Abstract
We obtain a qualitative characterization of the convergence rate of the averages (with respect to the Margulis measure) of C2 functions over the iterations of domains in unstable manifolds of a topologically mixing C3 Anosov diffeomorphism with oriented invariant foliations. For this purpose, we extend the constructions of Margulis and Bufetov and introduce holonomy invariant families of finitely additive measures on unstable leaves and a Banach space in which holonomy invariant measures correspond to the (generalized) eigenfunctions of the transfer operator with biggest eigenvalues.
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Acknowledgments
The author is a winner of the contest “Young Russian Mathematics” and expresses deep gratitude to the jury and trustees of the contest.
Funding
This work was supported by the 2018–2019 program “Scientific Foundation of the National Research University Higher School of Economics” (project no. 18-05-0019) and by the 5–100 Russian Academic Excellence Project for support of the leading universities of the Russian Federation.
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Russian Text © The Author (s), 2019. Published in Funktsional’ nyi Analiz i Ego Prilozheniya, 2019, Vol. 53, No. 3, pp. 92–97.
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Zubov, D.I. Finitely Additive Measures on the Unstable Leaves of Anosov Diffeomorphisms. Funct Anal Its Appl 53, 232–236 (2019). https://doi.org/10.1134/S0016266319030092
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DOI: https://doi.org/10.1134/S0016266319030092