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Topological conditional entropy for amenable group actions. (English) Zbl 1346.37038

Summary: We introduce the topological conditional entropy for countable discrete amenable group actions and establish a variational principle for it.

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B40 Topological entropy
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References:

[1] Bowen, Lewis, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32, 2, 427-466 (2012) · Zbl 1257.37007 · doi:10.1017/S0143385711000253
[2] Bowen, Rufus, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184, 125-136 (1973) · Zbl 0274.54030
[3] Boyle, Mike; Downarowicz, Tomasz, The entropy theory of symbolic extensions, \linebreak Invent. Math., 156, 1, 119-161 (2004) · Zbl 1216.37004 · doi:10.1007/s00222-003-0335-2
[4] Burguet, David, A direct proof of the tail variational principle and its extension to maps, Ergodic Theory Dynam. Systems, 29, 2, 357-369 (2009) · Zbl 1160.37320 · doi:10.1017/S0143385708080425
[5] Goodwyn, L. Wayne, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23, 679-688 (1969) · Zbl 0186.09804
[6] Goodman, T. N. T., Relating topological entropy and measure entropy, Bull. London Math. Soc., 3, 176-180 (1971) · Zbl 0219.54037
[7] Huang, Wen; Ye, Xiangdong; Zhang, Guohua, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261, 4, 1028-1082 (2011) · Zbl 1235.37008 · doi:10.1016/j.jfa.2011.04.014
[8] Ledrappier, F., A variational principle for the topological conditional entropy. Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Lecture Notes in Math. 729, 78-88 (1979), Springer: Berlin:Springer
[9] Li, Yuan; Chen, Ercai; Cheng, Wen-Chiao, Tail pressure and the tail entropy function, Ergodic Theory Dynam. Systems, 32, 4, 1400-1417 (2012) · Zbl 1260.37007 · doi:10.1017/S0143385711000204
[10] Liang, Bingbing; Yan, Kesong, Topological pressure for sub-additive potentials of amenable group actions, J. Funct. Anal., 262, 2, 584-601 (2012) · Zbl 1243.37017 · doi:10.1016/j.jfa.2011.09.020
[11] Misiurewicz, M., A short proof of the variational principle for a \(Z^n_+\)action on a compact space, Bull. Acad. Polon. Sci. S\'er. Sci. Math. Astronom. Phys., 24, 12, 1069-1075 (1976) · Zbl 0351.54036
[12] Misiurewicz, Micha{\l }, Topological conditional entropy, Studia Math., 55, 2, 175-200 (1976) · Zbl 0355.54035
[13] Moulin Ollagnier, Jean, Ergodic theory and statistical mechanics, Lecture Notes in Mathematics 1115, vi+147 pp. (1985), Springer-Verlag: Berlin:Springer-Verlag · Zbl 0558.28010
[14] Ornstein, Donald S.; Weiss, Benjamin, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48, 1-141 (1987) · Zbl 0637.28015 · doi:10.1007/BF02790325
[15] Pesin, Ya. B.; Pitskel{\cprime }, B. S., Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen., 18, 4, 50-63, 96 (1984)
[16] Walters, Peter, An introduction to ergodic theory, Graduate Texts in Mathematics 79, ix+250 pp. (1982), Springer-Verlag: New York:Springer-Verlag · Zbl 0958.28011
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