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Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization of nonuniform hyperbolicity and show that it is equivalent to invertibility of the same operator on a larger, Fréchet space. We apply it to obtain a condition for a diffeomorphism on the boundary of the set of Anosov diffeomorphisms to be nonuniformly hyperbolic. Finally, we generalize the Shadowing lemma in the same context. [ABSTRACT FROM AUTHOR]

Let f be a C2-diffeomorphism on a closed surface. In this article, we show that if f has the C2-stably inverse shadowing property with respect to the class of continuous methods then (i) f is Kupka-Smale, and (ii) if the periodic points are dense in the non-wandering set Ω(f) and there is a dominated splitting on the closure of periodic points of the saddle type Ph(f), then f satisfies both Axiom A and the strong transversality condition. [ABSTRACT FROM AUTHOR]

It is shown how known results for autonomous difference equations can be adapted to definitions of semi-hyperbolicity and bi-shadowing generalized to nonautonomous difference equations with Lipschitz continuous mappings. In particular, invertibility and smoothness of the mappings are not required and, for greater applicability, the mappings are allowed to act between possibly different Banach spaces. [ABSTRACT FROM AUTHOR]