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A new proof of the dimension gap for the Gauss map

Part of: Classical measure theory Dynamical systems with hyperbolic behavior 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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References

Baker, S. and Jurga, N.. Maximising Bernoulli measures and dimension gaps for countable branched systems. To appear in Ergodic Theory and Dynamical Systems.Google Scholar
Bowen, R.. Ergodic theory of axiom a diffeomorphisms. Lecture Notes in Math. 470 (Springer, Berlin, Heidelberg, 1975).Google Scholar
Cornfeld, I. P., Fomin, S. V. and Sinai, Y. G. E.. Ergodic theory. A Series of Comprehensive Studies in Mathematics. 245 (Springer Science and Business Media, 2012).Google Scholar
Kifer, Y., Peres, Y. and Weiss, B.. A dimension gap for continued fractions with independent digits. Israel J. Math. 124 (2001), 61–76.CrossRefGoogle Scholar
Liverani, C.. Decay of correlations. Ann. of Math. 142(2) (1995), 239–301.CrossRefGoogle Scholar
Mauldin, D. and Urbanski, M.. Graph directed Markov systems: Geometry and dynamics of limit sets. Cambridge Tracts in Math. 148 (Cambridge University Press, Cambridge, 2003). xii+281 pp. ISBN: 0-521-82538-5.Google Scholar
Mauldin, D. and Urbański, M.. Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125(1) (2001), 93–130.CrossRefGoogle Scholar
Ruelle, D.. Thermodynamic Formalism: the Mathematical Structure of Equilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 2004).Google Scholar
Viana, M.. Stochastic dynamics of deterministic systems. 21 (IMPA, Rio de Janeiro, 1997).Google Scholar
Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.CrossRefGoogle Scholar