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There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus

Part of: Ergodic theory Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  18 March 2021

VIVIANE BALADI Open the ORCID record for VIVIANE BALADI [Opens in a new window]
VIVIANE BALADI*
Affiliation:
Laboratoire de Probabilités, Statistique et Modélisation, CNRS, Sorbonne Université, Université de Paris, 4, Place Jussieu, 75005Paris, France
*
e-mail: baladi@lpsm.paris
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Abstract

We show that the ergodic integrals for the horocycle flow on the two-torus associated by Giulietti and Liverani with an Anosov diffeomorphism either grow linearly or are bounded; in other words, there are no deviations. For this, we use the topological invariance of the Artin–Mazur zeta function to exclude resonances outside the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools used in the proof. As a bonus, we show that for any $C^\infty $ Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and $C^\infty $ observables decay with a rate strictly smaller than $e^{-h_{\mathrm {top}}(F)}$ . We compare our results with very recent related work of Forni.

Keywords

low dimensional dynamicssmooth dynamicstransfer operatorszeta functions

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 37C40: Smooth ergodic theory, invariant measures 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 37A25: Ergodicity, mixing, rates of mixing
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 500 - 513
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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