Last time we saw that bifurcations of quadratic tangencies associated to fat horseshoes, , are complicated because of persistent tangencies, but, by the works of C. Moreira and J.-C. Yoccoz (see here and here), one realizes that, from the heuristic point of view, the “critical locus” (i.e., the region where the tangencies destroying the hyperbolicity show up) is very small, i.e., its Hausdorff dimension is close to zero, if the initial horseshoe is only slightly fat, i.e., is close to . In particular, one could imagine that bifurcations quadratic tangencies of slightly fat horseshoes could lead to a local dynamics on satisfying some form of weak (non-uniform) hyperbolicity for most despite the fact that doesn’t verify strong (uniform) hyperbolicity conditions in general.
In their tour-de-force work (of 217 pages!), J. Palis and J.-C. Yoccoz were able to formalize this crude heuristic argument by showing (among several other things) the following result in the context of heteroclinic bifurcations of slightly fat horseshoes.
Let be a smooth diffeomorphism of a compact surface possessing a uniformly hyperbolic horseshoe displaying a heteroclinic quadratic tangency, that is, contains two periodic points and with distinctorbits such that and have a quadratic tangency (i.e., a contact of order ) at some point . Let be a sufficiently small neighborhood of and let be a sufficiently small neighborhood of . Denote by be a sufficiently small neighborhood of and, as usual, let’s organize into depending on the relative positions of the continuations of and near .
Finally, let us denote by and the stable and unstable dimensions of the horseshoe of .
Remark 1 At first sight, there is no reason to restrict our attention to heteroclinic tangencies in the previous theorem. In fact, as we’ll see later (cf. Remark 9), for certain technical reasons, the arguments of J. Palis and J.-C. Yoccoz can treat only heteroclinic tangencies. Of course, the authors believe that this is merely an artifact of their methods, but unfortunately they don’t know how to modify the proofs to also include the case of homoclinic tangencies.
Remark 2 (“Pedagogical” remark) For those who like to see videos and can understand Portuguese, it is worth to note that J.-C. Yoccoz gave a series of lectures at IMPA in 2009 around his works with S. Marmi and P. Moussa, A. Avila and J. Bochi, and J. Palis, and his lectures were recorded by IMPA’s staff. In particular, one can find links for all these materials (including some lecture notes taken by Aline Cerqueira) here.
Concerning the statement of this result, let’s us comment first on the condition (1). As a trivial remark, note that this condition includes the case of thin horseshoes, but this is not surprising as any reasonable definition of “non-uniformly hyperbolic horseshoe” must include uniformly hyperbolic horseshoes as particular examples. Of course, this remark is not particularly interesting because the case of thin horseshoes was already treated by S. Newhouse, J. Palis and F. Takes (cf. the third post in this series), so that the condition (1) is really interesting in the regime of fat horseshoes . Here, one can get a clear idea about (1) by assuming or (i.e., by breaking the natural symmetry between and ), and by noticing that the boundary of the region determined by (1)is the union of two ellipses meeting the diagonal at the point as indicated in the figure below:
In this figure, we used the horizontal axis for the variable and the vertical axis for the variable . Also, we pointed out, for sake of comparison, two famous families of dynamical systems lying outside the scope of Theorem 1, namely the Hénon maps , , and the standard family , . Indeed, these important examples of dynamical systems can’t be studied by the current methods of J. Palis and J.-C. Yoccoz because they display homoclinic/heteroclinic bifurcations associated to “very fat horseshoes”:
- in the case of Hénon maps, the “horseshoes” have “stable dimension” and a very small unstable dimension for certain parameters , and
- in the case of the standard family, one has horseshoes with arbitrarily close to for large values of (see, e.g., this post here).
Now, let us start to explain the meaning of non-uniformly hyperbolic horseshoe in Theorem 1. As we explained in the first post of this series, a (uniformly hyperbolic) horseshoe of a surface diffeomorphism is a saddle-like object in the sense that is not an attractor nor a repellor, that is, both its stable set
and unstable set
have zero Lebesgue measure . Here, is the -dimensional Lebesgue measure of . In a similar vein, J. Palis and J.-C. Yoccoz (cf. Theorem 7 of their article) showed that their non-uniformly hyperbolic horseshoes are saddle-like objects:
Theorem 2 Under the same assumptions of Theorem 1, one has that
for most .
Actually, the statement of Theorem 7 of their article contains a slightly more precise explanation of the non-uniformly hyperbolic features of (for most ): it is possible to show that supports geometric Sinai-Ruelle-Bowen (SRB) measures with non-zero Lyapunov exponents, that is, is a non-uniformly hyperbolic object in the sense of the so-called Pesin theory. Unfortunately, a detailed explanation of these terms (i.e., SRB measures, Lyapunov exponents, Pesin theory) is out of the scope of these notes and we refer the curious reader to the original articles R. Bowen and D. Ruelle, Y. Sinai, the books of A. Katok and B. Hasselblatt and C. Bonatti, L. Diaz and M. Viana, and the links attached to these keywords for more informations.
