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RECURRENCE, DIMENSION AND ENTROPY

Published online by Cambridge University Press:  24 August 2001

AI-HUA FAN
Affiliation:
Département de Mathématiques, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France; ai-hua.fan@mathinfo.u-picardie.fr
DE-JUN FENG
Affiliation:
Department of Applied Mathematics, Tsinghua University, Beijing 100084, China Center for Advanced Study, Tsinghua University, Beijing, China; dfeng@math.tsinghua.edu.cn
JUN WU
Affiliation:
Department of Mathematics, Wuhan University, Wuhan 430072, China; wujunyu@public.wh.hb.cn
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Abstract

Let ([sum ]A, T) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Φ:[sum ]ARd be a continuous function. Denote by σΦ(x) the ergodic limit limn→∞n−1 [sum ]n−1j=0 Φ(Tjx) when the limit exists. Possible ergodic limits are just mean values ∫ Φdμ for all T-invariant measures. For any possible ergodic limit α, the following variational formula is proved:

[formula here]

where hμ denotes the entropy of μ and htop denotes topological entropy. It is also proved that unless all points have the same ergodic limit, then the set of points whose ergodic limit does not exist has the same topological entropy as the whole space [sum ]A

Type
Research Article
Copyright
The London Mathematical Society 2001

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RECURRENCE, DIMENSION AND ENTROPY
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