Hausdorff dimension OF dynamical Dophantine approximation associated with ergodic mixing systems

TL;DR

This paper estimates the Hausdorff dimension of dynamical coverings in ergodic mixing systems, providing formulas under different mixing speeds and extending known results to new dynamical maps on the torus.

Contribution

It introduces formulas for Hausdorff dimension in dynamical systems with varying mixing speeds and extends existing results to the two-dimensional torus.

Findings

Derived formulas for Hausdorff dimension under polynomial and super-polynomial mixing speeds.

Extended results from circle doubling map to times 2, times 3 map on the 2D torus.

Established conditions for exact-dimensional and non-exact-dimensional measures.

Abstract

In this article, we estimate the Hausdorff dimension of dynamical coverings with respect to mixing ergodic systems. More precisely, if the ergodic measure is exact-dimensionnal, we establish a formula provided that the system is polynomially fast mixing and if the measure is not exact-dimensionnal, we establish a similar result under super-polynomial speed of mix assumpetion. As an application of our result, we extend the result of Fan-Shmeling-Troubetzkoy for the doubling map on the circle to the case of the times 2, times 3 map on the two dimensional torus.