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Adv. Math., 2014, Volume 260, Pages 84–129 (Mi admat9)  

On the modulus of continuity for spectral measures in substitution dynamics

A. I. Bufetovabcde, B. Solomyakf

a Rice University, Houston, TX, USA
b The Institute for Information Transmission Problems, Moscow, Russia
c Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France
d Steklov Institute, Moscow, Russia
e National Research University Higher School of Economics, Moscow, Russia
f Box 354350, Department of Mathematics, University of Washington, Seattle, WA, USA

Abstract: The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the "fractal" structure of these measures. The main results are, first, a Holder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hölder estimate for self-similar suspension flows; and, third, a Hölder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hölder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an "arithmetic-Diophantine" proposition, which has other applications. In Appendix A this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.

DOI: https://doi.org/10.1016/j.aim.2014.04.004


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Received: 21.08.2013
Accepted:06.04.2014
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