Conservative flows with various types of shadowing

TL;DR

This paper investigates the robustness of shadowing properties in conservative flows, showing that certain shadowing properties imply strong hyperbolic structures like dominated splitting and Anosov flows.

Contribution

It establishes new links between shadowing properties and hyperbolic structures in incompressible and Hamiltonian flows, highlighting their implications for flow robustness.

Findings

Average and asymptotic shadowing imply dominated splitting.

Limit shadowing implies the flow is Anosov.

Results apply to both incompressible and Hamiltonian flows.

Abstract

In the present paper we study the C1-robustness of the three properties: average shadowing, asymptotic average shadowing and limit shadowing within two classes of conservative flows: the incompressible and the Hamiltonian ones. We obtain that the first two properties guarantee dominated splitting (or partial hyperbolicity) on the whole manifold, and the third one implies that the flow is Anosov.