Marked length spectrum rigidity for Anosov magnetic surfaces

TL;DR

This paper proves that on closed surfaces, conjugate Anosov magnetic systems with volume-preserving isotopic conjugacy and same cohomology class are actually isometric, extending previous rigidity results to magnetic systems.

Contribution

It extends the length spectrum rigidity result to magnetic systems on surfaces, showing isometry under specific conjugacy conditions.

Findings

Conjugate Anosov magnetic systems with volume-preserving isotopy are isometric.

The result generalizes previous rigidity theorems to magnetic settings.

Magnetic forms in the same cohomology class lead to isometric metrics.

Abstract

We show that if $M$ is a closed, connected, oriented surface, and two Anosov magnetic systems on $M$ are conjugate by a volume-preserving conjugacy isotopic to the identity, with their magnetic forms in the same cohomology class, then the metrics are isometric. This extends the recent result by Guillarmou, Lefeuvre, and Paternain to the magnetic setting.