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Entropy and its variational principle for non-compact metric spaces

Published online by Cambridge University Press:  24 August 2009

MAURO PATRÃO
MAURO PATRÃO*
Affiliation:
Universidade de Brasília, Brasíla-DF, Brasil (email: mpatrao@mat.unb.br)
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Abstract

In the present paper, we introduce a natural extension of Adler, Konheim and MacAndrew topological entropy for proper maps of locally compact separable metrizable spaces and prove a variational principle which states that this topological entropy, the supremum of the Kolmogorov–Sinai entropies and the minimum of the Bowen entropies always coincide. We apply this variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the Bowen formula for the Bowen entropy of an automorphism of a non-compact Lie group with respect to some invariant metric is just an upper bound for its topological entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1529 - 1542
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Adler, R., Konheim, A. and MacAndrew, H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[2]Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
[3]Ferraiol, T., Patrão, M. and Seco, L.. Jordan decomposition and dynamics on flag manifolds. Preprint, 2008.Google Scholar
[4]Gurevich, B.. Topological entropy of a countable Markov chain. Soviet Math. Dokl. 10 (1969), 911–915.Google Scholar
[5]Handel, M. and Kitchens, B.. Metrics and entropy for non-compact spaces. Israel J. Math. 91 (1995), 253–271.CrossRefGoogle Scholar
[6]Helgason, S.. Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York, 1978.Google Scholar
[7]Knapp, A. W.. Lie Groups Beyond an Introduction (Progress in Mathematics, 140). Birkhäuser, Boston, 2004.Google Scholar
[8]Mostow, G. D.. Factor spaces of solvable groups. Ann. of Math.  60(1) (1954), 1–27.CrossRefGoogle Scholar
[9]Walters, P.. An Introduction to Ergodic Theory. Springer, Berlin, 1981.Google Scholar