Vol. 2, No. 4, 2020

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Resonant spaces for volume-preserving Anosov flows

Mihajlo Cekić and Gabriel P. Paternain

Vol. 2 (2020), No. 4, 795–840

We consider Anosov flows on closed 3-manifolds preserving a volume form Ω. Following Dyatlov and Zworski (Invent. Math. 210:1 (2017), 211–229) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class [ιXΩ] (where X is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott–Ruelle resonance multiplicities under the assumption of semisimplicity. We prove various results regarding semisimplicity on 1-forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows. We also study non-null-homologous deformations of contact Anosov flows, and we show that there is always a splitting Pollicott–Ruelle resonance on 1-forms and that semisimplicity persists in this instance. These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in the resonant spaces and in the Ruelle zeta function.

Anosov flow, resonances, dynamical zeta functions
Mathematical Subject Classification
Primary: 37D20
Received: 28 November 2019
Revised: 3 July 2020
Accepted: 4 August 2020
Published: 25 February 2021
Mihajlo Cekić
Laboratoire de Mathématiques d’Orsay
CNRS, Université Paris-Saclay
Gabriel P. Paternain
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom