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On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces

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

Published online by Cambridge University Press:  07 December 2021

GIOVANNI FORNI
GIOVANNI FORNI*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD, USA
*
e-mail: [email protected]
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Abstract

We prove that the asymptotics of ergodic integrals along an invariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, is determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [Parabolic dynamics and anisotropic Banach spaces. J. Eur. Math. Soc. (JEMS)21(9) (2019), 2793–2858] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1, e^{h_{\mathrm {top}}})$ .

Keywords

Anosov diffeomorphismsdeviation of ergodic averages for invariant foliationsRuelle asymptotics

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37A25: Ergodicity, mixing, rates of mixing 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 855 - 880
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

In memory of Anatole Katok, with admiration and gratitude

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