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Smooth orbit equivalence rigidity for dissipative geodesic flows

Javier Echevarría Cuesta

TL;DR

This work proves smooth orbit equivalence rigidity for dissipative Gaussian thermostats and Anosov magnetic flows on closed surfaces with negative thermostat curvature. By lifting Guillarmou's framework to thermostats, it shows that an orbit equivalence isotopic to the identity enforces a conformal relation between background metrics, up to a diffeomorphism isotopic to the identity, and ties the thermostat forms via exactness conditions when certain closedness hypotheses hold. The strategy hinges on recovering the complex structure from the orbit data using Torelli's theorem and a period-matrix preservation argument for holomorphic differentials, supported by a novel pairing between thermostat geodesics and fibrewise currents, and overcoming divergence via an invariant current space and microlocal analysis. A key technical ingredient is the attenuated tensor tomography for Gaussian thermostats with negative curvature, enabling control over Fourier modes and enabling the holomorphic-extension and period-preservation machinery. Together, these results extend geodesic and magnetic rigidity to the broader thermostat setting and clarify how conformal changes and thermostat-forms interact under smooth orbit equivalences, with further open questions on higher Fourier-degree thermostats.

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.

Smooth orbit equivalence rigidity for dissipative geodesic flows

TL;DR

This work proves smooth orbit equivalence rigidity for dissipative Gaussian thermostats and Anosov magnetic flows on closed surfaces with negative thermostat curvature. By lifting Guillarmou's framework to thermostats, it shows that an orbit equivalence isotopic to the identity enforces a conformal relation between background metrics, up to a diffeomorphism isotopic to the identity, and ties the thermostat forms via exactness conditions when certain closedness hypotheses hold. The strategy hinges on recovering the complex structure from the orbit data using Torelli's theorem and a period-matrix preservation argument for holomorphic differentials, supported by a novel pairing between thermostat geodesics and fibrewise currents, and overcoming divergence via an invariant current space and microlocal analysis. A key technical ingredient is the attenuated tensor tomography for Gaussian thermostats with negative curvature, enabling control over Fourier modes and enabling the holomorphic-extension and period-preservation machinery. Together, these results extend geodesic and magnetic rigidity to the broader thermostat setting and clarify how conformal changes and thermostat-forms interact under smooth orbit equivalences, with further open questions on higher Fourier-degree thermostats.

Abstract

Let be a smooth closed oriented surface. Gaussian thermostats on 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 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 isotopic to the identity. We also give a relationship between the thermostat forms themselves. Finally, we prove the same result for Anosov magnetic flows.
Paper Structure (26 sections, 35 theorems, 130 equations, 1 figure)

This paper contains 26 sections, 35 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy
  4. Organization of the paper
  5. Preliminaries
  6. Unit tangent bundle of the surface
  7. Geometry of $SM$
  8. Appearance of the divergence
  9. Divergence and smooth orbit equivalences
  10. Vertical Fourier expansions
  11. Complex geometry
  12. Complex structures.
  13. Fiberwise holomorphic distributions
  14. Torelli's theorem
  15. Hyperbolic dynamics
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $(M, g, \lambda)$ and $(M, \tilde{g}, \tilde{\lambda})$ be two Anosov magnetic systems. If there is a smooth orbit equivalence isotopic to the identity between them, then there exists a smooth diffeomorphism $\psi : M\to M$ isotopic to the identity such that $\psi^*\tilde{g}=e^{2f}g$ for some $f

Figures (1)

  • Figure 1: The relevant subbundles of the tangent and cotangent bundles.

Theorems & Definitions (62)

  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 52 more