Posted by: matheuscmss | June 8, 2011

Non-uniformly hyperbolic horseshoes have Hausdorff dimension < 2

Today I gave a talk in the conference “Beyond Uniform Hyperbolicity 2011” (at CIRM, Marseille, France) about a joint work (still in progress) with Jacob Palis and Jean-Christophe Yoccoz on the geometry of certain objects called non-uniformly hyperbolic horseshoes.

More precisely, let f:M\to M be a smooth diffeomorphism displaying a heteroclinic tangency associated to two periodic orbits \mathcal{O}(p_s) and \mathcal{O}(p_u) inside a horseshoe K, i.e., K is a hyperbolic set contaning two periodic points p_s, p_u in distinct orbits such that the stable manifold W^s(p_s) of p_s intersects the unstable manifold W^u(p_u) of p_u in a tangential (i.e., non-transverse) way at some point q\in M-K. In general, the presence of such tangencies can be used as a mechanism to produce open sets of non-hyperbolic diffeomorphisms, as it was noticed by S. Newhouse (e.g., see this post here for more discussion on this). Nevertheless, if the stable and unstable dimensions d_s and d_u of the horseshoe K (i.e., the Hausdorff dimensions of K\cap W^s(x) and  K\cap W^u(x) for any x\in K; these dimensions are well-defined for horseshoes, i.e., they are independent of x) verify

d_s+d_u<1

then hyperbolicity is “prevalent” after unfolding the tangency, as it was shown by Newhouse, Palis and Takens (here and here): namely, the relative density of hyperbolic diffeomorphisms along generic 1-parameter families g_t unfolding the tangency with g_0=g tends to one when the size $\varepsilon>0$ parameter interval [-\varepsilon,\varepsilon] goes to zero.

On the other hand, after the works of J. Palis and J.-C. Yoccoz, and C. G. Moreira (Gugu) and J.-C. Yoccoz, we know that this nice scenario is false when d_s+d_u>1 (and actually stable tangencies is a property with positive relative density).

Nevertheless, in a recent tour-de-force paper (of 217 pages), J. Palis and J.-C. Yoccoz were able to show that, even though hyperbolicity doesn’t have full relative density after bifurcations of horseshoes with d_s+d_u>1, the set of diffeomorphisms displaying non-uniformly hyperbolic horseshoes have full relative density if the initial horseshoe K is only slightly fat.

The precise condition for slight fatness is not particularly inspiring

(d_s+d_u)^2+(\max\{d_s,d_u\})^2<d_s+d_u+\max\{d_s,d_u\}

and it comes from very delicate arguments related to an inductive scheme of parameter exclusion (or selection).

As a by-product, it is not very easy to describe precisely what a non-uniformly hyperbolic horseshoe is, but one can at least say that it is a saddle-like object (much like a horseshoe) in the sense that the area (two-dimensional Lebesgue measure) of its stable and unstable sets (i.e., the set of points converging to the object in the future and the past resp.) is zero, that is, non-uniformly hyperbolic horseshoes are definitely neither attractors nor repellors.

In my talk, I discussed some progress towards the computation of the Hausdorff dimension of the stable and unstable sets of the non-uniformly hyperbolic horseshoes constructed by Palis and Yoccoz. In particular, in this work we don’t perform further parameter exclusions to get our results! Also, I quickly mentioned that we get the “expected” Hausdorff dimension for these sets for certain values of d_s, d_u. Of course, I plan to comment more on this in the future (when our work is more developed), but, for the time being, the curious reader can consult my slides here for an illustrated discussion.


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