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Diophantine approximation by orbits of expanding Markov maps

Published online by Cambridge University Press:  07 February 2012

LINGMIN LIAO  and
STÉPHANE SEURET
LINGMIN LIAO
Affiliation:
LAMA, CNRS UMR 8050, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: lingmin.liao@u-pec.fr, seuret@u-pec.fr)
STÉPHANE SEURET
Affiliation:
LAMA, CNRS UMR 8050, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: lingmin.liao@u-pec.fr, seuret@u-pec.fr)
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Abstract

In 1995, Hill and Velani introduced the ‘shrinking targets’ theory. Given a dynamical system ([0,1],T), they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well approximated by orbits {Tnx}n≥0, where Tis an expanding Markov map with a finite partition supported by [0,1]. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 585 - 608
Copyright
©2012 Cambridge University Press

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