensional manifolds. Their main result says that every C1 vector field may be approximated by another which either is uniformly hyperbolic with no-cycles, has a homoclinic tangency, or has a singular cycle. By cycle we mean a finite family of equilibrium points and periodic orbits cyclically related by intersections of their stable and unstable manifolds. The cycle is called singular if it does involve equilibria (not just regular periodic orbits). Singular cycles are studied in Section 9.2. This result is an important step towards the case r = 1 of the following reformulation of Conjecture 5.12 for flows: Conjecture 5.14 (Palis [341]). Every vector field in a 3-dimensional manifold may be Cr approximated by another which either is uniformly hyperbolic, has a homoclinic tangency, or has a singular (Lorenz-like) attractor. More generally, in any dimension, the union of all vector fields which are uniformly hyperbolic or else have a homoclinic tangency, a heterodimensional cycle, or a singular cycle, should be Cr dense. The basis for proving the main result is a continuous-time version of Theorem 5.1: Arroyo, Rodriguez-Hertz prove that if a compact invariant set of a C2 flow on a three-dimensional manifold has a dominated splitting, contains no equilibria, and all the periodic orbits in it are hyperbolic saddles, then it may be decomposed as the union of a hyperbolic set and finitely many tori supporting an irrational flow. Let us point out that in this context, dominated splitting is defined in terms of the linear Poincare flow. See Appendix B and [33, Proposition 3.3]. We close with a few more conjectures inspired by the previous results. Partially hyperbolic systems with invariant splitting TM = Eu 0 Ec 0 Es with one-dimensional central direction Ec are, in a sense, closest to being hyperbolic among all non-hyperbolic diffeomorphisms. Hence, one may expect a description of the dynamics to be easier and more complete in this case. In this spirit, we formulate Conjecture 5.15. Generically, partially hyperbolic diffeomorphisms whose central direction has dimension 1 admit a filtration separating the nonwandering set into finitely many transitive pieces. A more ambitious formulation would require the same conclusion for partially hyperbolic diffeomorphisms with splitting TM = EU®E1®- • -®El®Ea, where every El', i = 1 , . . . , / is one-dimensional and, as before, Eu is uniformly expanding and Es is uniformly contracting. On the other hand, Conjecture 5.15 might be easier when the invariant splitting contains only two sub-bundles:
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5 Non-Critical Dynamics and Hyperbolicity
Conjecture 5.16. Let / be a C2 diffeomorphism such that the limit set L(f) has a dominated splitting E®F where F has dimension 1 and E is uniformly contracting. Then the limit set L(f) is the union of finitely many hyperbolic circles on which / is an irrational rotation, of (possibly infinitely many) periodic points contained in a finite union of periodic intervals, and finitely many pairwise disjoint homoclinic classes, each of them containing at most finitely many non-hyperbolic periodic points. This corresponds to an extension of Theorems 2.5 and 5.6 to the general codimension-1 case. The definition of dominated splitting is in Definition B.I. Crovisier [142] has obtained some progress in the direction of this conjecture, in the case when there is a unique non-hyperbolic periodic point.
6 Heterodimensional Cycles and Blenders
We call heterodimensional cycle the geometric configuration corresponding to two periodic points P and Q of Morse unstable index (dimension of the unstable bundle) p < q, respectively, such that the stable manifold of each point meets the unstable manifold of the other. In most cases we consider q = p -f-1. Such cycles were first considered by Newhouse, Palis in [323], and were systematically studied in the series of papers [151, 152, 155, 156, 157, 158, 161]. This elementary phenomenon, which occurs for diffeomorphisms in any dimension larger than 2, is one of the main mechanisms for non-hyperbolicity as we have already mentioned in Section 1.4. For instance, the homoclinic classes of P and Q may coincide, for open sets in parameter space, in which case they can not be hyperbolic. See Section 6.1. In fact, a great variety of dynamical phenomena may arise in the unfolding of these cycles, depending in particular on the geometry of the heteroclinic intersections, as we also comment at the end of that section. At the heart of heterodimensional cycles and the kinds of dynamics they yield, are certain transitive hyperbolic sets that we call blenders. This was introduced in [71], motivated by [152]. Topologically blenders are Cantor sets, but their distinctive geometric feature is that their embedding in the manifold is such that their stable set behaves as if it were of dimension higher than its actual topological dimension. More precisely, if p is the unstable index of the blender F, then the closure of the (n — p)—dimensional stable manifold of a periodic point P G F (in other words, the stable set of F) may intersect an unstable manifold of dimension p — 1 in a robust fashion. In Chapter 7 we shall use this property as a key ingredient in a semi-local argument for building robustly transitive dynamics. In Section 6.2 we present an affine model of a blender, and then explore various generalizations of this model. In Section 6.3 we show that the unfolding of heterodimensional cycles yields blenders, leading to periodic orbits of different indices being heteroclinically related in a robust way.
