Dynamical diophantine approximation

Summary, in English

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.

• Author: Ai-Hua Fan; Jörg Schmeling; Serge Troubetzkoy
• Department/s: Mathematics (Faculty of Engineering); Dynamical systems
• Publishing year: 2008
• Language: English
• Publication/Series: Historielärarnas Förenings Årsskrift
• Links: http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.4203v1.pdf
• Document type: Journal article
• Publisher: Historielärarnas förening
• Topic: Mathematics
• Keywords: Dynamical Systems (math.DS); Number Theory (math.NT); Probability (math.PR)
• Status: Submitted
• Research group: Dynamical systems
• ISBN/ISSN/Other: ISSN: 0439-2434