On the modulus of continuity for spectral measures in substitution dynamics

Alexander I. Bufetov, Boris Solomyak

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

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 Hölder 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.

Original languageEnglish
Pages (from-to)84-129
Number of pages46
JournalAdvances in Mathematics
Volume260
DOIs
StatePublished - 1 Aug 2014
Externally publishedYes

Bibliographical note

Funding Information:
B. Solomyak is supported in part by NSF grant DMS-0968879 . He was also partially supported by the Forschheimer Fellowship and the ERC AdG 267259 grant at the Hebrew University of Jerusalem when working on this project.

Funding Information:
Work on this project was begun when the authors were visiting the Centre International de Rencontres Mathématiques in Luminy in the framework of the “Recherches en Binôme” programme. We are deeply grateful to the Centre for the warm hospitality. A. Bufetov has been supported by the A*MIDEX project (no. ANR-11-IDEX-0001-02 ) funded by the “Investissements d'Avenir” French Government program, managed by the French National Research Agency ( ANR ). A. Bufetov has also been supported in part by the Grant MD-2859.2014.1 of the President of the Russian Federation , by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, by the ANR under the project “VALET” (grant no. ANR-13-JS01-0010 ) of the Programme JCJC SIMI 1 , and by the RFBR grants 11-01-00654 , 12-01-31284 , 12-01-33020 , 13-01-12449 .

Keywords

  • Bernoulli convolution
  • Hoelder continuity
  • Spectral measure
  • Substitution dynamical system

Fingerprint

Dive into the research topics of 'On the modulus of continuity for spectral measures in substitution dynamics'. Together they form a unique fingerprint.

Cite this