Lipschitz Shadowing for Flows

TL;DR

This paper establishes a precise equivalence between Lipschitz shadowing properties of flows and classical stability notions of the underlying vector fields, linking geometric shadowing with dynamical stability.

Contribution

It proves that Lipschitz shadowing characterizes structural stability and Lipschitz periodic shadowing characterizes $ ext{Omega}$-stability for flows generated by smooth vector fields.

Findings

Lipschitz shadowing is equivalent to structural stability.

Lipschitz periodic shadowing is equivalent to $ ext{Omega}$-stability.

Provides a geometric criterion for classical stability concepts.

Abstract

Let $\phi$ be the flow generated by a smooth vector field $X$ on a smooth closed manifold. We show that the Lipschitz shadowing property of $\phi$ is equivalent to the structural stability of $X$ and that the Lipschitz periodic shadowing property of $\phi$ is equivalent to the $\Omega$-stability of $X$.