2019, 15: 133-141. doi: 10.3934/jmd.2019016

The work of Lewis Bowen on the entropy theory of non-amenable group actions (Brin Prize article)

Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, 4 Place Jussieu, 75252 Paris Cedex 05, France

Received  April 05, 2019 Published  May 2019

We present the achievements of Lewis Bowen, or, more precisely, his breakthrough works after which a theory started to develop. The focus will therefore be made here on the isomorphism problem for Bernoulli actions of countable non-amenable groups which he solved brilliantly in two remarkable papers. Here two invariants were introduced, which led to many developments.

Citation: Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
References:
[1]

A. Alpeev, On Pinsker factors for Rokhlin entropy, J. Math. Sci. (N.Y.), 209 (2015), 826-829.  doi: 10.1007/s10958-015-2529-8.

[2]

T. Austin and P. Burton, Uniform mixing and completely positive sofic entropy, to appear in J. Anal. Math.

[3]

T. Austin, The geometry of model spaces for probability-preserving actions of sofic groups, Anal. Geom. Metr. Spaces, 4 (2014), 160-186.  doi: 10.1515/agms-2016-0006.

[4]

T. Austin, Additivity properties of sofic entropy and measures on model spaces, Forum Math. Sigma, 4 (2016), e25, 79 pp. doi: 10.1017/fms.2016.18.

[5]

L. Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math. (2), 171 (2010), 1387-1400.  doi: 10.4007/annals.2010.171.1387.

[6]

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.

[7]

L. Bowen, The ergodic theory of free group actions: Entropy and the f-invariant, Groups Geom. Dyn., 4 (2010), 419-432.  doi: 10.4171/GGD/89.

[8]

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

[9]

L. Bowen, Every countably infinite group is almost Ornstein, in Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 67–78. doi: 10.1090/conm/567/11234.

[10]

L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.

[11]

L. Bowen, Finitary random interlacements and the Gaboriau-Lyons problem, preprint, arXiv: 1707.09573v3.

[12]

L. Bowen, Sofic homological invariants and the weak Pinsker property, arXiv: 1807.08191.

[13]

E. Gordon and A. Vershik, Groups that are locally embeddable in the class of finite groups, Algebra i Analiz, 9 (1997), 71-97. 

[14]

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

[15]

B. Hayes, Fuglede-Kadison determinants and sofic entropy, Geom. Funct. Anal., 26 (2016), 520-606.  doi: 10.1007/s00039-016-0370-y.

[16]

B. Hayes, Mixing and spectral gap relative to Pinsker factor for sofic groups, in Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F. R. Jones' 60th Birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., 46, Austral. Nat. Univ., Canberra, 2017,193–221.

[17]

B. Hayes, Sofic entropy of Gaussian actions, Ergodic Theory Dynam. Systems, 37 (2017), 2187-2222.  doi: 10.1017/etds.2016.6.

[18]

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

[19]

D. Kerr, Bernoulli actions of sofic groups have completey positive entropy, Israel J. Math., 202 (2014), 461-474.  doi: 10.1007/s11856-014-1077-0.

[20]

D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672.  doi: 10.4171/GGD/142.

[21]

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.

[22]

D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, Amer. J. Math., 135 (2013), 721-761.  doi: 10.1353/ajm.2013.0024.

[23]

J. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Probability, 3 (1975), 1031-1037.  doi: 10.1214/aop/1176996230.

[24]

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.

[25]

D. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), 2 (1980), 161-164.  doi: 10.1090/S0273-0979-1980-14702-3.

[26]

S. Popa, Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu, 5 (2006), 309-332.  doi: 10.1017/S1474748006000016.

[27]

B. Seward, Bernoulli shifts with base of equal entropy are isomorphic, arXiv: 1805.08279, 2018.

[28]

B. Seward, Krieger's finite generator theorem for ergodic actions of countable groups I, Invent. Math., 215 (2019), 265-310.  doi: 10.1007/s00222-018-0826-9.

