On the modulus of continuity for spectral measures in substitution dynamics

Alexander I. Bufetov; Boris Solomyak

doi:10.1016/j.aim.2014.04.004

On the modulus of continuity for spectral measures in substitution dynamics

Alexander I. Bufetov
, Boris Solomyak

Aix-Marseille Université
RAS - Steklov Mathematical Institute
Russian Academy of Sciences
Higher School of Economics
Rice University
University of Washington

Research output: Contribution to journal › Article › peer-review

39 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 language	English
Pages (from-to)	84-129
Number of pages	46
Journal	Advances in Mathematics
Volume	260
DOIs	https://doi.org/10.1016/j.aim.2014.04.004
State	Published - 1 Aug 2014
Externally published	Yes

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 .

Funding

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. 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 .

Funders	Funder number
National Science Foundation	DMS-0968879
Directorate for Mathematical and Physical Sciences	0968879
European Commission	267259
Agence Nationale de la Recherche	MD-2859.2014.1
Russian Foundation for Basic Research	13-01-12449, 11-01-00654, 12-01-33020, 12-01-31284
Russian Academy of Sciences	ANR-13-JS01-0010

Keywords

Bernoulli convolution
Hoelder continuity
Spectral measure
Substitution dynamical system

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10.1016/j.aim.2014.04.004

Cite this

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title = "On the modulus of continuity for spectral measures in substitution dynamics",

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 H{\"o}lder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-H{\"o}lder estimate for self-similar suspension flows; and, third, a H{\"o}lder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-H{\"o}lder 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.",

keywords = "Bernoulli convolution, Hoelder continuity, Spectral measure, Substitution dynamical system",

author = "Bufetov, \{Alexander I.\} and Boris Solomyak",

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 Math{\'e}matiques in Luminy in the framework of the “Recherches en Bin{\^o}me” 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 . ",

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N2 - 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.

AB - 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.

KW - Bernoulli convolution

KW - Hoelder continuity

KW - Spectral measure

KW - Substitution dynamical system

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SP - 84

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JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

On the modulus of continuity for spectral measures in substitution dynamics

Abstract

Bibliographical note

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Keywords

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