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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. Bowen, Weak isomorphisms between Bernoulli shifts, Israel J. Math., 183 (2011), 93-102. doi: 10.1007/s11856-011-0043-3.  Google Scholar

[5]

M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc., 1 (1999), 109-197. doi: 10.1007/PL00011162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632. doi: 10.4171/GGD/200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604. doi: 10.1016/j.aim.2013.05.005.  Google Scholar

[12]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.  Google Scholar

[13]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. doi: 10.1007/BF02810577.  Google Scholar

[14]

D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459. doi: 10.1090/noti974.  Google Scholar

[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.  Google Scholar

[16]

V. A. Rohlin, Generators in ergodic theory, Vest. Leningrad Univ., 18 (1963), 26-32.  Google Scholar

[17]

V. Rohlin and Y. Sinai, The structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR, 141 (1961), 1038-1041.  Google Scholar

[18]

D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150. doi: 10.2307/121130.  Google Scholar

[19]

A. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.  Google Scholar

[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.  Google Scholar

[21]

B. Weiss, Sofic groups and dynamical systems, Sankhya Series A, 62 (2000), 350-359.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. Bowen, Weak isomorphisms between Bernoulli shifts, Israel J. Math., 183 (2011), 93-102. doi: 10.1007/s11856-011-0043-3.  Google Scholar

[5]

M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc., 1 (1999), 109-197. doi: 10.1007/PL00011162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632. doi: 10.4171/GGD/200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604. doi: 10.1016/j.aim.2013.05.005.  Google Scholar

[12]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 227-262.  Google Scholar

[13]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24. doi: 10.1007/BF02810577.  Google Scholar

[14]

D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459. doi: 10.1090/noti974.  Google Scholar

[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.  Google Scholar

[16]

V. A. Rohlin, Generators in ergodic theory, Vest. Leningrad Univ., 18 (1963), 26-32.  Google Scholar

[17]

V. Rohlin and Y. Sinai, The structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR, 141 (1961), 1038-1041.  Google Scholar

[18]

D. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150. doi: 10.2307/121130.  Google Scholar

[19]

A. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.  Google Scholar

[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.  Google Scholar

[21]

B. Weiss, Sofic groups and dynamical systems, Sankhya Series A, 62 (2000), 350-359.  Google Scholar

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