Marked magnetic action rigidity

TL;DR

This paper investigates whether the magnetic action spectrum uniquely determines the metric and magnetic field in Anosov magnetic flows, providing partial answers in local and conformal settings.

Contribution

It extends the marked length rigidity conjecture to magnetic systems, establishing conditions under which the spectrum determines the system.

Findings

Spectrum determines the metric and 1-form locally for close systems.

Spectrum determines the metric and 1-form within the same conformal class.

Abstract

An exact magnetic system over a closed manifold $M$ consists of a pair $(g,\alpha)$, where $g$ is a Riemannian metric and $\alpha$ is a 1-form encoding a magnetic field. In this context, we consider a generalization of the marked length rigidity conjecture: does the marked magnetic action spectrum of magnetic systems with Anosov magnetic flow determine the metric and the 1-form, up to a natural obstruction? In this article we answer this question in two settings: 1) locally for systems with close metrics and 1-forms and 2) for metrics in the same conformal class.