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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

NATALIA JURGA
NATALIA JURGA*
Affiliation:
School of Mathematics and Statistics, Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY 16 9SS e-mail: natalia-j@live.co.uk
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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, 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

Primary: 37C45: Dimension theory of dynamical systems
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 28A80: Fractals
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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