[29]

B. Seward, The Koopman representation and positive Rokhlin entropy, arXiv: 1804.05270, 2018.

[30]

B. Seward, Positive entropy actions of countable groups factor onto Bernoulli shifts, to appear in J. Amer. Math. Soc.

[31]

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

[32]

B. Weiss, Entropy and actions of sofic groups, Discrete Contin. Dynam. Syst. Ser. B, 20 (2015), 3375-3383.  doi: 10.3934/dcdsb.2015.20.3375.

show all references

References:
[1]

A. Alpeev, On Pinsker factors for Rokhlin entropy, J. Math. Sci. (N.Y.), 209 (2015), 826-829.  doi: 10.1007/s10958-015-2529-8.

[2]

T. Austin and P. Burton, Uniform mixing and completely positive sofic entropy, to appear in J. Anal. Math.

[3]

T. Austin, The geometry of model spaces for probability-preserving actions of sofic groups, Anal. Geom. Metr. Spaces, 4 (2014), 160-186.  doi: 10.1515/agms-2016-0006.

[4]

T. Austin, Additivity properties of sofic entropy and measures on model spaces, Forum Math. Sigma, 4 (2016), e25, 79 pp. doi: 10.1017/fms.2016.18.

[5]

L. Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math. (2), 171 (2010), 1387-1400.  doi: 10.4007/annals.2010.171.1387.

[6]

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.

[7]

L. Bowen, The ergodic theory of free group actions: Entropy and the f-invariant, Groups Geom. Dyn., 4 (2010), 419-432.  doi: 10.4171/GGD/89.

[8]

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

[9]

L. Bowen, Every countably infinite group is almost Ornstein, in Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 67–78. doi: 10.1090/conm/567/11234.

[10]

L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.

[11]

L. Bowen, Finitary random interlacements and the Gaboriau-Lyons problem, preprint, arXiv: 1707.09573v3.

[12]

L. Bowen, Sofic homological invariants and the weak Pinsker property, arXiv: 1807.08191.

[13]

E. Gordon and A. Vershik, Groups that are locally embeddable in the class of finite groups, Algebra i Analiz, 9 (1997), 71-97. 

[14]

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

[15]

B. Hayes, Fuglede-Kadison determinants and sofic entropy, Geom. Funct. Anal., 26 (2016), 520-606.  doi: 10.1007/s00039-016-0370-y.

[16]

B. Hayes, Mixing and spectral gap relative to Pinsker factor for sofic groups, in Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F. R. Jones' 60th Birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., 46, Austral. Nat. Univ., Canberra, 2017,193–221.

[17]

B. Hayes, Sofic entropy of Gaussian actions, Ergodic Theory Dynam. Systems, 37 (2017), 2187-2222.  doi: 10.1017/etds.2016.6.

[18]

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

[19]

D. Kerr, Bernoulli actions of sofic groups have completey positive entropy, Israel J. Math., 202 (2014), 461-474.  doi: 10.1007/s11856-014-1077-0.

[20]

D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672.  doi: 10.4171/GGD/142.

[21]

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.

[22]

D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, Amer. J. Math., 135 (2013), 721-761.  doi: 10.1353/ajm.2013.0024.

[23]

J. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Probability, 3 (1975), 1031-1037.  doi: 10.1214/aop/1176996230.

[24]

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.

[25]

D. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), 2 (1980), 161-164.  doi: 10.1090/S0273-0979-1980-14702-3.

[26]

S. Popa, Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu, 5 (2006), 309-332.  doi: 10.1017/S1474748006000016.

[27]

B. Seward, Bernoulli shifts with base of equal entropy are isomorphic, arXiv: 1805.08279, 2018.

[28]

B. Seward, Krieger's finite generator theorem for ergodic actions of countable groups I, Invent. Math., 215 (2019), 265-310.  doi: 10.1007/s00222-018-0826-9.

[29]

B. Seward, The Koopman representation and positive Rokhlin entropy, arXiv: 1804.05270, 2018.

[30]

B. Seward, Positive entropy actions of countable groups factor onto Bernoulli shifts, to appear in J. Amer. Math. Soc.

[31]

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

[32]

B. Weiss, Entropy and actions of sofic groups, Discrete Contin. Dynam. Syst. Ser. B, 20 (2015), 3375-3383.  doi: 10.3934/dcdsb.2015.20.3375.

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