Pinching conditions, linearization and regularity of Axiom A flows

TL;DR

This paper investigates the regularity of Axiom A flows, showing that under certain pinching and integrability conditions, the flow exhibits regular distortion and is locally Lipschitz conjugate to its linearization.

Contribution

It establishes conditions under which Axiom A flows have regular distortion and are Lipschitz conjugate to their linearization, advancing understanding of their spectral and geometric properties.

Findings

Flow has regular distortion under pinched spectrum conditions.

Flow is Lipschitz conjugate to linearization over the pinched unstable bundle.

Spectral properties influence geometric regularity of the flow.

Abstract

In this paper we study a certain regularity property of Axiom A flows over basic sets related to diameters of balls in Bowen's metric, which we call regular distortion along unstable manifolds. The motivation to investigate the latter comes from the study of spectral properties of Ruelle transfer operators. We prove that if the bottom of the spectrum of the tangent map of the flow over unstable manifolds is point-wisely pinched and integrable, then the flow has regular distortion along unstable manifolds over the basic set. In the process, under the same conditions, we show that locally the flow is Lipschitz conjugate to its linearization over the `pinched part' of the unstable tangent bundle.