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A new proof of the dimension gap for the Gauss map
Part of:
Dynamical systems with hyperbolic behavior
Classical measure theory
Smooth dynamical systems: general theory
Published online by Cambridge University Press: 15 June 2021
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.
MSC classification
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 43 - 71
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society
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