Posted by: matheuscmss | September 20, 2012

Homoclinic/heteroclinic bifurcations: fat horseshoes

Before continuing with our discussion of homoclinic/heteroclinic bifurcations associated to periodic points inside horseshoes, let me make a few “organizational” comments about the current series of posts. By the time I announced this series, I was planning to post 5 texts because this is how the content was “thematically organized” in my mind. However, it turns out that this thematic organization is not very good because it makes the current post extremely short and the next “final” one very big. So, after struggling a little bit with the texts, I decided that I’ll keep this post very short, but I’ll break up the “final text” – announced under the title “Homoclinic/heteroclinic bifurcations: slightly fat horseshoes” – into two posts, namely, “Homoclinic/heteroclinic bifurcations: slightly fat horseshoes A” (where we will discuss the Palis-Yoccoz article on non-uniformly hyperbolic horseshoe) and “Homoclinic/heteroclinic bifurcations: slightly fat horseshoes B” (a sort of research announcement where some results about the Hausdorff dimension of non-uniformly horseshoes are outlined).

Now, let us get back to the main topic.

Consider again the setting (and notations) from the previous posts: {f:M\rightarrow M} is a {C^2}-diffeomorphism of a surface {M} with a horseshoe {K} and a periodic point {p\in K} whose stable and unstable manifolds have a quadratic tangency at some point {q\in M-K}. Consider again {U} a sufficiently small neighborhood of {K} and {V} a sufficiently small neighborhood of the orbit of {q}, and let’s fix {\mathcal{U}} a sufficiently small {C^2}-neighborhood of {f} organized into the open sets {\mathcal{U}_-} and {\mathcal{U}_+} and the codimension {1} hypersurface {\mathcal{U}_0} depending on the relative positions of {W^s(p_g)} and {W^u(p_g)} near {V}.

Organization of the parameter space {\mathcal{U}}.

Assume further that the quadratic tangency of {f=g_0\in\mathcal{U}_0} is associated to a fat horseshoe {K=K_f}, i.e., suppose that {\textrm{HD}(K)=d_s^0+d_u^0>1}. In particular, by the continuity of the Hausdorff dimension of horseshoes (see this article of J. Palis and M. Viana for more details), if we choose {\mathcal{U}} sufficiently small, then {\textrm{HD}(K_g)>1} for {g\in\mathcal{U}}. By reviewing the arguments in the previous post, we see that the intersections and/or arithmetic differences of the regular Cantor sets {K^s=K^s(g_0)} and {K^u=K^u(g_0)} will hint what we should expect for the local dynamics of {g\in\mathcal{U}_+}.

Regular Cantors sets {K^s} and {K^u} in the line of tangencies {\ell}.

Here, the following result due to J. M. Marstrand is very inspiring:

Proposition 1 (J. M. Marstrand) Let {C\subset\mathbb{R}^2} be a subset with {\textrm{HD}(C)>1}. Then, for Lebesgue almost every {\lambda\in\mathbb{R}}, the set {\pi_{\lambda}(C)\subset \mathbb{R}} has positive Lebesgue measure, where {\pi_{\lambda}(x,y):=x-\lambda y}.

For a proof of this result using potential theory, see Theorem 2 at page 64 of Palis-Takens book (alternatively, the reader may consult this post).

In our context, we can apply Marstrand’s theorem to {C=K^s(g_0)\times K^u(g_0)} because, by hypothesis, {\textrm{HD}(C)\geq d_s^0+d_u^0>1}. By doing so, we get that for Lebesgue almost every {\lambda\in\mathbb{R}} the arithmetic difference {K^s(g_0)\ominus\lambda K^u(g_0) = \pi_{\lambda}(K^s(g_0)\times K^u(g_0))} has positive Lebesgue measure. In particular, if one can produce a {2}parameter family {g_{\lambda,t}\in\mathcal{U}} such that {g_{\lambda,0}\in\mathcal{U}_0}, {K^s(g_{\lambda,t})=K^s(g_0)} and {K^u(g_{\lambda,t})=\lambda K^u(g_0) + t} for all {\lambda} close to {1}, then one would get that for almost every {\lambda} close to {1}, the stable and unstable laminations of {K_{g_{\lambda,t}}} meet tangentially in the region {V} near {q} for a set of parameters {t} of positive Lebesgue measure.

