Abstract
Abstract 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.
Highlights
Let M be a closed smooth Riemannian manifold with dimM ≥, and let Di (M) be the space of di eomorphisms of M endowed with the C topology
We show that if f has the C robustly asymptotic orbital shadowing property it is an Anosov di eomorphism
Robinson [34] and Sakai [37] proved that f belongs to the C interior of the set of all di eomorphisms having the shadowing property if and only if it is structurally stable
Summary
Let M be a closed smooth Riemannian manifold with dimM ≥ , and let Di (M) be the space of di eomorphisms of M endowed with the C topology. Lee and Wen [20] proved that for a C generic f , if f has the shadowing property on a locally maximal chain transitive set it is hyperbolic. The remarkable result of this problem, Lee [29] proved that for C generic f of a two dimensional smooth manifold M, if f has the average shadowing property or the asymptotic average shadowing property it is Anosov. Since the asymptotic orbital shadowing property is a general notion of the limit shadowing property, such as those results before, we will prove the following. To prove Theorem B, we will show that for a C generic f , if f has the asymptotic orbital shadowing property it is a transitive Anosov di eomorphism
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
