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We introduce a notion of shadowing property for actions of finitely generated groups and study its basic properties. We formulate and prove a shadowing lemma for actions of nilpotent groups. We construct an example of a faithful linear action of a solvable Baumslag–Solitar group and show that the shadowing property depends on quantitative characteristics of hyperbolicity. Finally, we show that any linear action of a non-abelian free group does not have the shadowing property. [ABSTRACT FROM AUTHOR]

Let Diff(M) be the space of diffeomorphisms of a closed C ∞ manifold M endowed with the C 1-topology, and denote by ℛ(f) the chain recurrent set of f ∈ Diff(M). In this article, we show that C 1-generically f |ℛ(f) has the s-limit shadowing property if and only if f satisfies both Axiom A and no-cycle condition. [ABSTRACT FROM PUBLISHER]

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]