In particular, since the presence of tangencies prevents hyperbolicity, this hints that, in the context of fat horseshoes {K=K_f}, {f\in\mathcal{U}_0}, the statement of the theorem of Newhouse, Palis and Takens that {\Lambda_g} is a horseshoe for most {g\in\mathcal{U}_+} (cf. the previous post) may fail along certain {2}-parameter families {g_{\lambda,t}\in\mathcal{U}}.

This idea was pursued in this work here where J. Palis and J.-C. Yoccoz showed (in 1994) the following result. Let

\displaystyle \mathcal{T}:=\left\{g\in\mathcal{U}: \textrm{the stable and unstable laminations of } K_g \textrm{ meet } \right.

\displaystyle \left. \textrm{ tangentially somewhere in } V \right\}

be the locus of tangencies. Then, for any smooth {2}-parameter family {(g_{\lambda,t})_{|\lambda|<\lambda_0, |t|<t_0}\in\mathcal{U}} such that, for each {|\lambda|<\lambda_0},

  • {f_{\lambda}=g_{\lambda,0}\in\mathcal{U}_0}, {g_{\lambda,t}\in\mathcal{U}_-} for all {-t_0<t<0} and {g_{\lambda,t}\in\mathcal{U_+}} for all {0<t<t_0},
  • {g_{\lambda,t}} is transverse to {\mathcal{U}_0} at {f_{\lambda}:=g_{\lambda,0}},
  • the eigenvalues {|\alpha(\lambda)|<1} and {|\beta(\lambda)|>1} of the derivative of {g_{\lambda,0}} at the hyperbolic periodic point {p_{g_{\lambda,0}}} vary non-trivially with {\lambda} in the sense that their derivatives {\alpha'(0),\beta'(0)} at {\lambda=0} don’t vanish,
  • the horseshoe {K_{f_\lambda}} is fat (i.e., its Hausdorff dimension is larger than 1),

there exists a constant {c=c(g_{\lambda,t})>0} with the following property. For Lebesgue almost every{|\lambda|<\lambda_0}, it holds

\displaystyle \limsup\limits_{\varepsilon\rightarrow0}\frac{\textrm{Leb}(\{t\in(0,t_0): g_{\lambda,t}\in\mathcal{T}\})}{\varepsilon}>c.

More recently, C. G. Moreira and J.-C. Yoccoz studied in this article here (from 2001) the geometry of the intersections {K\cap K'} of regular Cantor sets {K} and {K'}, and they showed in this article here (from 2010) how the key ideas from their paper from 2001 can be extended (with some non-trivial technical work) to show that the subset of {g\in\mathcal{U}_+} with stable tangencies in the region {V} (near {q}) has positive density in the setting of bifurcations of fat horseshoes. More precisely, let {\textrm{int}(\mathcal{T})} be the locus of stable tangencies, that is, {\textrm{int}(\mathcal{T})} is the interior of {\mathcal{T}}.

Theorem 2 (C. G. Moreira and J.-C. Yoccoz (2010)) Suppose that {\textrm{HD}(K)>1} for {f\in\mathcal{U}_0}. Then, there exists an open and dense subset {\mathcal{U}_0^*} of {\mathcal{U}_0} such that any smooth {1}-parameter family {(g_t)_{|t|<t_0}} passing by {g_0\in\mathcal{U}_0^*} transversely to {\mathcal{U}_0} meets the locus of stable tangencies {\textrm{int}(\mathcal{T})} with positive (inferior) density in the sense that

\displaystyle \liminf\limits_{\varepsilon\rightarrow0}\frac{\textrm{Leb}(\{t\in(0,\varepsilon):\, g_t\in\textrm{int}(\mathcal{T})\})}{\varepsilon}>0

