Horocycle averages on closed manifolds and transfer operators

We study the ergodic integrals of the horocycle flows $h_{ρ}$ of $C^{r}$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^{3}$ contact Anosov flow with orientable strong-stable distribution $E_{-}$ , we show that $| \frac{1}{T} \int_{0}^{T} φ \circ h_{ρ} (x) d ρ - μ (φ) | \leq \frac{C}{T^{𝜖}} ∥ φ ∥_{C^{3}}$ for some $𝜖 > 0$ , with $μ$ the invariant measure of $h_{ρ}$ . 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.

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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

Vol. 4, No. 3, 2022

Alexander Adam and Viviane Baladi

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors