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In this paper, we show the followings: (i) If a volume preserving diffeomorphism f belongs to the C¹-interior of the set of all volume preserving diffeomorphims having the ergodic shadowing property then it is transitive Anosov. Moreover, (ii) if a C¹-generic volume-preserving diffeomorphism f has the ergodic shadowing property then it is transitive Anosov. [ABSTRACT FROM AUTHOR]

Let ${\it\gamma}$ be a hyperbolic closed orbit of a $C^{1}$ vector field $X$ on a compact $C^{\infty }$ manifold $M$ and let $H_{X}({\it\gamma})$ be the homoclinic class of $X$ containing ${\it\gamma}$. In this paper, we prove that if a $C^{1}$-persistently expansive homoclinic class $H_{X}({\it\gamma})$ has the shadowing property, then $H_{X}({\it\gamma})$ is hyperbolic. [ABSTRACT FROM AUTHOR]

Let f be a diffeomorphism on a C∞ closed surface. In this paper, we show that if f has the C2-stably limit shadowing property, then we have the following: (i) f satisfies the Kupka-Smale condition; (ii) if P(f) is dense in the nonwandering set Ω(f) and if there is a dominated splitting on Ps(f), then f satisfies both Axiom A and the strong transversality condition. [ABSTRACT FROM AUTHOR]

We prove that forC1generic diffeomorphisms, every measure-expansive locally maximal homoclinic class is hyperbolic. [ABSTRACT FROM PUBLISHER]

Let M be a closed smooth manifold M, and let f: M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f. We show that if f has the C-stably average shadowing property on Λ, then Λ admits a dominated splitting. [ABSTRACT FROM AUTHOR]

Abstract: Let p be a hyperbolic periodic point of a diffeomorphism f on a closed manifold M. Introducing here the notion of -persistently continuum-wise expansivity, we show that if the homoclinic class of f associated to p is -persistently continuum-wise expansive then (i) it admits a dominated splitting and (ii) it is hyperbolic provided it satisfies the chain condition. Moreover, we show that the chain recurrent set of f is -persistently continuum-wise expansive if and only if f satisfies both Axiom A and the no-cycle condition. [Copyright &y& Elsevier]