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6 Heterodimensional Cycles and Blenders
6.1 Heterodimensional cycles Consider a closed manifold M and a diffeomorphism / defined on M and having a pair of hyperbolic saddles P and Q with different indices, that is, different dimensions of their unstable subspaces. Assume WS(P) and WU(Q) have non-empty intersection, and the same for WU(P) and WS(Q). We say that / has a heterodimensional cycle associated to P and Q. Of course, heterodimensional cycles can only exist in dimension bigger or equal to 3. 6.1.1 Explosion of homoclinic classes The main result in this section depicts heterodimensional cycles as a powerful mechanism for explosion of homoclinic classes of periodic points and creation of robust non-hyperbolic transitive sets. The assumptions are not minimal, we shall see later that this kind of behavior is typical of a large class of cycles. For simplicity, we suppose that the periodic points P and Q in the cycle are actually fixed points. In addition, we always assume •
(codimension 1) the saddles P and Q have indices p and q = _p 4-1, respectively, • (quasi-transversality) the manifolds WS(P) and WU(Q) intersect transversely, and the intersection between WU(P) and WS(Q) is quasi-transverse: TXWS{Q) C\TXWU{P) = {0} for every intersection point x. The next theorem asserts that the homoclinic classes of P and Q often explode, and become intermingled (non-empty intersection) when the cycle is unfolded. A parametrized family (ft)t£[-i,i] °f diffeomorphisms unfolds generically a heterodimensional cycle of / = /o if there are open disks K? C Wu(Pt) and K* c Ws(Qt), depending continuously on t, such that KQ n KQ contains a point of quasi-transverse intersection, and the distance between K* and K™ increases with positive velocity when t increases. Here Pt and Qt denote the continuations for ft of the periodic points P and Q. Theorem 6.1 (Diaz [152]). There is a non-empty open set of parametrized C°° families of diffeomorphisms (ft)t€[-i,i] unfolding generically a heterodimensional cycle of f = /o ; such that for all small positive t 1. the transverse intersection between Ws(Pt) and Wu(Qt) is contained in the homoclinic class of Qt; 2. and the homoclinic class of Pt is contained in the homoclinic class of Qt. Let d be the dimension of the ambient manifold M. The key fact behind Theorem 6.1 is that, for every small positive t, the (d—_p)-dimensional manifold Ws(Pt) is contained in the closure of Ws(Qt). This makes that the stable manifold of Qt, which has dimension (d — p—1), behaves in fact as a manifold of dimension d — p, that is one unit bigger. This type of phenomenon has been synthesized in the notion of blender, which we discuss in Section 6.2.
6.1 Heterodimensional cycles
109
The proof of the theorem, which we are going to sketch next, relies on a reduction of the dynamics to a family of iterated functions systems on the interval. Besides the standing hypotheses of codimension 1 and quasitransversality above, the main conditions defining the open set in the theorem are connectedness of the transverse intersection 7, non-criticality or partial hyperbolicity of the dynamics on 7, and small distortion of the transition from the neighborhood of Q to that of P. This will be explained in the sequel, but let us point out right away that these conditions may hold simultaneously for a parametrized family ft and for its inverse ff1: one gets open sets of families for which the homoclinic classes of Pt and Qt coincide after the bifurcation. See Remark 6.3 below.
