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$\unicode[STIX]{x1D6FA}$be a compact, forward invariant and partially hyperbolic set of${\mathcal{R}}$such that${\mathcal{R}}:\unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6FA}$is onto. The$\unicode[STIX]{x1D6FF}$-shadow$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$of$\unicode[STIX]{x1D6FA}$is the union of the sets$$\begin{eqnarray}W_{\unicode[STIX]{x1D6FF}}^{s}(G)=\{F:\operatorname{dist}({\mathcal{R}}^{i}F,{\mathcal{R}}^{i}G)\leq \unicode[STIX]{x1D6FF}\text{for every }i\geq 0\},\end{eqnarray}$$where$G\in \unicode[STIX]{x1D6FA}$. Suppose that$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$has transversal empty interior, that is, for every$C^{1+\text{Lip}}$$n$-dimensional manifold$M$transversal to the distribution of dominated directions of$\unicode[STIX]{x1D6FA}$and sufficiently close to$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$we have that$M\cap W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$has empty interior in$M$. Here$n$is the finite dimension of the strong unstable direction. We show that if$\unicode[STIX]{x1D6FF}^{\prime }$is small enough then$$\begin{eqnarray}\mathop{\bigcup }_{i\geq 0}{\mathcal{R}}^{-i}W_{\unicode[STIX]{x1D6FF}^{\prime }}^{s}(\unicode[STIX]{x1D6FA})\end{eqnarray}$$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.