Dynamics of Ostrowski skew-product: 1. Limit laws and Hausdorff dimensions

Abstract

We present a dynamical study of Ostrowski’s map based on the use of transfer operators. The Ostrowski dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and intervals as fibers) and produces expansions of real numbers with respect to an irrational base given by continued fractions. By studying spectral properties of the associated transfer operators, we show that the absolutely continuous invariant measure of the Ostrowski dynamical system has exponential mixing properties. We deduce a central limit theorem for random variables of an arithmetic nature, and motivated by applications in inhomogeneous Diophantine approximation, we also get Bowen–Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a bounded digit set.