In order to further explain the structure of , we’ll briefly describe in the sequel some elements of the proof of Theorem 1.
2. A global view on Palis-Yoccoz induction scheme
As usual, we wish to understand the local dynamics of on the neighborhood indicated in the picture above, that is, we want to investigate the structure of the set
for most parameters .
In this direction, we consider and we look at the parameter interval . Very roughly speaking, the scheme of J. Palis and J.-C. Yoccoz has the following structure: besides , one has two extra parameters and chosen such that
Then, one proceeds inductively:
- at the 1st stage, one defines and one divides the interval into candidates subintervals;
- then, one apply an exam called strong regularity to each candidate subinterval: the good subintervals (passing the exam) are kept while the bad ones are discarded;
- after that, one goes to the next stages, that is, one takes the good intervals from the th stage, subdivide them into subintervals of size , apply the strong regularity exam to each subinterval and one keeps the good subintervals and discard the bad subintervals.
Of course, the strong regularity of an interval is a property about the (non-uniform) hyperbolic features of for all parameters , and the choice of the set of properties defining the strong regularity must be extremely careful: it should not be too weak (otherwise one doesn’t get hyperbolicity) nor too strong (otherwise there is a risk that no interval is good at some stage).
Actually, as we’ll see later, for each candidate interval , J. Palis and J.-C. Yoccoz construct a class of so-called (-persistent) affine-like iterates of , and they “test” by strong regularity of by examining the features of the class .
Remark 3 It is worth to point out that the class is unique, but this is shown in the article only a posteriori. Also, a nice feature of the arguments of J. Palis and J.-C. Yoccoz is that they are time-symmetric, that is, the dynamical estimates for the past and the future are the same (i.e., one has only to do half of the computations). In particular, those readers with some familiarity with Hénon maps know that the past behavior is very different from the future behavior (due to strong dissipation) and this morally explains why the methods of their article are not directly useful in the case of Hénon maps.
After this very approximative description of Palis-Yoccoz inductive scheme, it is clear that one of the key ideas is to carefully setup the notion of strong regularity property. However, before discussing this subject, we need to make some preparations: firstly we need to localize the dynamics, secondly we need to introduce the affine-like iterates, and thirdly we need to introduce the class .
2.1. Localization of the dynamics
The local dynamics of for has the following appearance:
As it is highlighted in this picture, after unfolding the tangency, we get two regions and called unstable and stable parabolic tongues bounded by the pieces of and near . The transition time from the unstable tongue to the stable tongue under the dynamics is a large but fixed integer depending only on .
Using this, we can organize the local dynamics of on as follows. Firstly, we select a finite Markov partition of the horseshoe into compact disjoint rectangles , , i.e., by fixing a convenient system of coordinates, we write in such a way that:
- the derivative of expands (uniformly) the horizontal direction and contracts (uniformly) the vertical direction,
- is the maximal invariant set of the interior of , i.e., ,
- is a Markov partition, i.e., , , and there exists an integer with for all .
Secondly, we denote by . Then, in this setting, it is not hard to see that the local dynamics of on is given by
- the uniformly hyperbolic maps , related to the horseshoe (they are called uniformly hyperbolic because the horizontal direction is uniformly expanded and the vertical direction is uniformly contracted by their derivatives), and
- the folding map making the transition between parabolic tongues.
In this context, by letting , we have that is the maximal invariant set of , i.e., .
This localization of the dynamics of on to the region is useful because it allows to think of in terms of an iterated system of maps, i.e., we approach the points of by looking at the domains and the images of the compositions (i.e., certain -iterates) of the uniformly hyperbolic maps and the folding map .
By thinking in this way, we see that the points in the domains and/or images of -iterates (composition) with affine-like features, that is, -iterates whose derivates expand the horizontal direction and contract the vertical direction, will contribute to the hyperbolicity of . In other words, it is desirable to get as much affine-like iterates as possible. Of course, the -iterates obtained by composition of transition maps related to the horseshoe have affine-like features (by definition), so that one risks to lose the affine-like property only when one considers compositions with the folding map (because the folding map mixes up the horizontal and vertical directions).
In particular, this suggests that strong regularity property (whatever this means) has something to do with the consecutive passages through the critical region given by the parabolic tongues and . However, before pursuing this direction, let’s formalize the notion of affine-like iterates (following this article here).
2.2. Affine-like maps
A vertical strip is a region of the form
and a horizontal strip is a region of the form
Intuitively, we wish to call “affine-like” a map from a vertical strip to a horizontal strip “approximately” contracting the vertical direction and expanding horizontal direction such as the one depicted in the figure below.
