Entropy and actions of sofic groups

Previous Article
Directional complexity and entropy for lift mappings
DCDS-B Home
This Issue
Next Article
Brief survey on the topological entropy

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
Related Papers

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.

[1]	Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90.
[2]	John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367
[3]	Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012
[4]	André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
[5]	Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.
[6]	Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779
[7]	Benjamin Hellouin de Menibus, Hugo Maturana Cornejo. Necessary conditions for tiling finitely generated amenable groups. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2335-2346. doi: 10.3934/dcds.2020116
[8]	John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743
[9]	Richard Sharp. Distortion and entropy for automorphisms of free groups. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 347-363. doi: 10.3934/dcds.2010.26.347
[10]	Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[11]	A. Yu. Ol'shanskii and M. V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. Electronic Research Announcements, 2001, 7: 63-71.
[12]	Yunping Wang, Ercai Chen, Xiaoyao Zhou. Mean dimension theory in symbolic dynamics for finitely generated amenable groups. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022050
[13]	Kengo Matsumoto. Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 841-862. doi: 10.3934/dcds.2021139
[14]	Richard Miles, Michael Björklund. Entropy range problems and actions of locally normal groups. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 981-989. doi: 10.3934/dcds.2009.25.981
[15]	Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323
[16]	Daniele Mundici. The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 537-549. doi: 10.3934/dcds.2008.21.537
[17]	Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
[18]	Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753
[19]	Andrea Bonfiglioli, Ermanno Lanconelli. Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1587-1614. doi: 10.3934/cpaa.2012.11.1587
[20]	Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 34-39.

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Benjamin Weiss
on Google Scholar
Benjamin Weiss

[Back to Top]