One-to-Oneness, Persistence, and Hyperbolicity
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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,  and ). The more general case of (compact) hyperbolic sets that are invariant under (not necessarily injective) maps was also carefully studied (see Ruelle , Shub  and Palis and Takens ). The persistence and smoothness of (compact) hyperbolic invariant manifolds for RFDE were considered in detail in Magalhães  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(, ).
KeywordsBanach Space Invariant Manifold Compact Manifold Unstable Manifold Exponential Dichotomy
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