One-to-Oneness, Persistence, and Hyperbolicity

Abstract

The persistence of a (compact) normally hyperbolic manifold and the existence of its stable and unstable manifolds, corresponding to the action of C ^r maps or semiflows, r ≥ 1, are well known facts (see, for instance, [102] and [176]). The more general case of (compact) hyperbolic sets that are invariant under (not necessarily injective) maps was also carefully studied (see Ruelle [176], Shub [189] and Palis and Takens [163]). The persistence and smoothness of (compact) hyperbolic invariant manifolds for RFDE were considered in detail in Magalhães [132] based on skew-product semiflows defined for RFDE, locally around hyperbolic invariant manifolds, and their spectral properties, associated with exponential dichotomies, following the lines of work developed by Sacker and Sell for flows([186], [187]).