Smooth orbit equivalence rigidity for dissipative geodesic flows

TL;DR

This paper establishes that smooth orbit equivalences between certain dissipative geodesic flows on surfaces imply conformal equivalence of the underlying metrics, extending rigidity results to non-preserving measure flows.

Contribution

It proves a new orbit equivalence rigidity result for dissipative Gaussian thermostats and magnetic flows on surfaces with negative thermostat curvature.

Findings

Orbit equivalence implies conformal metric equivalence.

Results extend to Anosov magnetic flows.

Provides relationships between thermostat forms and background metrics.

Abstract

Let $M$ be a smooth closed oriented surface. Gaussian thermostats on $M$ correspond to the geodesic flows arising from metric connections, including those with non-zero torsion. These flows may not preserve any absolutely continuous measure. We prove that if two Gaussian thermostats on $M$ with negative thermostat curvature are related by a smooth orbit equivalence isotopic to the identity, then the two background metrics are conformally equivalent via a smooth diffeomorphism of $M$ isotopic to the identity. We also give a relationship between the thermostat forms themselves. Finally, we prove the same result for Anosov magnetic flows.