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

TL;DR

This paper analyzes the Ostrowski skew-product map using transfer operators, revealing its spectral properties, invariant measures, mixing behavior, and implications for Hausdorff dimensions and limit laws in Diophantine approximation.

Contribution

It provides a spectral and dynamical analysis of Ostrowski's map, establishing invariant measures, mixing rates, and dimension estimates, which are novel in understanding its complex behavior.

Findings

Existence of an absolutely continuous invariant measure with a piecewise holomorphic density.

Exponential mixing properties of the Ostrowski dynamical system.

Central limit theorem and Hausdorff dimension estimates for related sets.

Abstract

We present a dynamical and spectral study of Ostrowski's map based on the use of transfer operators. The Ostroswki dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and interval 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 operator, we show that the Ostroswki dynamical system admits an absolutely continuous invariant measure with piecewise holomorphic density and exponential mixing properties. We deduce a central limit theorem for associated random variables of an arithmetic nature and motivated by applications in inhomogeneous Diophantine approximation, we get Bowen--Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a certain bounded digit set.