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Entropy for expansive algebraic actions of residually finite groups

Published online by Cambridge University Press:  26 May 2010

LEWIS BOWEN
LEWIS BOWEN*
Affiliation:
Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843-3368, USA (email: lpbowen@math.tamu.edu)
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Abstract

We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 703 - 718
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Abert, M. and Weiss, B.. Bernoulli actions are weakly contained in any free action. In preparation.Google Scholar
[2]Bowen, L.. A measure-conjugacy invariant for actions of free groups. Ann. of Math. (2) to appear.Google Scholar
[3]Bowen, L.. The ergodic theory of free group actions: entropy and the f-invariant. Groups, Geometry and Dynamics, arXiv:0902.0174, to appear.Google Scholar
[4]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217–245.CrossRefGoogle Scholar
[5]Deninger, C.. Fuglede–Kadison determinants and entropy for actions of discrete amenable groups. J. Amer. Math. Soc. 19 (2006), 737–758.CrossRefGoogle Scholar
[6]Deninger, C. and Schmidt, K.. Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Ergod. Th. & Dynam. Sys. 27 (2007), 769–786.CrossRefGoogle Scholar
[7]Fuglede, B. and Kadison, R. V.. Determinant theory in finite factors. Ann. of Math. (2) 55 (1952), 520–530.CrossRefGoogle Scholar
[8]Kechris, A. S.. Global Aspects of Ergodic Group Actions (Mathematical Surveys and Monographs, 160). American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
[9]Kechris, A. S. and Tsankov, T.. Amenable actions and almost invariant sets. Proc. Amer. Math. Soc. 136(2) (2008), 687–697 (electronic).CrossRefGoogle Scholar
[10]Losert, V. and Rindler, H.. Almost invariant sets. Bull. London Math. Soc. 13(2) (1981), 145–148.CrossRefGoogle Scholar
[11]Lind, D., Schmidt, K. and Ward, T.. Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101 (1990), 593–629.CrossRefGoogle Scholar
[12]Lind, D. and Ward, T.. Automorphisms of solenoids and p-adic entropy. Ergod. Th. & Dynam. Sys. 8 (1988), 411–419.CrossRefGoogle Scholar
[13]Royden, H. L.. Real Analysis, 3rd edn. Macmillan, New York, 1988.Google Scholar
[14]Schmidt, K.. Amenability, Kazhdan’s property (T), strong ergodicity and invariant means for ergodic group actions. Ergod. Th. & Dynam. Sys. 1(2) (1981), 223–236.CrossRefGoogle Scholar
[15]Yuzvinskii, S. A.. Computing the entropy of a group endomorphism. Sib. Mat. Z. 8 (1967), 230–239; Sib. Math. J. 8 (1968) 172–178 (Engl. transl.).Google Scholar