Furthermore, denoting by {\mathcal{H}:=\{g\in\mathcal{U}:\,\Lambda_g \textrm{ is a horseshoe}\}}, one has that {(g_t)_{|t|<t_0}} meets {\textrm{int}(\mathcal{T})\cup\mathcal{H}} with full density in the sense that

\displaystyle \lim\limits_{\varepsilon\rightarrow0}\frac{\textrm{Leb}(\{t\in(0,\varepsilon):\, g_t\in\textrm{int}(\mathcal{T})\cup\mathcal{H}\})}{\varepsilon}=1

It is likely that this theorem will be the subject of a few posts in this blog in the future, but for now let us only contemplate what this result is telling to us: it makes clear that, in the context of fat horseshoes (i.e., {\textrm{HD}(K)>1} for {f\in\mathcal{U}_0}), it is not true that {\Lambda_g} is a horseshoe for most {g\in\mathcal{U}_+}!

We can summarize the discussion so far in this series of posts with the following two phrases:

  • by the results of S. Newhouse, J. Palis and F. Takens, in the context of thin horseshoes (i.e., {\textrm{HD}(K)<1} for {f\in\mathcal{U}_0}), {\Lambda_g} is a horseshoe for most {g\in\mathcal{U}_+}, and
  • by Theorem 2, in the context of fat horseshoes (i.e., {\textrm{HD}(K)>1} for {f\in\mathcal{U}_0}), {\Lambda_g} has persistent tangencies with positive “probability” and {\Lambda_g} is not a horseshoe for most {g\in\mathcal{U}_+}.

Of course, these phrases provide a fairly complete picture as far as the uniform hyperbolicity of {\Lambda_g} (for most {g\in\mathcal{U}_+}). However, there is no reason to stop here. Indeed, it is very well known nowadays that there is life (and even a book! :)) beyond uniform hyperbolicity, that is, even when {\Lambda_g} is not uniform hyperbolic, it could be that {\Lambda_g} verifies weaker forms of hyperbolicity (such as non-uniform hyperbolicity, partial hyperbolicity, existence of dominated splittings, etc.) that are still good enough to deduce some qualitative properties of the dynamics of {g} on {\Lambda_g}.

In this direction, even though we saw that bifurcations of quadratic tangencies associated to fat horseshoes, {\textrm{HD}(K)=d_s^0+d_u^0>1}, are complicated because of persistent tangencies, by a closer inspection of the works of C. Moreira and J.-C. Yoccoz from 2001 and 2010, one realizes that the regular Cantor sets {K^s(g)} and {K^u(g)} for {g\in\mathcal{U}_+} are usually expected to intersect in a set {K^s(g)\cap K^u(g)} of Hausdorff dimension close to {d_s^0+d_u^0-1=\textrm{HD}(K)-1}. Thus, from the heuristic point of view, the “critical locus{K^s(g)\cap K^u(g)} (i.e., the region where the tangencies destroying the hyperbolicity show up) is very small, i.e., its Hausdorff dimension is close to zero, if the initial horseshoe {K} is only slightly fat, i.e., {\textrm{HD}(K)>1} is close to {1}. In particular, one could imagine that bifurcations quadratic tangencies of slightly fat horseshoes could lead to a local dynamics on {\Lambda_g} satisfying some form of weak (non-uniform) hyperbolicity for most {g\in\mathcal{U}_+} despite the fact that {\Lambda_g} doesn’t verify strong (uniform) hyperbolicity conditions in general.

In an impressive work (of 217 pages!), J. Palis and J.-C. Yoccoz were able to formalize this crude heuristic argument by showing (among several other things) that {\Lambda_g} is a non-uniformly hyperbolic horseshoe for most {g\in\mathcal{U}_+} in the context of heteroclinic bifurcations of slightly fat horseshoes. However, a reasonable discussion of this work surely merits an entire post, and so, for today, let’s stop here while postponing the introduction of the so-called non-uniformly hyperbolic horseshoes for the next post.


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