Formally, we define:
Definition 3 We say that a map from a vertical strip to a horizontal strip is weakly affine-like whenever admits an implicit representation , i.e., we can write and . Equivalently, is weakly affine-like if and only if the projection from the graph of to is a diffeomorphism.
This definition of affine-like maps in terms of implicit representations is somewhat folkloric in Dynamical Systems, and it was used by J. Palis and J.-C. Yoccoz because it is technically easier to estimate than as the symmetry between past and future is more evident, and are “contractive” maps.
In what follows, we will denote the derivatives of and by . Following J. Palis and J.-C. Yoccoz, we will consider exclusively with weakly affine-like maps satisfying the following hyperbolicity conditions:
Definition 4 A weakly affine-like map is called affine-likeif its implicit representation verifies:
- Cone condition: and where , and
- Bounded distortion condition: are uniformly bounded by some constant .
Here, the constants and are fixed once and for all depending only on .
Informally, the cone condition says that contracts the vertical direction and expands the horizontal direction, and the bounded distortion condition says that the derivative of behaves in the same way in all scales.
For later use, we introduce the following notion:
Definition 5 The widths of the domain and the image of an affine-like map with implicit representation are
Once we dispose of the notion of affine-like iterates, we’re ready to introduce the class whose “strong regularity” will be tested later.
2.3. Simple and parabolic compositions of affine-like maps and the class
Coming back to the interpretation of the dynamics on as an iterated system of maps given by compositions of and the folding map , we see that the following two ways of composing affine-like maps are particularly interesting in our context.
Definition 6 Let and be two affine-like maps such that . Then, the simple composition is the affine-like map with domain and image shown in the figure below.
Remark 4 By direct inspection of definitions, one can check that where the implied constant depends only on (by means of the constants in the cone condition).
The composition of two transition maps and associated to the horseshoe is the canonical example of simple composition.
In particular, if we wish to understand , it is not a good idea to work only with simple compositions, that is, we must include some passages through the parabolic tongues. This is formalized by the following notion.
Definition 7 Denote by and the rectangles of the Markov partition of containing the parabolic tongues and . Let and be two affine-like maps such that , resp. , passes near the parabolic tongue , resp. , i.e., crosses and crosses . We define the parabolic compositions of and as follows. Firstly, we compare with the parabolic-like strip and we say that the parabolic composition of and is possible if the intersection has two connected components and as shown (in black) in Figure 2 below. Then, assuming that the parabolic composition of and is possible, we define their parabolic compositions as the two weakly affine-like maps and shown in Figure 2below obtaining by concatenating , the folding map and in the strips , , , .
As it is indicated in the figure above, the parabolic composition comes with an important parameter measuring the distance between the vertical strip and the tip of the parabolic-like strip , or, equivalently, the horizontal strip and the tip of the parabolic-like strip .
Remark 5 By direct inspection of definitions, one can check that .
In this notation, the class is defined as follows.
Definition 8 is the class of affine-like iterates of , , closed under all simple compositions and certain parabolic compositions. More precisely, contains only parabolic compositions satisfying certain transversalityconditions such as
Remark 6 In fact, the transversality conditions on parabolic compositions imposed by J. Palis and J.-C. Yoccoz involves 6 conditions besides the one on the parameter given above.
For later use, we denote by an affine-like iterate taking a vertical strip to a horizontal strip after iterations of .
At this stage, we are ready to discuss the strong regularity property for .
2.4. Critical strips, bicritical dynamics and strong regularity
Definition 9 We say that , resp. , is -critical when , resp. is not -transverse to the parabolic tongue , resp. , i.e., the distance between , resp. to the “tip” of , resp. , is smaller , resp. for some .
Definition 10 We say that an element is -bicritical if and are -critical.
In other words, a bicritical corresponds to some part of the dynamics starting at some close to the tip of and ending at some close to the tip of , that is, a bicritical corresponds to a return of the critical region to itself.
Of course, one way of getting hyperbolicity for is to control the bicritical dynamics, i.e., bicritical elements .
Definition 11 Given , we say that a candidate parameter is -regularif
for every -bicritical element .
Remark 7 In their article, J. Palis and J.-C. Yoccoz choose depending only on the stable and unstable dimensions and of the initial horseshoe and the hyperbolicity strength of the periodic points and involved in the heteroclinic tangency. See Equation (5.19) of Palis-Yoccoz article for the precise requirements on .
Intuitively, a candidate parameter interval is -regular if the bicritical dynamics seen through is confined to very small strips and . Unfortunately, the condition of -regularity is not enough to run the induction scheme of J. Palis and J.-C. Yoccoz, and they end up by introducing a more technical condition called strong regularity. However, for the sake of this text, let’s ignore this issue by pretending that strong regularity is -regularity for some adequate parameter .
