Shy shadows of infinite-dimensional partially hyperbolic invariant sets

Abstract

Let $\mathcal{R}$ be a strongly compact $C^2$ map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative $D_F\mathcal{R}$ is dense for every $F$. Let $\Omega$ be a compact, forward invariant and partially hyperbolic set of $\mathcal{R}$ such that $\mathcal{R}:\Omega \to \Omega$ is onto. The $\delta$-shadow $W_{\delta }^s(\Omega )$ of $\Omega$ is the union of the sets $$W_{\delta }^s(G)=\{F:dist({\mathcal{R}}^{i}F,{\mathcal{R}}^{i}G)\le \delta ,foreveryi\ge 0\},$$ where $G\in \Omega$. Suppose that $W_{\delta }^s(\Omega )$ has transversal empty interior, that is, for every $C^{1+Lip}$ $n$-dimensional manifold $M$ transversal to the distribution of dominated directions of $\Omega$ and sufficiently close to $W_{\delta }^s(\Omega )$ we have that $M\cap W_{\delta }^s(\Omega )$ has empty interior in $M$. Here $n$ is the finite dimension of the strong unstable direction. We show that if ${\delta }^{\prime }$ is small enough then $${\cup }_{i\ge 0}{\mathcal{R}}^{-i}{W}_{{\delta }^{\prime }}^s(\Omega )$$ intercepts a $C^k$-generic finite dimensional curve inside the Banach space in a set of parameters with zero Lebesgue measure, for every $k\ge 0$. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.

37 pages, 2 figures. To appear in Ergodic Theory and Dynamical Systems

References in corpus (2)

• Solenoidal attractors with bounded combinatorics are shy
• Global hyperbolicity of renormalization for C^r unimodal mappings

Cited by in corpus (2)

• Quasisymmetric rigidity in one-dimensional dynamics
• The boundary of chaos for interval mappings