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Shy shadows of infinite-dimensional partially hyperbolic invariant sets



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.



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