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November  2008, 21(4): 1103-1128. doi: 10.3934/dcds.2008.21.1103

Generic points in systems of specification and Banach valued Birkhoff ergodic average

Aihua Fan 1, , Lingmin Liao 2, and  Jacques Peyrière 3,

1. 

Department of Mathematics, Wuhan University, 430072 Wuhan, China
2. 

CNRS UMR 6140, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039 Amiens Cedex 1, France
3. 

Université Paris-Sud, CNRS UMR 8628, Mathématique bât. 425, 91405 Orsay Cedex, France

Received  May 2007 Revised  April 2008 Published  May 2008

We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.
Keywords: Variational principle., Generic points, Topological entropy, Systems of specification.
Mathematics Subject Classification: Primary: 37B40, 37B45; Secondary: 37A0.
Citation: Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103
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