Abstract
Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C ∗-algebra.
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References
Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)
Berg, K.R.: Convolution of invariant measures, maximal entropy. Math. Syst. Theory 3, 146–150 (1969)
Bowen, L.: Sofic entropy and amenable groups. Ergod. Theory Dyn. Syst., to appear. arXiv:1008.1354
Bowen, L.: Entropy for expansive algebraic actions of residually finite groups. Ergod. Theory Dyn. Syst. doi:10.1017/S0143385710000179
Bowen, L.: Measure conjugacy invariants for actions of countable sofic groups. J. Am. Math. Soc. 23, 217–245 (2010)
Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)
Brown, N.P., Ozawa, N.: C ∗-Algebras and Finite-Dimensional Approximations. Graduate Studies in Mathematics, vol. 88. American Mathematical Society, Providence (2008)
Deninger, C.: Fuglede-Kadison determinants and entropy for actions of discrete amenable groups. J. Am. Math. Soc. 19, 737–758 (2006)
Deninger, C., Schmidt, K.: Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Ergod. Theory Dyn. Syst. 27, 769–786 (2007)
Gottschalk, W.: Some general dynamical notions. In: Recent Advances in Topological Dynamics. Lecture Notes in Math., vol. 318, pp. 120–125. Springer, Berlin (1973)
Gromov, M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1, 109–197 (1999)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. I. Elementary Theory. Pure and Applied Mathematics, vol. 100. Academic Press, San Diego (1983)
Kerr, D., Li, H.: Bernoulli actions and infinite entropy. arXiv:1005.5143. Groups Geom. Dyn., to appear
Kerr, D., Li, H.: Soficity, amenability, and dynamical entropy. arXiv:1008.1429. Am. J. Math., to appear
Kolmogorov, A.N.: A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR 119, 861–864 (1958)
Li, H., Compact group automorphisms, addition formulas and Fuglede-Kadison determinants. arXiv:1001.0419
Lind, D., Schmidt, K., Ward, T.: Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101, 593–629 (1990)
Sinai, Ya.G.: On the concept of entropy for a dynamical system. Dokl. Akad. Nauk SSSR 124, 768–771 (1959)
Weiss, B.: Sofic groups and dynamical systems. In: Ergodic Theory and Harmonic Analysis, Mumbai, 1999. Sankhyā Ser. A, vol. 62, pp. 350–359 (2000)
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Kerr, D., Li, H. Entropy and the variational principle for actions of sofic groups. Invent. math. 186, 501–558 (2011). https://doi.org/10.1007/s00222-011-0324-9
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DOI: https://doi.org/10.1007/s00222-011-0324-9