Logarithmic Fourier decay for self conformal measures

Amir Algom; Federico Rodriguez Hertz; Zhiren Wang

doi:10.1112/jlms.12608

Logarithmic Fourier decay for self conformal measures

Amir Algom, Federico Rodriguez Hertz, Zhiren Wang

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

1 Scopus citations

Abstract

We prove that the Fourier transform of a self conformal measure on (Formula presented.) decays to 0 at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS that both acts linearly on its attractor and contracts by scales that are not Diophantine. Our key technical result is an effective version of a local limit Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that is of independent interest.

Original language	English (US)
Pages (from-to)	1628-1661
Number of pages	34
Journal	Journal of the London Mathematical Society
Volume	106
Issue number	2
DOIs	https://doi.org/10.1112/jlms.12608
State	Published - Sep 2022

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1112/jlms.12608

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abstract = "We prove that the Fourier transform of a self conformal measure on (Formula presented.) decays to 0 at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS that both acts linearly on its attractor and contracts by scales that are not Diophantine. Our key technical result is an effective version of a local limit Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that is of independent interest.",

author = "Amir Algom and {Rodriguez Hertz}, Federico and Zhiren Wang",

note = "Funding Information: The authors thank Tuomas Sahlsten and Kasun Fernando for some helpful discussions. The authors are also grateful to the anonymous referee for a very thorough reading and many helpful suggestions which greatly improved the presentation of the paper Publisher Copyright: {\textcopyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",

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AU - Rodriguez Hertz, Federico

AU - Wang, Zhiren

N1 - Funding Information: The authors thank Tuomas Sahlsten and Kasun Fernando for some helpful discussions. The authors are also grateful to the anonymous referee for a very thorough reading and many helpful suggestions which greatly improved the presentation of the paper Publisher Copyright: © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

PY - 2022/9

Y1 - 2022/9

N2 - We prove that the Fourier transform of a self conformal measure on (Formula presented.) decays to 0 at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS that both acts linearly on its attractor and contracts by scales that are not Diophantine. Our key technical result is an effective version of a local limit Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that is of independent interest.

AB - We prove that the Fourier transform of a self conformal measure on (Formula presented.) decays to 0 at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS that both acts linearly on its attractor and contracts by scales that are not Diophantine. Our key technical result is an effective version of a local limit Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that is of independent interest.

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Logarithmic Fourier decay for self conformal measures

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