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

Fan, Ai-Hua ; Schmeling, Jörg LU and Troubetzkoy, Serge (2013) In Proceedings of the London Mathematical Society 107. p.1173-1219
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 of R+, consider the intervals (T-n x - r(n) (mod 1), T-n x + r(n) (mod 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 that are multifractal. Such a principle was proved by Beresnevich and Velani [Ann. Math. 164 (2006) 971-992] for monofractal measures. In the symbolic language, we completely describe the combinatorial structure of a typical relatively short... (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 of R+, consider the intervals (T-n x - r(n) (mod 1), T-n x + r(n) (mod 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 that are multifractal. Such a principle was proved by Beresnevich and Velani [Ann. Math. 164 (2006) 971-992] 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 dyadic Diophantine approximation by numbers belonging to a given (dyadic) Diophantine class. (Less)
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type
Contribution to journal
publication status
published
subject
in
Proceedings of the London Mathematical Society
volume
107
pages
1173 - 1219
publisher
LONDON MATH SOC, BURLINGTON HOUSE PICCADILLY, LONDON, ENGLAND W1V 0NL
external identifiers
  • wos:000326748500006
  • scopus:84890238925
ISSN
0024-6115
DOI
10.1112/plms/pdt005
language
English
LU publication?
yes
id
ee8a9922-299f-4e0b-9f96-4d5da70696ea (old id 4201015)
date added to LUP
2016-04-01 11:04:30
date last changed
2022-04-28 06:53:09
@article{ee8a9922-299f-4e0b-9f96-4d5da70696ea,
  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 of R+, consider the intervals (T-n x - r(n) (mod 1), T-n x + r(n) (mod 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 that are multifractal. Such a principle was proved by Beresnevich and Velani [Ann. Math. 164 (2006) 971-992] 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 dyadic Diophantine approximation by numbers belonging to a given (dyadic) Diophantine class.}},
  author       = {{Fan, Ai-Hua and Schmeling, Jörg and Troubetzkoy, Serge}},
  issn         = {{0024-6115}},
  language     = {{eng}},
  pages        = {{1173--1219}},
  publisher    = {{LONDON MATH SOC, BURLINGTON HOUSE PICCADILLY, LONDON, ENGLAND W1V 0NL}},
  series       = {{Proceedings of the London Mathematical Society}},
  title        = {{A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation}},
  url          = {{http://dx.doi.org/10.1112/plms/pdt005}},
  doi          = {{10.1112/plms/pdt005}},
  volume       = {{107}},
  year         = {{2013}},
}