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

Published online by Cambridge University Press:  18 March 2021

VIVIANE BALADI*
Affiliation:
Laboratoire de Probabilités, Statistique et Modélisation, CNRS, Sorbonne Université, Université de Paris, 4, Place Jussieu, 75005Paris, France

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.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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