A new proof of the dimension gap for the Gauss map

Abstract

Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map $T(x)=\frac{1}{x}\mathrm{mod}1$ satisfy a `dimension gap' meaning that for some $c>0$, ${\sup }_{\mathbf{p}}\dim {\mu }_{\mathbf{p}}<1-c$, where ${\mu }_{\mathbf{p}}$ denotes the (pushforward) Bernoulli measure for the countable probability vector $\mathbf{p}$. 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.

arXiv comment30 pages, 1 figure

References in corpus (1)

• Maximising Bernoulli measures and dimension gaps for countable branched systems