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Let ff be a C2{C}^{2}-diffeomorphism with Axiom A and no cycle condition on a two-dimensional smooth manifold. In this article, we prove that if ff is C2{C}^{2}-robustly weak measure expansive, then it is Q2{Q}^{2}-Anosov. Moreover, we expand the results of the C2{C}^{2}-diffeomorphism case into the C2{C}^{2}-vector field on a three-dimensional smooth manifold. Let XX be a C2{C}^{2}-vector field with Axiom A and no cycle condition. We prove that if XX is C2{C}^{2}-robustly weak measure expansive, then it is Q2{Q}^{2}-Anosov.

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.

In this paper, we consider positively weak measure expansive homeomorphisms and flows with the shadowing property on a compact metric space X. Moreover, we prove that if a homeomorphism (or flow) has a positively weak expansive measure and the shadowing property on its nonwandering set, then its topological entropy is positive.

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].