A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation
Författare
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 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.
Avdelning/ar
- Matematik LTH
- Dynamical systems
Publiceringsår
2013
Språk
Engelska
Sidor
1173-1219
Publikation/Tidskrift/Serie
Proceedings of the London Mathematical Society
Volym
107
Dokumenttyp
Artikel i tidskrift
Förlag
LONDON MATH SOC, BURLINGTON HOUSE PICCADILLY, LONDON, ENGLAND W1V 0NL
Ämne
- Mathematics
Status
Published
Forskningsgrupp
- Dynamical systems
ISBN/ISSN/Övrigt
- ISSN: 0024-6115