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 $mathcal{R}colon Omegarightarrow Omega$ is onto. The $delta$-shadow $W^s_delta(Omega)$ of $Omega$ is the union of the sets $$W^s_delta(G)= {Fcolon dist(mathcal{R}^iF, mathcal{R}^iG) leq delta, for every igeq 0 },$$ 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 $Mcap W^s_delta(Omega)$ 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 $$cup_{igeq 0}mathcal{R}^{-i}W^s_{delta} (Omega)$$ intercepts a $C^k$-generic finite dimensional curve inside the Banach space in a set of parameters with zero Lebesgue measure, for every $kgeq 0$. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.
Download