# A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation

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1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée 2 Centre for Mathematical Sciences Lund Institute of Technology Lund University 3 IML - Institut de mathématiques de Luminy 4 CPT - Centre de Physique Théorique - UMR 6207 5 CPT - Centre de Physique Théorique - UMR 7332

Abstract : Let $\mu$ be a Gibbs measure of the doubling map $T$ of the circle. For a $\mu$-generic point $x$ and a given sequence $\{r n\} \subset \R^+$, consider the intervals $T^nx - r n \pmod 1, T^nx + r n \pmod 1$. In analogy to the classical Dvoretzky covering of the circle we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures which are multifractal. Such a principle was shown by Beresnevich and Velani \cite{BV} only for monofractal measures. In the symbolic language we completely describe the combinatorial structure of a typical relatively short sequence, in particular we can describe the occurrence of -atypical- relatively long words. Our results have a direct and deep number-theoretical interpretation via inhomogeneous diadic diophantine approximation by numbers belonging to a given diadic diophantine class.

Autor: Ai-Hua Fan - Joerg Schmeling - Serge Troubetzkoy -

Fuente: https://hal.archives-ouvertes.fr/

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