Hausdorff dimension OF dynamical Dophantine approximation associated with ergodic mixing systems

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.