6.1.2 A simplified example To explain the ideas of the proof of Theorem 6.1 while avoiding technical difficulties, we consider here a simple situation of a cycle in M3, which displays the main features of the general case we are dealing with.
Fig. 6.1. Model heterodimensional cycle
We assume the restriction of ft to a domain C = [—1,4] x [—2, 2] x [—2,2] satisfies (see Figure 6.1): Partial hyperbolicity and product structure along the transverse intersection: (a) On some domain [—1,4] x [—2,2] x [-a, a] whose image contains some [0,3] x [—b,b] x [—2,2] the diffeomorphism ft is independent of t and coincides with the product of its restriction to the X-axis by a hyperbolic affine map, such that Y and Z are the strong stable and strong unstable directions: Z © X 0 Y is a dominated splitting for ft. (b) The points Qt = (0,0,0) and Pt = (3,0,0) are hyperbolic saddles of indices 2 and 1, respectively. The connection 7 = (0,3) x {(0,0)} is included
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(c) The restrictions of ft to the neighborhoods UQ = [— 1,1] x [—2, 2] x [—a, a] and Up = [2,4] x [—2,2] x [—a, a] are affine maps, and the eigenvalues of ft at Qt and Pt in the X-direction are j3 > 1 and 0 < A < 1, respectively. Product structure along the quasi-transverse orbit and generic unfolding of the cycle: (d) Consider the points A = (3,0,1) eWu(P0) and B = (0,-1,0) e WS(QO). There is £ > 0 such that fo(A) = B, and so A and B are heteroclinic points for f = fo- Moreover, there is a small neighborhood UA of A where /Q coincides with the translation (x,y,z) H-> (x — 3, y — 1, z — 1). (e) The restriction of / / to C/^ coincides with the translation
By the first two conditions, (0, 3) x [—2,2] x {0} is contained in the stable manifold of Pt, and (0, 3) x {0} x [—2, 2] is contained in the unstable manifold of Qt- Then A and Bt = (£, — 1, 0) are transverse homoclinic points of Pt, and At = (3 —£, 0,1) and B are transverse homoclinic points of Qt, for every small positive t. Key properties of the diffeomorphism ft may be derived from properties of the dynamics it induces in the quotient space by the sum Y 0 Z of the strong stable and the strong unstable directions. This quotient dynamics consists of an iterated functions system with two generators: the restriction of / to the X-axis, also denoted by / , and the quotient
of the restriction of / / to UA, defined for small t > 0. Since we are interested in the dynamics in a neighborhood of the cycle, it is natural to restrict / to [0,3] and gt to [3 - t, 3]. See Figure 6.2.
0
3-t
3
Fig. 6.2. Iterated functions system with 2 generators
The dynamics of the iterated functions system is, essentially, determined by the smooth conjugacy class and by the Mather invariant [288] of the interval
6.1 Heterodimensional cycles
111
map / . The former is described by the pair of eigenvalues (/3, A). Let us recall the definition of the latter. By assumption, in a neighborhood of Q = 0 the transformation / is the time-1 map of the flow defined by the affine vector field XQ(X) — (log/3) x -j^. Similarly, in a neighborhood of P — 3 the transformation / is the time-1 of the vector field Xp(x) = (log A) (x — 3) J^. For x G (0, 3) close to Q, consider n big enough so that fn(x) is close to P, and then write
for some /i(x) G M. Since the vector fields XQ and Xp are /-invariant, the function /i(-) does not depend on n, and it satisfies fi(x) = /i(/(x)) for all x. Hence, the Mather invariant fi(-) is a smooth function on the circle (0,3)//. It corresponds to what [152] calls the transition from Q to P. As part of defining the open set in the statement of Theorem 6.1, we also assume that the transition has small distortion, in other words, that \i is close to 1. The precise condition appears in Lemma 6.2 below, involving also the eigenvalues (3 and A. Observe that \x is constant equal to 1 if the restriction of / to [0,3] is the time-1 of a smooth vector field. We now sketch the proof of Theorem 6.1 for our simplified model. Denote by Dfp = [3 — t, 3 — At] the fundamental domain of / at distance t from P. Let st — f~nt (3 - t) be the unique backward iterate of 3 — t under / in the interval (/3~1t,t]. Then let DQ = [stj/?st] be the corresponding fundamental domain of / close to Q. We are going to construct a return map to this last domain, for the iterated function system. See Figure 6.3. 3-t
ht\
fnt\ D\>
Fig. 6.3. A return map to DQ
By construction, / n t (Dg) = DfP. Consider the increasing surjective map
Given a point x G DQ there exists a unique integer k(x) > 0 such that fk^ o ht(x) is in DQ. This defines a return map to the fundamental domain DfQ. Due to the jumps of fc(-), the map fk(x) o ht(x) has a sequence of discontinuities
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accumulating at the point x = /3~1st. We get rid of these discontinuities, though not the one at the accumulating point, by identifying the endpoints of the domain DfQ: so that it becomes a circle. To make this precise, consider the projection 7T: (0,1] -> S1 = R/Z,
TT(X)
= | ^ |
(mod 1).
