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Generic points of shift-invariant measures in the countable symbolic space

Published online by Cambridge University Press:  21 February 2018

AI–HUA FAN ,
MING–TIAN LI  and
JI–HUA MA
AI–HUA FAN
Affiliation:
School of Mathematics and Statistics, Central China Normal University, 152, Luoyu Road, 430079 Wuhan, China LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039 Amiens Cedex 1, France. e-mail: ai-hua.fan@u-picardie.fr
MING–TIAN LI
Affiliation:
College of Mathematics and Informatics, Fujian Normal University, Keji Road 350017 Fuzhou, China. e-mail: limtwd@fjnu.edu.cn
JI–HUA MA
Affiliation:
Department of Mathematics, Wuhan University, 299 Bayi Road, 430072 Wuhan, China. e-mail: jhma@whu.edu.cn
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Abstract

We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 2 , March 2019 , pp. 381 - 413
Copyright
Copyright © Cambridge Philosophical Society 2018 

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