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Hausdorff dimension OF dynamical Dophantine approximation associated with ergodic mixing systems

E. Daviaud

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.

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.

Paper Structure

This paper contains 24 sections, 28 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, mixing properties of ergodic systems
  3. Recall on geometric measure theory
  4. Hausdorff content, measure and dimension
  5. Multifractal analysis
  6. Mixing properties of ergodic systems
  7. Statement of the result and preliminaries
  8. General theorems
  9. Application to $\times m,\times n$ and self-affine measures on Bedford-McMullen carpets
  10. Proof of Proposition \ref{['LocdimMix']}
  11. The case $\delta<\frac{1}{\overline{\dim}_H \mu}$
  12. Proof of Theorem \ref{['mainthm']}
  13. Recall on the mass transference principle for finite measure
  14. Full measure statement for every induced map
  15. Some useful lemmas and recalls
  16. ...and 9 more sections

Key Result

Proposition 2.2

Let $(T,\mu)$ be an ergodic system which is $\Sigma$ mixing. Then for every $y$ such that $0<\underline{\dim}(\mu,y)\leq \overline{\dim}(\mu,y)<+\infty,$ for $\mu$-almost every $x$,

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Remark 3.3
  • ...and 45 more