Let ct = ft(st) = 7r(/?st), and denote by TT^T1 the inverse of TT: (st,/3 st] —> S1. The return map is given by
Rt =7Tohton-1:
S1 -> S1.
This circle map Rt has a unique discontinuity, at ct. Since ht(st) — 0 and k(x) goes to infinity when x goes to st inside Dg, we have that the derivative DRt(z) goes to +oo when z —> ct from the right. Moreover, Rt(z) "goes to —oo", meaning that the graph winds infinitely many times around the circle. L e m m a 6.2. Suppose the Mather function /i(-) and the eigenvalues X, (3 satisfy
/i(x)A|logA|
> 1
Then, for every small t > 0, the map Rt is uniformly expanding: \DRt(z)\ > 1 for all z ^ ct. This lemma implies that for any open interval / C [0,3] there is an element (/) of the iterated functions system G(f,gt) generated by / and gt such that (/) contains Q = 0. Indeed, consider the projection J = TT(/). Since the map Rt is expanding, there is n > 1 such that R™{J) contains the discontinuity ct. Then, if ip is the element of G(f,9t) corresponding to iJJ1, we have that ijj{I) / = ht o I/J and to recall that ht(st) = 0. contains st. Then it suffices to take From the previous paragraph we get that the stable manifold of Q intersects any disk A transverse to the stable manifold of P. Indeed, suppose A is a vertical disk / x {y} x [—2, 2] for some interval / C [0, t] and some y G [—2, 2]. Consider the image of A under the map <j> (extended to the ambient space): the previous conclusion means that this image intersects {0} x [—2,2] x {0}. Since this last segment is contained in the stable manifold of Q, it follows that A n WS(Q) is nonempty. For a general disk A transverse to WS(P), the assertion follows after observing that the forward orbit of A contains disks arbitrarily close to vertical disks as before. As an immediate consequence we get that the connection curve 7 = (0,3) x {(0,0)} is contained in the homoclinic class of Q. The other claim in Theorem 6.1 follows from a variation of the same argument. To finish our sketch of the proof of the theorem, we give the proof of Lemma 6.2: Proof. Consider z G S1 and let x = 7rt~1(z). Observe that DRt(z) by the relation
is defined
6.1 Heterodimensional cycles
D(gt o D(XQ(x)) Recalling that Dfnt(XQ(x)) and Xp, one has
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= DRt(z) • XQ(gt o /"*(*)).
= /j,(x) Xp(fnt(x))
and the expressions of XQ
»{x) I logA| (r*(x) - 3) = DiJt(z) (log/?)&(/"'(a:)). Since, by definition, fnt(x) e Dlp = [3 - t, 3 - At], one has (/™'(x) - 3) [-£, -At]. Thus5 t (/ n *(x)) = f ' ( i ) - 3 + t e [O,£-At]. Hence (log/3)
