Volume Preserving Diffeomorphisms with Inverse Shadowing

TL;DR

This paper establishes that for volume-preserving diffeomorphisms on a closed manifold, having inverse shadowing properties in the C1-interior is equivalent to being Anosov, linking dynamical stability with shadowing behaviors.

Contribution

It proves the equivalence between inverse shadowing properties' C1-interior and the Anosov condition for volume-preserving diffeomorphisms.

Findings

Inverse shadowing properties are equivalent to the Anosov condition in the C1-interior.

Volume-preserving diffeomorphisms with these properties are structurally stable.

The results unify shadowing concepts with hyperbolic dynamics.

Abstract

Let f be a volume-preserving diffeomorphism of a closed C^\infty n-dimensional Riemannian manifold M: In this paper, we prove the equivalence between the following conditions: (a) f belongs to the C1-interior of the set of volume-preserving diffeoeomorphisms which satisfy the inverse shadowing property with respect to the continuous methods. (b) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the weak inverse shadowing property with respect to the continuous methods. (c) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the orbital inverse shadowing property with respect to the continuous methods, (d) f is Anosov.