In what follows, we will consider the same setting of the previous post, and we will discuss a recent improvement obtained in collaboration with Jacob Palis and Jean-Christophe Yoccoz (on a work still in progress). In particular, all statements below concern the dynamics of where and is a strongly regular parameter in the sense of the article of J. Palis and J.-C. Yoccoz: in other words, in the sequel, we don’t have to exclude further parameters in order to get our (slightly improved) statements, and, indeed, our arguments will be based on soft analysis.
In this post we will prove the following result:
Logically, this result slightly improves Theorem 2 (of J. Palis and J.-C. Yoccoz) in the previous post saying that the -dimensional Lebesgue measures of and are zero (because any subset of the compact -dimensional manifold with Hausdorff dimension strictly smaller than has zero -dimensional Lebesgue measure).
The plan for the rest of this post (the last one of this series) is the following: in the next section we will prove Theorem 1, and in the final section we will make some comments on further results obtained in collaboration with Jacob and Jean-Christophe.
1. Hausdorff dimension of the stable sets of non-uniformly hyperbolic horseshoes
Let’s start by nicely decomposing the stable set . Using the notations of of the previous post of this series, we can write
Since is a diffeomorphism, it follows from the basic properties of the Hausdorff dimension (see item (e) of Proposition 3 of this post here) that
Now, we separate into its good (non-uniformly hyperbolic) part and its exceptional part as follows:
In other words, the good part of consists of points passing by the nice set of stable curves and the exceptional set is, by definition, the complement of the good part.
The set is the “nice” (non-uniformly hyperbolic) part of the dynamics and hence it is not surprising that J. Palis and J.-C. Yoccoz showed in Section 10 of their work that has Hausdorff dimension where is close to the stable dimension of the initial horseshoe .
Now, we follow the discussion of Section 11.7 of Palis-Yoccoz article to decompose by looking at successive passages through parabolic cores of strips. More concretely, given an element , we define the parabolic core of as
Here, a child of is a such that but there is no with . The geometry of a parabolic core of is depicted below:
By checking the definitions (of good part and exceptional set), it is not hard to convince oneself that can be decomposed as
where and the sets are inductively defined as , , (cf. Equations (11.57) to (11.63) of Palis-Yoccoz paper). Here, we say that is admissible if .
As the picture above indicates, the fact that the points of passes by successive parabolic cores imposes strong conditions over the elements : for instance, the parabolic core of any is non-empty, the horizontal bands are always critical and, because we’re dealing where is a strongly -regular parameter, the following estimate holds:
where and and are the three main parameters in Palis-Yoccoz induction scheme.
This lemma is crucial for our purposes because it says that the exceptional set is “confined” into regions whose widths are decaying in a double exponential way to zero. Note that this is in sharp contrast with the case of the stable set of the initial horseshoe (which is confined into regions whose widths are going exponentially to zero): in other words, this lemma is a quantitative way of saying that the set is exceptional when compared with the stable lamination of the horseshoe .
In any event, the lemma above allow us to estimate the Hausdorff -measure of . By definition, we know that a certain -iterate of is contained in the parabolic core . On the other hand, as it is shown in the figure below, we know that is contained in a vertical strip of width and heigth (see Proposition 62 of Palis-Yoccoz article).
We divide this vertical strip containing into squares of sides of lengths and we analyze individually their evolution under the dynamics by an inductive procedure.
Remark 1 Of course, this crude partition of into squares of dimensions aligned along a vertical strip is motivated by the fact that we don’t want to keep track of the fine geometry of because it gets complicated very fast (as a few iterations of the figure above shows).
More precisely, at the -th step of our procedure, we have squares of dimensions inside . We fix one of these squares and we note that sends this square into a vertical strip of width and height because is affine-like.
