A new proof of the dimension gap for the Gauss map

N. Jurga

doi:10.1017/S0305004121000104

A new proof of the dimension gap for the Gauss map

N. Jurga^*

^*Corresponding author for this work

Pure Mathematics

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Abstract

In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a 'dimension gap' meaning that for some c > 0, sup_p dim μ_p < 1-c, where μ_p denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

Original language	English
Number of pages	29
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	FirstView
DOIs	https://doi.org/10.1017/S0305004121000104
Publication status	Published - 15 Jun 2021

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Jurga_2021_MC-CPS_New-proof-Gauss
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the final published version of the work, which was originally published at https://doi.org/10.1017/S0305004121000104.
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title = "A new proof of the dimension gap for the Gauss map",

abstract = "In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a 'dimension gap' meaning that for some c > 0, supp dim μp < 1-c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.",

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note = "Funding: This paper was written while the author was supported by a Leverhulme Trust Research Project Grant (RF-2016-194).",

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N1 - Funding: This paper was written while the author was supported by a Leverhulme Trust Research Project Grant (RF-2016-194).

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Y1 - 2021/6/15

N2 - In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a 'dimension gap' meaning that for some c > 0, supp dim μp < 1-c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

AB - In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a 'dimension gap' meaning that for some c > 0, supp dim μp < 1-c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

UR - http://arxiv.org/pdf/1806.00841

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JF - Mathematical Proceedings of the Cambridge Philosophical Society

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A new proof of the dimension gap for the Gauss map

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