Hyperbolicity, transitivity and the two-sided limit shadowing property

TL;DR

This paper characterizes the set of diffeomorphisms with the two-sided limit shadowing property, showing it coincides with transitive Anosov diffeomorphisms on closed manifolds, and explores its implications for dynamical systems.

Contribution

It provides a characterization of the $C^1$-interior of diffeomorphisms with the property, linking it to transitive Anosov diffeomorphisms and reducing a major conjecture in the field.

Findings

The $C^1$-interior of diffeomorphisms with the property equals transitive Anosov diffeomorphisms.

All codimension-one Anosov diffeomorphisms have the property.

$C^1$-generic diffeomorphisms with the property are transitive Anosov.

Abstract

We explore the notion of two-sided limit shadowing property introduced by Pilyugin \cite{P1}. Indeed, we characterize the $C^1$-interior of the set of diffeomorphisms with such a property on closed manifolds as the set of transitive Anosov diffeomorphisms. As a consequence we obtain that all codimention-one Anosov diffeomorphisms have the two-sided limit shadowing property. We also prove that every diffeomorphism $f$ with such a property on a closed manifold has neither sinks nor sources and is transitive Anosov (in the Axiom A case). In particular, no Morse-Smale diffeomorphism have the two-sided limit shadowing property. Finally, we prove that $C^1$-generic diffeomorphisms on closed manifolds with the two-sided limit shadowing property are transitive Anosov. All these results allow us to reduce the well-known conjecture about the transitivity of Anosov diffeomorphisms on closed manifolds to prove that the set of diffeomorphisms with the two-sided limit shadowing property coincides with the set of Anosov diffeomorphisms.