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

Alexander I. Bufetov, Boris Solomyak

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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.

Original languageEnglish
Pages (from-to)205-259
Number of pages55
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - 1 Feb 2018

Bibliographical note

Publisher Copyright:
© 2018, Hebrew University of Jerusalem.


Acknowledgements. We are deeply grateful to Giovanni Forni for our useful discussions of the preliminary version of the paper, and to the referee for constructive criticism and helpful suggestions. The research of A. Bufetov on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 647133 (ICHAOS). It was also supported by the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the programme “Investisse-ments d’Avenir” of the Government of the French Republic, managed by the French National Research Agency (ANR), by the Grant MD 5991.2016.1 of the President of the Russian Federation and by the Russian Academic Excellence Project ‘5-100’. B. Solomyak has been supported by NSF grants DMS-0968879, DMS-1361424, and the Israel Science Foundation grant 396/15.

FundersFunder number
National Science FoundationDMS-0968879, DMS-1361424
Horizon 2020 Framework Programme647133, ANR-11-IDEX-0001-02
European Research Council
Agence Nationale de la RechercheMD 5991.2016.1
Israel Science Foundation396/15


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