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 1** * Let be a diffeomorphism from the closed unit ball of into its image.*Let and be two constants such that and for all .

Then, for each , the -dimensional Hausdorff measure at scale of satisfies

* *

**Remark 1** * In 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 2** * In 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

where .

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:

**Claim.** For each and , one has

**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.

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