Resonant forms at zero for dissipative Anosov flows

Cekić, Mihajlo; Paternain, Gabriel P

doi:10.2140/gt.2025.29.3635

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Resonant forms at zero for dissipative Anosov flows

Mihajlo Cekić and Gabriel P Paternain

Geometry & Topology 29 (2025) 3635–3716

DOI: 10.2140/gt.2025.29.3635

Abstract

We study resonant differential forms at zero for transitive Anosov flows on $3$ -manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai–Ruelle–Bowen 3-forms, $Ω_{SRB}^{\pm}$ , and the cohomology classes $[ι_{X} Ω_{SRB}^{\pm}]$ (where $X$ is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant 1-forms. When both classes vanish we associate to the flow a helicity that naturally extends the classical notion associated with null-homologous volume-preserving flows. We provide a general theory that includes horocyclic invariance of resonant $1$ -forms and SRB-measures as well as the local geometry of the maps $X \mapsto [ι_{X} Ω_{SRB}^{\pm}]$ near a null-homologous volume-preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys (1992). For these flows we compute explicitly all resonant 1-forms at zero, we show that $[ι_{X} Ω_{SRB}^{\pm}] = 0$ and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is $- χ (M)$ , the same as in the geodesic flow case. In contrast, we show that if $(M, g)$ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic $1$ -form has a Ruelle zeta function whose order of vanishing at zero is $- χ (M) - 1$ .

To the memory of Will Merry.

Keywords

dissipative Anosov flow, SRB measure, winding cycle, Ruelle zeta function, quasi-Fuchsian flow, thermostat, microlocal analysis, helicity, coupled vortex equations

Mathematical Subject Classification

Primary: 35P05, 37C30, 37D40, 53C65, 58C40

References

Publication

Received: 16 November 2022

Revised: 8 July 2024

Accepted: 4 January 2025

Published: 10 October 2025

Proposed: Anna Wienhard

Seconded: David Fisher, Leonid Polterovich

Authors

Mihajlo Cekić
Institut für Mathematik University of Zurich Zürich Switzerland

Gabriel P Paternain
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge United Kingdom	Department of Mathematics University of Washington Seattle, WA United States

Open Access made possible by participating institutions via Subscribe to Open.

Volume 29, issue 7 (2025)