After this brief discussion of strong regularity, it is time to come back to Palis-Yoccoz induction scheme in order to say a few words about the dynamics of for belonging to strongly regular intervals.
2.5. Dynamics of strong regular parameters
As it is explained in Sections 10 and 11 of their article, J. Palis and J.-C. Yoccoz can reasonably control the dynamics of for strongly regular parameters : these are the parameters where is a decreasing sequence of strongly regular intervals .
Remark 8 It is interesting to notice that the strongly regular parameters of Palis-Yoccoz are not defined a priori, i.e., one has to perform the entire induction scheme before putting the hands on them. This is in contrast with the so-called Jakobson theorem, a sort of -dimensional version of Theorem 1, where the strongly regular parameters are known since the beginning of the argument.
Of course, before starting the analysis of strongly regular parameters, one needs to ensure that such parameters exist, that is, one want to know whether there are parameters left from the parameter exclusion scheme of J. Palis and J.-C. Yoccoz. This issue is carefully treated in Section 9 (of 50 pages!) of Palis-Yoccoz article, where the authors estimate the relative speed of strips associated to elements when the parameter moves, and, by induction, they are able to control the measure of bad (not strongly regular) intervals: as it turns out, the measure of the set of bad intervals is , so that the strongly regular parameters have almost full measure in , i.e., (cf. Corollary 15 of Palis-Yoccoz article).
Remark 9 In order to get some strongly regular parameter, one has to ensure that the initial interval is strongly regular (otherwise, one ends up by excluding in the first step of Palis-Yoccoz induction scheme, so that one has no parameters to play with in the next rounds of the induction!). Here, J. Palis and J.-C. Yoccoz makes use of the technical assumption that one is unfolding a heteroclinic tangency: indeed, the idea is that the formation of bicritical elements takes a long time in heteroclinic tangencies because the points in the critical region should pass near first, then near and only then they can return to the critical region again; of course, in the case of homoclinic tangencies, it may happen that bicritical elements pop up quickly and this is why one can’t include homoclinic tangencies in the statement of Theorem 1.
From now on, let us fix a strongly regular parameter, and let’s study for . Keeping this goal in mind, we introduce the collection of all affine-like iterates of coming from the strongly regular intervals . Using the class , we can define the class of stable curves, i.e., the class of curves coming from intersections of decreasing sequences of vertical strips serving as domain of affine-like iterates of , that is, . Also, we put the set of points of in some stable curve.
These stable curves were introduced by analogy with uniformly hyperbolic horseshoes: indeed, the stable lamination of can be recovered from the transitions maps by looking at the decreasing sequences of domains of simple compositions of these transitions maps.
From the nice features of strong regular parameters, it is possible to prove that the class is a -lamination and one can use to induce a dynamical system isomorphic to a Bernoulli map with infinitely many branches (cf. Subsection 10.4 of Palis-Yoccoz article). Here, is the set of stable curves not contained in infinitely many prime elements of (we say that an element is prime if it can’t be obtained by simple composition of shorter elements (shorter meaning )). In particular, as it is shown in Subsections 10.5, 10.6, 10.7, 10.8, 10.9, 10.10 of Palis-Yoccoz article, is a non-uniformly hyperbolic dynamical system (in a very precise sense). Of course, by the symmetry between past and future (see Remark 3), one also has an analog non-uniformly hyperbolic dynamical system on unstable curves, so that inherits a natural non-uniformly hyperbolic part consisting of points whose and iterates never escape and .
Therefore, if we can show that the size of the sets of the points of escaping and/or is relatively small compared to the non-uniformly hyperbolic part of , then we can say that is a non-uniformly hyperbolic horseshoe. Here, J. Palis and J.-C. Yoccoz set up in Section 11 of their article a series of estimates towards showing that the points of escaping and/or are exceptional: for instance, they show Theorem 2 that the -dimensional Lebesgue measure of is zero because this property is true for the non-uniformly hyperbolic part of (by the “usual” hyperbolic theory) and the set of points of escaping and/or are “rare” in the sense that their -dimensional Lebesgue measure contribute as an “error term” to the the non-uniformly hyperbolic part of .
At this point, our overview of Palis-Yoccoz induction scheme is complete. Closing this post, we would like to make two comments. Firstly, as it is pointed out in page 14 of Palis-Yoccoz article, the philosophy that is constituted of a non-uniformly hyperbolic part and an exceptional set makes them expect that one could improve the information on the geometry of and/or . As it turns out, we will discuss in the next (final) post of this series some recent results in this direction. Finally, the condition (1) is not expected to be sharp by any means, but it seems that the strongly regular parameters are not sufficient to go beyond (1), so that, as some arguments from singularity theory seem to indicate, it is likely that one has to exclude further parameters in order to improve Theorem 1.