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
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.