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Let f : M → M be a diffeomorphism of compact smooth Riemannian manifold M, an let Λ ⊂ M be a closed f-invariant set. We obtain conditions for Λ to be topologically stable which is called Λ -topologically stable. Moreover, we prove that if f is C 1 robustly Λ -topologically stable then Λ satisfies star condition for f. Then in the above, if a closed f-invariant set Λ is chain transitive (or transitive) then it is hyperbolic for f. [ABSTRACT FROM AUTHOR]

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]

In this paper, we introduce the notion of the asymptotic orbital pseudo-orbit tracing property for a vector field X of a compact smooth manifold M. We show that if a vector field X has the C 1 robust asymptotic orbital pseudo-orbit tracing property, then it is Anosov, and if a C 1 generic vector field X has the asymptotic orbital pseudo-orbit tracing property, then it is Anosov. Moreover, we also show our results for divergence-free vector fields and Hamiltonian systems. [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 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]

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]

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]

In the study, we consider continuum-wise expansiveness for the homoclinic class of a kind of C 1 -robustly expansive dynamical system. First, we show that if the homoclinic class H (p , f) , which contains a hyperbolic periodic point p, is R-robustly continuum-wise expansive, then it is hyperbolic. For a vector field, if the homoclinic class H (γ , X) does not include singularities and is R-robustly continuum-wise expansive, then it is hyperbolic. [ABSTRACT FROM AUTHOR]