Shy shadows of infinite-dimensional partially hyperbolic invariant sets

TL;DR

This paper investigates the measure-theoretic properties of stable sets near infinite-dimensional partially hyperbolic invariant sets, showing that under certain conditions, their preimages intersect generic curves in a measure-zero set.

Contribution

It extends finite-dimensional results on the Lebesgue measure of stable laminations to infinite-dimensional Banach space dynamical systems.

Findings

Preimages of stable sets intersect generic curves in measure-zero sets.

Under certain transversality conditions, stable laminations have zero Lebesgue measure.

Results generalize finite-dimensional stable lamination measure properties to infinite dimensions.

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 $\mathcal{R}\colon \Omega\rightarrow \Omega$ is onto. The $\delta$-shadow $W^s_\delta(\Omega)$ of $\Omega$ is the union of the sets $$W^s_\delta(G)= \{F\colon dist(\mathcal{R}^iF, \mathcal{R}^iG) \leq \delta, \ for \ every \ i\geq 0 \},$$ where $G \in \Omega$. Suppose that $W^s_\delta(\Omega)$ 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 $W^s_\delta(\Omega)$ we have that $M\cap W^s_\delta(\Omega)$ has empty interior in $M$. Here $n$ is the finite dimension of the strong unstable direction. We show that if $\delta'$ is small enough then $$\cup_{i\geq 0}\mathcal{R}^{-i}W^s_{\delta'} (\Omega)$$ intercepts a $C^k$-generic finite dimensional curve inside the Banach space in a set of parameters with zero Lebesgue measure, for every $k\geq 0$. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.