Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows

TL;DR

This paper investigates how volume-preserving perturbations of geodesic flows on negatively curved surfaces affect measure disintegration, revealing conditions under which the perturbation remains a flow or becomes atomic.

Contribution

It establishes a dichotomy for perturbations of geodesic flows based on measure disintegration properties, linking absolute continuity to flow structure preservation.

Findings

If Liouville measure disintegrates Lebesgue-wise, the perturbation is a smooth flow.

Otherwise, the measure disintegration is atomic.

The results connect measure disintegration with flow rigidity.

Abstract

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.