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In this paper, we will assume MM to be a compact smooth manifold and f:M→Mf:M\to M to be a diffeomorphism. We herein demonstrate that a C1{C}^{1} generic diffeomorphism ff is Axiom A and has no cycles if ff is asymptotic measure expansive. Additionally, for a C1{C}^{1} generic diffeomorphism ff, if a homoclinic class H(p,f)H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point pp of ff is asymptotic measure-expansive, then H(p,f)H\left(\hspace{0.08em}p,f) is hyperbolic of ff.

Let f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.

In this paper, we show that for generic C1, if a flow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.

Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C1 generic diffeomorphism f, if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

We show that if a vector field X has the C1 robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a generic C1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

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]

We show that if a C¹ generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C¹ generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles. [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]

It is known that if a closed invariant set Λ of a diffeomorphism f of a compact smooth manifold M is hyperbolic then f has the pseudo orbit tracing property. For a weak notion of hyperbolicity, in [4] if a transitive set of a diffeomorphism f of the three dimensional manifold M has a partially hyperbolic structure then f does not have the pseudo orbit tracing property. From the results, we consider a compact smooth manifold M which the dimension is greater than three. It is a general version of the three dimensional manifold M. More detail, we show that if a transitive diffeomorphism f of a compact smooth manifold M is partially hyperbolic, then f does not have the pseudo orbit tracing property. Moreover, if f is partially hyperbolic on M then f does not have the asymtotic and the ergodic pseudo orbit tracing properties. [ABSTRACT FROM AUTHOR]

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

Let f : M → M be a diffeomorphism of a finite dimension, smooth compact Riemannian manifold M. In this paper, we demonstrate that if a diffeomorphism f lies within the C 1 interior of the set of all chain recurrence class-topologically stable diffeomorphisms, then the chain recurrence class is hyperbolic. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for a generically C 1 vector field X of a compact smooth manifold M, if a homoclinic class H (γ , X) which contains a hyperbolic closed orbit γ is measure expansive for X then H (γ , X) is hyperbolic. [ABSTRACT FROM AUTHOR]

We show that if a differentiable map f of a compact smooth Riemannian manifold M is C 1 robustly positive continuum-wise expansive, then f is expanding. Moreover, C 1 -generically, if a differentiable map f of a compact smooth Riemannian manifold M is positively continuum-wise expansive, then f is expanding. [ABSTRACT FROM AUTHOR]