C1-stably shadowable conservative diffeomorphisms are Anosov

TL;DR

This paper proves that symplectomorphisms and volume-preserving diffeomorphisms that are C1-stably shadowable must be Anosov, establishing a strong link between shadowing stability and hyperbolic dynamics.

Contribution

It demonstrates that C1-stable shadowability implies Anosov behavior for symplectomorphisms and volume-preserving diffeomorphisms, a new result in dynamical systems theory.

Findings

C1-stably shadowable symplectomorphisms are Anosov

C1-stably shadowable volume-preserving diffeomorphisms are Anosov

Establishes a connection between shadowing stability and hyperbolic structure

Abstract

In this short note we prove that if a symplectomorphism f is C1-stably shadowable, then f is Anosov. The same result is obtained for volume-preserving diffeomorphisms.