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Topological pressure and the variational principle for actions of sofic groups

Published online by Cambridge University Press:  25 July 2012

NHAN-PHU CHUNG
NHAN-PHU CHUNG*
Affiliation:
Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, USA (email: phuchung@buffalo.edu)
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Abstract

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We introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 5 , October 2013 , pp. 1363 - 1390
Copyright
Copyright © 2012 Cambridge University Press 

References

[1]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217–245.Google Scholar
[2]Bowen, L.. Sofic entropy and amenable groups. Ergod. Th. & Dynam. Sys. 32(2) (2012), 427–466.CrossRefGoogle Scholar
[3]Ceccherini-Silberstein, T. and Coornaert, M.. Cellular Automata and Groups (Springer Monographs in Mathematics). Springer, Berlin, 2010.Google Scholar
[4]Elek, G. and Szabó, E.. Hyperlinearity, essentially free actions and $L^2$-invariants. The sofic property. Math. Ann. 332(2) (2005), 421–441.Google Scholar
[5]Elek, G. and Szabó, E.. On sofic groups. J. Group Theory 9(2) (2006), 161–171.Google Scholar
[6]Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197.CrossRefGoogle Scholar
[7]Keller, G.. Equilibrium States in Ergodic Theory (London Mathematical Society Student Texts, 42). Cambridge University Press, Cambridge, 1998.Google Scholar
[8]Kerr, D. and Li, H.. Entropy and the variational principle for actions of sofic groups. Invent. Math. 186(3) (2011), 501–558.CrossRefGoogle Scholar
[9]Kerr, D. and Li, H.. Soficity, amenability, and dynamical entropy. Amer. J. Math. to appear; arXiv:1008.1429.Google Scholar
[10]Li, H.. Sofic mean dimension. arXiv:1105.0140v1.Google Scholar
[11]Lindenstrauss, E. and Weiss, B.. Mean topological dimension. Israel J. Math. 115 (2000), 1–24.Google Scholar
[12]Misiurewicz, M. A.. A short proof of the variational principle for a ${\mathbb Z}^n_+$ action on a compact space. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 24(12) (1976), 1069–1075.Google Scholar
[13]Ollagnier, J. M. and Pinchon, D.. The variational principle. Studia Math. 72(2) (1982), 151–159.Google Scholar
[14]Ollagnier, J. M.. Ergodic Theory and Statistical Mechanics (Lecture Notes in Mathematics, 1115). Springer, Berlin, 1985.Google Scholar
[15]Paterson, A. L. T.. Amenability (Mathematical Surveys and Monographs, 29). American Mathematical Society, Providence, RI, 1988.CrossRefGoogle Scholar
[16]Pesin, Y. B.. Dimension theory in dynamical systems. Contemporary Views and Applications (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, IL, 1997.Google Scholar
[17]Pesin, Y. B. and Pitskel’, B. S.. Topological pressure and the variational principle for noncompact sets. Funktsional. Anal. i Prilozhen. 18(4) (1984), 50–63 . In Russian, translated in Funct. Anal. Appl. 18(4) (1984), 307–318.Google Scholar
[18]Pestov, V. G.. Hyperlinear and sofic groups: a brief guide. Bull. Symbolic Logic 14(4) (2008), 449–480.Google Scholar
[19]Ruelle, D.. Statistical mechanics on a compact set with ${\mathbb Z}^\nu $ action satisfying expansiveness and specification. Trans. Amer. Math. Soc. 187 (1973), 237–251.Google Scholar
[20]Stepin, A. M. and Tagi-Zade, A. T.. Variational characterization of topological pressure of the amenable groups of transformations. Dokl. Akad. Nauk SSSR 254(3) (1980), 545–549 . In Russian, translated in Sov. Math. Dokl. 22(2) (1980), 405–409.Google Scholar
[21]Tempelman, A. A.. Specific characteristics and variational principle for homogeneous random fields. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65(3) (1984), 341–365.Google Scholar
[22]Tempelman, A. A.. Ergodic Theorems for Group Actions. Informational and Thermodynamical Aspects (Mathematics and its Applications, 78). Kluwer Academic, Dordrecht, 1992. Translated and revised from the 1986 Russian original.Google Scholar
[23]Thom, A.. Sofic groups and Diophantine approximation. Comm. Pure Appl. Math. 61(8) (2008), 1155–1171.Google Scholar
[24]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97(4) (1975), 937–971.Google Scholar
[25]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.Google Scholar
[26]Weiss, B.. Sofic groups and dynamical systems. Ergodic Theory and Harmonic Analysis (Mumbai, 1999). Sankhyā Ser. A 62 (2000), 350–359.Google Scholar
[27]Zhang, G.. Local variational principle concerning entropy of a sofic group action. J. Funct. Anal. 262(4) (2012), 1954–1985.Google Scholar