Remark 2 Here, we are using the affine-like property of to say that essentially it is an affine map whose linear part consists of multiplying the vertical direction by and the horizontal direction by . Evidently, this is not completely true as is non-linear, but the bounded distortion conditions (implicitly imposed in the definition of affine-like maps) ensure that, up to a (multiplicative) constant factor , behaves like an affine map. Of course, this provides a reasonably good control of the geometry, but we should be careful when inductively analyzing the composition of for . Indeed, the bounded distortion property ensures only that the composition of for behaves like an affine map up to a multiplicative factor of growing exponentially with . In other words, the distortion (“difference”) between and an affine map becomes significant as . In general, this is an important technical problem, but, fortunately, in our situation this will not be a big issue because Lemma 1 ensures that the size of the exceptional set decreases doubly exponentially fast () and this turns out to be sufficient to control the exponential growth of the distortion. In particular, we will ignore this minor distortion issue and we will pretend that the maps are affine.
Again, we divide this vertical strip into squares of sides of length (similarly to Figure 2). Of course, during each step of this backward inductive procedure, we need to verify the compatibility condition . In the present case, this compatibility condition is automatically satisfied in view of the estimate of Lemma 1.
In particular, at the final step of this argument, we obtain a covering of by a collection of squares of sides of length . Thus, we have that
Let us take a real number very close to and rewrite the previous expression as
Applying again Lemma 1, we can bound this expression by
where . However, the hypothesis (H4) of Palis-Yoccoz article forces , so that for any . It follows that
Now we use two facts derived in the pages 204 and 205 of Palis-Yoccoz paper. Firstly, they show that the number of admissible sequences with fixed extremities and is (see Equation (11.77) of their article). Secondly, the sum is bounded because the results of the Subsection 11.5.9 of their article show that converges provided that .
Remark 3 It is hidden in the estimate for the fundamental (moral) fact that the critical locus is expected to have Hausdorff dimension that we alluded to in the introduction of the previous post of this series.
Putting these two facts together with (2), we see that the “Hausdorff -measure of at scale ” satisfies
Because as , this proves that . Hence, the proof of Theorem 1 is complete.
2. Final comments on further results
The attentive reader observed that the arguments of the previous subsection were based on soft analysis of the geometry of the exceptional set. Indeed, every time the shape of the parabolic cores was ready to get complicated, we divide it into squares and we analyzed the evolution of individual squares. In particular, every time we saw some parabolic geometry, we covered the “tip of the parabola” by a black box (square) and we forgot about the finer details of in this region. Of course, it is not entirely surprising that this kind of soft estimate works to show , but it is too crude if one wishes to compute the actual value of .
In particular, if one desires to prove that is really exceptional so that is close to the expected dimension , one has to somehow “face” the geometry of and its successive passages through parabolic cores .
In the forthcoming article by Jacob, Jean-Christophe and myself, we improve the soft strategy above without entering too much into the fine geometry of by noticing that each is the image of the parabolic core with a map of the form (obtained by alternating compositions of affine-like iterates of and the folding map ) whose derivative and Jacobian can be reasonably controlled. Using this control, we can study the Hausdorff -measure of at certain fixed scales in terms of the geometry of and the bounds on the derivative and the Jacobian of the map sending into . By putting forward this estimate, we can show that has the expected Hausdorff dimension (namely ) in a certain subregion of values of stable and unstable dimensions and of the initial horseshoe. In Figure 3 below we depicted in wave texture the region inside the larger region where
Actually, in this picture we drew only because the other half of can be deduce by symmetry.
The intersection of the region with the diagonal (corresponding to the “conservative case”) can be explicitly computed:
Remark 4 The nomenclature “conservative” comes from the fact that the stable and unstable dimensions of any horseshoe of a area-preserving diffeomorphism coincide (see, e.g., this article of H. McCluskey and A. Manning).
In particular, by putting this together with Figure 3, we see that occupies slightly less than half of region given by Palis-Yoccoz condition (3).
Finally, let us remark that we get the expected Hausdorff dimensions for and (in region ), but the arguments can’t be used to get the expected Hausdorff dimension for . In fact, our constructions so far start from the future of and the past of where some geometric control is available, e.g., in the form of nice partitions, and then it tries to bring back the information, i.e., partitions, by analyzing the -iterates used in our way back to the present time. Of course, this works if we deal separately with the past or the future, but if we try to deal with both at the same time, we run into trouble because it is not obvious how the partitions coming from the future and the past intersect in the present time (due to the lack of transversality produced by -iterates related to the folding map ). Evidently, the question of getting the expected Hausdorff dimension for is natural and interesting, and we hope to address this issue in our forthcoming paper.