A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/4201015
- author
- Fan, Ai-Hua ; Schmeling, Jörg LU and Troubetzkoy, Serge
- organization
- publishing date
- 2013
- 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}}, }