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Dynamical diophantine approximation

Fan, Ai-Hua; Schmeling, Jörg LU and Troubetzkoy, Serge (2008) In [unknown]
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
submitted
subject
keywords
Dynamical Systems (math.DS), Number Theory (math.NT), Probability (math.PR)
in
[unknown]
publisher
Historielärarnas förening
ISSN
0439-2434
language
English
LU publication?
yes
id
aed515fc-f9a2-418f-bf5b-897a6d9a292b (old id 945878)
alternative location
http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.4203v1.pdf
date added to LUP
2008-03-04 13:57:17
date last changed
2016-09-13 15:31:21
@misc{aed515fc-f9a2-418f-bf5b-897a6d9a292b,
  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.},
  author       = {Fan, Ai-Hua and Schmeling, Jörg and Troubetzkoy, Serge},
  issn         = {0439-2434},
  keyword      = {Dynamical Systems (math.DS),Number Theory (math.NT),Probability (math.PR)},
  language     = {eng},
  publisher    = {ARRAY(0xb52f2e8)},
  series       = {[unknown]},
  title        = {Dynamical diophantine approximation},
  year         = {2008},
}