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 $\mathrm{\Omega }$ be a compact, forward invariant and partially hyperbolic set of $\mathcal{R}$ such that $\mathcal{R}\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to \mathrm{\Omega }$ is onto. The $\delta$-shadow $W_{\delta }^s(\mathrm{\Omega })$ of $\mathrm{\Omega }$ is the union of the sets $$W_{\delta }^s(G)=\{F\text{\hspace{0.17em}⁣}:dist({\mathcal{R}}^{i}F,{\mathcal{R}}^{i}G)\le \delta ,foreveryi\ge 0\},$$ where $G\in \mathrm{\Omega }$. Suppose that $W_{\delta }^s(\mathrm{\Omega })$ has transversal empty interior, that is, for every $C^{1+Lip}$ $n$-dimensional manifold $M$ transversal to the distribution of dominated directions of $\mathrm{\Omega }$ and sufficiently close to $W_{\delta }^s(\mathrm{\Omega })$ we have that $M\cap W^s_\delta(\Omega)$hasemptyinteriorin$M$\mathrm{.}Here$n$ is the finite dimension of the strong unstable direction. We show that if ${\delta }^{\mathrm{\prime }}$ is small enough then $${\cup }_{i\ge 0}{\mathcal{R}}^{-i}{W}_{{\delta }^{\mathrm{\prime }}}^s(\mathrm{\Omega })$$ 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.