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In this paper, we extend the notion of asymptotic measure expansivity for diffeomorphisms to flows on a compact metric space, and prove that there exists an asymptotic measure expansive flow in that space. Then we consider the hyperbolicity of the asymptotic measure expansive vector fields on a closed smooth Riemannian manifold M. More precisely, we prove that any C1 stably asymptotic measure expansive vector field on M satisfy Axiom A without cycles, and it is quasi-Anosov. On the other hand, we show that C1 generically, every asymptotic measure expansive vector field on M satisfies Axiom A without cycles. Moreover, we study the asymptotic measure expansivity of the homoclinic classes which are the delegate of the invariant subsets for given systems, prove that C1 stably and C1 generically, the asymptotic measure expansive homoclinic class is hyperbolic. Furthermore, we consider the hyperbolicity of asymptotic measure expansive divergence-free vector fields for C1 stably and C1 generic point of view. We also apply the our results to the divergence-free vector fields. [ABSTRACT FROM AUTHOR]

Let f be a diffeomorphism on a compact smooth Riemannian manifold M, and let Λ be a closed f-invariant transitive subset of M. In this paper, we show that f|Λ is C1-stably expansive if and only if Λ is a hyperbolic basic set. Moreover, C1-generically, a locally maximal transitive set Λ is expansive if and only if it is a hyperbolic basic set. [ABSTRACT FROM AUTHOR]

Let be a symplectic diffeomorphism on a closed -dimensional Riemannian manifold . In this paper, we show that is Anosov if any of the following statements holds: belongs to the -interior of the set of symplectic diffeomorphisms satisfying the limit shadowing property or belongs to the -interior of the set of symplectic diffeomorphisms satisfying the limit weak shadowing property or belongs to the -interior of the set of symplectic diffeomorphisms satisfying the s-limit shadowing property. [ABSTRACT FROM AUTHOR]

Let f: M → M( d ≥ 2) be a diffeomorphism on a compact C manifold on M. If a diffeomorphism f belongs to the C-interior of the set of all diffeomorphisms having the barycenter property, then f is Ω-stable. Moreover, if a generic diffeomorphism f has the barycenter property, then f is Ω-stable. We also apply our results to volume preserving diffeomorphisms. [ABSTRACT FROM AUTHOR]

Let be a -dimensional (), compact smooth Riemannian manifold endowed with a symplectic form . In this paper, we show that, if a symplectic diffeomorphism is -robustly measure expansive, then it is Anosov and a generic measure expansive symplectic diffeomorphism is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system is robustly measure expansive, then is Anosov. [ABSTRACT FROM AUTHOR]

Let f: M → M be a diffeomorphism on a closed smooth d( d ≥ 2)-dimensional manifold. For each $n\in \mathbb N$ , if f belongs to C -interior of the set of the n-expansive diffeomorphisms, then f satisfies quasi-Anosov. For C -generic f, if f is n-expansive then f satisfies both Axiom A and the no-cycle condition. [ABSTRACT FROM AUTHOR]

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 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 present the following: (1) if a volume-preserving diffeomorphism has C1-robustly various limit shadowing property, then it is Anosov; (2) C1-generically, if a volume-preserving diffeomorphism has various limit shadowing property, then it is Anosov. [ABSTRACT FROM AUTHOR]

We prove that C¹-generically, continuum-wise expansive diffeomorphisms satisfy both Axiom A and the no-cycle condition. Moreover, (i) if a volume-preserving diffeomorphism belongs to the C¹-interior of the set of all continuum-wise expansive volume-preserving diffeomorphisms then it is Anosov, and (ii) C¹-generically, every continuum-wise expansive volume-preserving diffeomorphism is transitive Anosov. [ABSTRACT FROM AUTHOR]

We prove that any C-stable weakly shadowable volume-preserving diffeomorphism de-fined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity. [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]

Let f be a diffeomorphism of a closed C ∞ manifold M, and let Λ ⊂ M be a closed f-invariant set. We show that if f |Λ is robustly chain transitive with orbital shadowing property, then Λ is a basic set. [ABSTRACT FROM PUBLISHER]

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]

Abstract: Let be a hyperbolic periodic saddle of a diffeomorphism of on a closed smooth manifold , and let be the homoclinic class of containing . In this paper, we show that if is locally maximal and every hyperbolic periodic point in is uniformly far away from being nonhyperbolic, and has the average shadowing property, then is hyperbolic. [Copyright &y& Elsevier]

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]