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The Hölder property for the spectrum of translation flows in genus two

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Abstract

The paper is devoted to generic translation flows corresponding to Abelian differentials with one zero of order two on flat surfaces of genus two. These flows are weakly mixing by the Avila–Forni theorem. Our main result gives first quantitative estimates on their spectrum, establishing the Hölder property for the spectral measures of Lipschitz functions. The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdős and Kahane.

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Authors and Affiliations

  1. Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 rue F. Joliot Curie, Marseille, France

    Alexander I. Bufetov

  2. Steklov Mathematical Institute of RAS, Moscow, Russia

    Alexander I. Bufetov

  3. Institute for Information Transmission Problems, Moscow, Russia

    Alexander I. Bufetov

  4. National Research University Higher School of Economics, Moscow, Russia

    Alexander I. Bufetov

  5. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Boris Solomyak

Authors
  1. Alexander I. Bufetov
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  2. Boris Solomyak
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Correspondence to Alexander I. Bufetov.

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To the memory of William Austin Veech (1938–2016)

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Bufetov, A.I., Solomyak, B. The Hölder property for the spectrum of translation flows in genus two. Isr. J. Math. 223, 205–259 (2018). https://doi.org/10.1007/s11856-017-1614-8

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  • DOI: https://doi.org/10.1007/s11856-017-1614-8

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