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Horocycle averages on closed manifolds and transfer operators

Alexander Adam and Viviane Baladi

Vol. 4 (2022), No. 3, 387–441
Abstract

We study the ergodic integrals of the horocycle flows hρ of Cr codimension one mixing Anosov flows. In dimension three, for any suitably bunched C3 contact Anosov flow with orientable strong-stable distribution E, we show that |1 T0Tφ hρ(x)d ρ μ(φ)| C T𝜖φC3 for some 𝜖 > 0, with μ the invariant measure of hρ. We thereby implement the toy model program of Giulietti–Liverani (2017) in the natural setting of geodesic flows in variable negative curvature, where nontrivial resonances exist.

Keywords
transfer operators, resonances, Anosov flow, horocycle flow
Mathematical Subject Classification
Primary: 37C30
Secondary: 37C20, 37D20
Milestones
Received: 21 May 2021
Revised: 6 December 2021
Accepted: 9 March 2022
Published: 9 November 2022
Authors
Alexander Adam
Aachen, Germany
Viviane Baladi
Sorbonne Université and Université Paris Cité
CNRS, Laboratoire de Probabilités, Statistique et Modélisation
F-75005 Paris
France