The Hölder property for the spectrum of translation flows in genus two
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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.
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- A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. (2006), 143᾿11.Google Scholar
- L. Barreira and Y. Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, Vol. 115, Cambridge University Press, Cambridge, 2007, Dynamics of systems with nonzero Lyapunov exponents.Google Scholar
- V. Berthé, W. Steiner and J. M. Thuswaldner, Geometry, dynamics, and arithmetic of s-adic shifts, arXiv:1410.0331, preprint (2014).Google Scholar
- A. I. Bufetov, Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichm üller flow on the moduli space of abelian differentials, J. Amer.Math. Soc. 19 (2006), 579᾿23.MathSciNetCrossRefMATHGoogle Scholar
- N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002, Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel.Google Scholar
- H. Furstenberg, Stationary processes and prediction theory, Annals of Mathematics Studies, No. 44, Princeton University Press, Princeton, N.J., 1960.Google Scholar
- P. Varjú, Absolute continuity of bernoulli convolutions for algebraic parameters, arXiv:1602.00261, preprint (2016).Google Scholar
- M. Viana, Interval exchange transformations and teichmüller flows, IMPA, preprint, 2008.Google Scholar