American Institute of Mathematical Sciences

Advanced
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.  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.  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.  doi: 10.1090/S0894-0347-09-00637-7.  Google Scholar

[4]

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

[5]

M. Gromov, Endomorphisms of symbolic algebraic varieties,, J. Eur. Math. Soc., 1 (1999), 109.  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.  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.  doi: 10.2307/1971117.  Google Scholar

[8]

D. Kerr, Sofic measure entropy via finite partitions,, Groups Geom. Dyn., 7 (2013), 617.  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.  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.  doi: 10.1007/s00222-011-0324-9.  Google Scholar

[11]

H. Li, Sofic mean dimension,, Adv. Math., 244 (2013), 570.  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.   Google Scholar

[13]

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

[14]

D. Ornstein, Newton's laws and coin tossing,, Notices Amer. Math. Soc., 60 (2013), 450.  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.  doi: 10.1007/BF02790325.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

A. Stepin, Bernoulli shifts on groups,, Dokl. Akad. Nauk SSSR, 223 (1975), 300.   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.  doi: 10.1007/BF02760797.  Google Scholar

[21]

B. Weiss, Sofic groups and dynamical systems,, Sankhya Series A, 62 (2000), 350.   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.  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.  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.  doi: 10.1090/S0894-0347-09-00637-7.  Google Scholar

[4]

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

[5]

M. Gromov, Endomorphisms of symbolic algebraic varieties,, J. Eur. Math. Soc., 1 (1999), 109.  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.  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.  doi: 10.2307/1971117.  Google Scholar

[8]

D. Kerr, Sofic measure entropy via finite partitions,, Groups Geom. Dyn., 7 (2013), 617.  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.  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.  doi: 10.1007/s00222-011-0324-9.  Google Scholar

[11]

H. Li, Sofic mean dimension,, Adv. Math., 244 (2013), 570.  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.   Google Scholar

[13]

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

[14]

D. Ornstein, Newton's laws and coin tossing,, Notices Amer. Math. Soc., 60 (2013), 450.  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.  doi: 10.1007/BF02790325.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

A. Stepin, Bernoulli shifts on groups,, Dokl. Akad. Nauk SSSR, 223 (1975), 300.   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.  doi: 10.1007/BF02760797.  Google Scholar

[21]

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

[1]

Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065
[2]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021008
[3]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374
[4]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[5]

Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021017
[6]

Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020033
[7]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[8]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274
[9]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395
[10]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[11]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127
[12]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[13]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015
[14]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404
[15]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266
[16]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317
[17]

Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014
[18]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002
[19]

Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021004
[20]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (189)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences