Shy shadows of infinite-dimensional partially hyperbolic invariant sets

Let R be a strongly compact C-2 map defined in an open subset of an infinitedimensional Banach space such that the image of its derivative DFR is dense for every F. Let Omega be a compact, forward invariant and partially hyperbolic set of R such that R: Omega -> Omega is onto. The delta hadow W-delta(s)(Omega) of Omega is the union of the sets W-delta(s)(Omega = {F : dist((RF)-F-i, R(i)G) <= delta for every i >= 0}, where G is an element of Omega. Suppose that W-delta(s)(Omega) has transversal empty interior, that is, for every C1+Lip n-dimensional manifold M transversal to the distribution of dominated directions of and sufficiently close to W-delta(s)(Omega) we have that M boolean AND W-delta(s)(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 boolean OR(i >= 0) (R-iW delta's)(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 >= 0. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.