Some of the partial advances obtained by Jacob Palis, Jean-Christophe Yoccoz and myself on the computation of Hausdorff dimensions of stable and unstable sets of non-uniformly hyperbolic horseshoes (announced in this blog post here and this survey article here) are based on the following lemma:

Lemma 1Let and be two constants such that and for all .Let be a diffeomorphism from the closed unit ball of into its image.Then, for each , the -dimensional Hausdorff measure at scale of satisfies

Remark 1In fact, this is not the version of the lemma used in practice by Palis, Yoccoz and myself. Indeed, for our purposes, we need the estimate

where is the ball of radius centered at the origin and is a diffeomorphism such that and for . Of course, this estimate is deduced from the lemma above by scaling, i.e., by applying the lemma to where is the scaling .

Nevertheless, we are not completely sure if we should write down an article just with our current partial results on non-uniformly hyperbolic horseshoes because our feeling is that these results can be significantly improved by the following heuristic reason.

In a certain sense, Lemma 1 says that one of the “worst” cases (where the estimate (1) becomes “sharp” [modulo the multiplicative factor ]) happens when is an affine hyperbolic conservative map (say ): indeed, since , the most “economical” way to cover using a countable collection of sets of diameters is basically to use squares of sizes (which gives an estimate ).

However, in the context of (expectional subsets of stable sets of) non-uniformly hyperbolic horseshoes, we deal with maps obtained by successive compositions of affine-like hyperbolic maps *and* a certain *folding* map (corresponding to “almost tangency” situations). In particular, we work with maps which are *very different* from affine hyperbolic maps and, thus, one can expect to get slightly *better* estimates than Lemma 1 in this setting.

In summary, Jacob, Jean-Christophe and I hope to improve the results announced in this survey here, so that Lemma 1 above will become a “deleted scene” of our forthcoming paper.

On the other hand, this lemma might be useful for other purposes and, for this reason, I will record its (short) proof in this post.

**1. Proof of Lemma 1 **

The proof of (1) is based on the following idea. By studying the intersection of with dyadic squares on , we can interpret the measure as a sort of -norm of a certain function. Since , we can control this -norm in terms of the and norms (by interpolation). As it turns out, the -norm, resp. -norm, is controlled by the features of the derivative , resp. Jacobian determinant , and this morally explains the estimate (1).

Let us now turn to the details of this argument. Denote by and its boundary. For each integer , let be the collection of *dyadic squares* of level , i.e., is the collection of squares of sizes with corners on the lattice .

Consider the following recursively defined cover of . First, let be the subset of squares such that

Next, for each , we define inductively as the subset of squares such that is *not* contained in some for , and intersects a *significant portion* of in the sense that

In other words, we start with and we look at the collection of dyadic squares of level intersecting it in a significant portion. If the squares in suffice to cover , we stop the process. Otherwise, we consider the dyadic squares of level not belonging to , we divide each of them into four dyadic squares of level , and we build a collection of such dyadic squares of level intersecting in a significant way the remaining part of not covered by , etc.

Remark 2In this construction, we are implicitly assuming that is not entirely contained in a dyadic square . In fact, if , then the trivial bound (for ) is enough to complete the proof of the lemma.

In this way, we obtain a countable collection covering such that and

By thinking of this expression as a -norm and by applying interpolation between the and norms, we obtain that

This reduces our task to estimate these and norms. We begin by observing that the -norm is easily controlled in terms of the Jacobian of (thanks to the condition (2)):

for any . In particular, we have that

From this estimate, we see that the -norm satisfies

Thus, we have just to estimate the series . We affirm that this series is controlled by the derivative of . In order to prove this, we need the following claim:

**Proof of Claim.** Note that can not contain : indeed, since for some dyadic square of level (and, thus, ), if , then , a contradiction with the definition of . Because we are assuming that is not contained in (cf. Remark 2) and we also have that intersects (a significant portion of) , we get that

For the sake of contradiction, suppose that . Since intersects , the -neighborhood of contains . This means that

- (a) either is contained in
- (b) or is disjoint from

However, we obtain a contradiction in both cases. Indeed, in case (a), we get that a dyadic square of level containing satsifies

a contradiction with the definition of . Similarly, in case (b), we obtain that

a contradiction with (2).

This completes the proof of the claim.

Coming back to the calculation of the series , we observe that the estimate (7) from the claim and the fact that imply:

By plugging this estimate into (6), we deduce that the -norm verifies

Finally, from (3), (4), (5) and (8), we conclude that

This ends the proof of the lemma.

What kind of diffeomorphisms you have in mind? Can you name an example of a diffeomorphism of the disk where Lemma 1 gives a non-trivial result? In fact, map where I can write the equation or compute will be pretty boring. I am not a dynamicist so I can’t think of many examples.

Even though the image is topologially a disk and the boundary is a differentiable, the boundary might be very complicated.

By:

MrCactu5 (@MonsieurCactus)on May 11, 2015at 8:46 pm

A prototype of the kind of diffeomorphisms that we have in mind is defined on a ball of doubly exponentially small radius (i.e., for some , ) “near” to the origin, where is an affine hyperbolic diffeomorphism (with and “close” to the origin), and is a folding map (given by the composition of a fold and a rotation ).

Here, the point is that is a “complicated” map (algebraic map of degree whose coefficients depend on parameters , ), but one can estimate the Hausdorff measure (at scale ) of thanks to (the remark after) Lemma 1.

Indeed, is area-preserving, and, furthermore, one can give a better bound to the derivative of than the trivial bound

(where is a bound for the derivative of the folding map ) because the derivative of the folding map “tends” to send (with some “parabolic” effect) the (horizontal) direction of maximal expansion for to (almost vertical) directions of almost maximal contraction for , and vice-versa. In particular, by performing the calculation of using this observation, one can show that

and this is the one of the important inputs used by Jacob, Jean-Christophe and I in our dynamical applications of the lemma.

I hope this explains why the lemma helps us in this setting (coming from dynamics): we want to estimate the Hausdorff measure (at fixed scale ) of a set with complicated geometry (of “algebraic complexity” ); instead of looking at the fine details of the geometry of this set, the lemma tells us that one can reasonably estimate the Hausdorff measure (at fixed scale) if there is a “dynamical reason” forcing the derivative and Jacobian of to satisfy some better bounds than the “trivial” ones.

In other terms, the lemma gives you good geometrical information (Hausdorff measure, optimal covers of sets) if you dispose of good dynamical/analytical information (non-trivial bounds on derivative and/or Jacobian).

By:

matheuscmsson May 12, 2015at 9:42 am