Entropy and actions of sofic groups

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December 2015, 20(10): 3375-3383. doi: 10.3934/dcdsb.2015.20.3375

Entropy and actions of sofic groups

Benjamin Weiss ^1,

1.	Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904

Received February 2015 Revised February 2015 Published September 2015

Abstract
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In recent years there has been a great deal of progress in the study of actions of countable groups. In particular, the concept of the entropy of an action has been extended to all sofic groups following the seminal work of Lewis Bowen. This survey is an invitation to these new developments. It includes a new proof of the analogue of Kolmogorov's theorem for sofic groups, namely that isomorphic Bernoulli shifts have the same base entropy.

Keywords: Kolmogorov's theorem., Entropy, amenable groups, sofic groups, Bernoulli shifts.

Mathematics Subject Classification: Primary: 37A15, 31A35; Secondary: 22F1.

Citation: Benjamin Weiss. Entropy and actions of sofic groups. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375

References:

[1]	R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.
[2]	L. Bowen, A measure-conjugacy invariant for free group actions, Annals of Math., 171 (2010), 1387-1400. doi: 10.4007/annals.2010.171.1387.
[3]	L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc., 23 (2010), 217-245. doi: 10.1090/S0894-0347-09-00637-7.
[4]	L. Bowen, Weak isomorphisms between Bernoulli shifts, Israel J. Math., 183 (2011), 93-102. doi: 10.1007/s11856-011-0043-3.
[5]	M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc., 1 (1999), 109-197. doi: 10.1007/PL00011162.
[6]	M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Math. Phys. Anal. Geom., 2 (1999), 323-415. doi: 10.1023/A:1009841100168.
[7]	M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math., 109 (1979), 397-406. doi: 10.2307/1971117.
[8]	D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632. doi: 10.4171/GGD/200.
[9]	D. Kerr, Bernoulli actions of sofic groups have completely positive entropy, Israel J. Math., 202 (2014), 461-474. doi: 10.1007/s11856-014-1077-0.
[10]	D. Kerr and H. Li, Entropy and the variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558. doi: 10.1007/s00222-011-0324-9.
[11]	H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604. doi: 10.1016/j.aim.2013.05.005.
[12]	E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.
[13]	E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. doi: 10.1007/BF02810577.
[14]	D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459. doi: 10.1090/noti974.
[15]	D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141. doi: 10.1007/BF02790325.
[16]	V. A. Rohlin, Generators in ergodic theory, Vest. Leningrad Univ., 18 (1963), 26-32.
[17]	V. Rohlin and Y. Sinai, The structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR, 141 (1961), 1038-1041.
[18]	D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150. doi: 10.2307/121130.
[19]	A. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.
[20]	J.-P. Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l'un est un schéma de Bernoulli, Israel J. Math., 21 (1975), 177-207. doi: 10.1007/BF02760797.
[21]	B. Weiss, Sofic groups and dynamical systems, Sankhya Series A, 62 (2000), 350-359.

show all references

References:

[1]	R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.
[2]	L. Bowen, A measure-conjugacy invariant for free group actions, Annals of Math., 171 (2010), 1387-1400. doi: 10.4007/annals.2010.171.1387.
[3]	L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc., 23 (2010), 217-245. doi: 10.1090/S0894-0347-09-00637-7.
[4]	L. Bowen, Weak isomorphisms between Bernoulli shifts, Israel J. Math., 183 (2011), 93-102. doi: 10.1007/s11856-011-0043-3.
[5]	M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc., 1 (1999), 109-197. doi: 10.1007/PL00011162.
[6]	M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Math. Phys. Anal. Geom., 2 (1999), 323-415. doi: 10.1023/A:1009841100168.
[7]	M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math., 109 (1979), 397-406. doi: 10.2307/1971117.
[8]	D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632. doi: 10.4171/GGD/200.
[9]	D. Kerr, Bernoulli actions of sofic groups have completely positive entropy, Israel J. Math., 202 (2014), 461-474. doi: 10.1007/s11856-014-1077-0.
[10]	D. Kerr and H. Li, Entropy and the variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558. doi: 10.1007/s00222-011-0324-9.
[11]	H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604. doi: 10.1016/j.aim.2013.05.005.
[12]	E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.
[13]	E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. doi: 10.1007/BF02810577.
[14]	D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459. doi: 10.1090/noti974.
[15]	D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141. doi: 10.1007/BF02790325.
[16]	V. A. Rohlin, Generators in ergodic theory, Vest. Leningrad Univ., 18 (1963), 26-32.
[17]	V. Rohlin and Y. Sinai, The structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR, 141 (1961), 1038-1041.
[18]	D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150. doi: 10.2307/121130.
[19]	A. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.
[20]	J.-P. Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l'un est un schéma de Bernoulli, Israel J. Math., 21 (1975), 177-207. doi: 10.1007/BF02760797.
[21]	B. Weiss, Sofic groups and dynamical systems, Sankhya Series A, 62 (2000), 350-359.

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