Bernoulli shifts with bases of equal entropy are isomorphic

Brandon Seward

doi:10.3934/jmd.2022011

Article Contents

2022, Volume 18: 345-362. Doi: 10.3934/jmd.2022011

This volume Previous Article Cocycle superrigidity from higher rank lattices to

$ {{\rm{Out}}}{(F_N)} $

Next Article Common preperiodic points for quadratic polynomials

Bernoulli shifts with bases of equal entropy are isomorphic

Brandon Seward

Department of Mathematics, University of California San Diego, 9500 Gilman Dr #0112, La Jolla, CA, 92093, USA

Received: February 09, 2021

Revised: December 08, 2021

The author was partially supported by ERC grant 306494 and Simons Foundation grant 328027 (P.I. Tim Austin)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We prove that if $ G $ is a countably infinite group and $ (L, \lambda) $ and $ (K, \kappa) $ are probability spaces having equal Shannon entropy, then the Bernoulli shifts $ G \curvearrowright (L^G, \lambda^G) $ and $ G \curvearrowright (K^G, \kappa^G) $ are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both $ \lambda $ and $ \kappa $ have at least $ 3 $ points in their support. We furthermore produce finitary isomorphisms in the case where both $ L $ and $ K $ are finite.

Keywords:

Mathematics Subject Classification: Primary: 37A35; Secondary: 28D20.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. Alpeev and B. Seward, Krieger's finite generator theorem for ergodic actions of countable groups Ⅲ, Ergodic Theory Dynam. Systems, 41 (2021), 2881-2917. doi: 10.1017/etds.2020.89.
[2]	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.
[3]	L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466. doi: 10.1017/S0143385711000253.
[4]	L. Bowen, Every countably infinite group is almost Ornstein, Dynamical Systems and Group Actions, 67–78 doi: 10.1090/conm/567/11234.
[5]	S. Gao and S. Jackson, Countable abelian group actions and hyperfinite equivalence relations, Invent. Math., 201 (2015), 309-383. doi: 10.1007/s00222-015-0603-y.
[6]	S. Jackson, A. S. Kechris and A. Louveau, Countable Borel equivalence relations, J. Math. Logic, 2 (2002), 1-80. doi: 10.1142/S0219061302000138.
[7]	M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math. (2), 109 (1979), 397-406. doi: 10.2307/1971117.
[8]	A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.
[9]	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.
[10]	D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, Amer. J. Math., 135 (2013), 721-761. doi: 10.1353/ajm.2013.0024.
[11]	D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672. doi: 10.4171/GGD/142.
[12]	J. C. Kieffer, A generalized Shannon–McMillan theorem for the action of an amenable group on a probability space, Ann. Prob., 3 (1975), 1031-1037. doi: 10.1214/aop/1176996230.
[13]	A. N. Kolmogorov, New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958), 861-864.
[14]	A. N. Kolmogorov, Entropy per unit time as a metric invariant for automorphisms, Dokl. Akad. Nauk SSSR, 124 (1959), 754-755.
[15]	D. V. Lytkina, Structure of a group with elements of order at most 4, Siberian Math. J., 48 (2007), 283-287. doi: 10.1007/s11202-007-0028-y.
[16]	A. S. Mamontov, Groups of exponent 12 without elements of order 12, Siberian Math. J., 54 (2013), 114-118. doi: 10.1134/S003744661301014X.
[17]	L. D. Mešalkin, A case of isomorphism of Bernoulli schemes, Dokl. Akad. Nauk SSSR, 128 (1959), 41-44.
[18]	A. Yu. Olshanskii, An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 309-321.
[19]	D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4 (1970), 337-348. doi: 10.1016/0001-8708(70)90029-0.
[20]	D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math., 5 (1970), 339-348. doi: 10.1016/0001-8708(70)90008-3.
[21]	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.
[22]	B. Petit, Deux schémas de Bernoulli d'alphabet dénombrable et d'entropie infinie sont finitairement isomorphes, Z. Wahrsch. Verw. Gebiete, 59 (1982), 161-168. doi: 10.1007/BF00531740.
[23]	S. Schneider and B. Seward, Locally nilpotent groups and hyperfinite equivalence relations, preprint, arXiv: 1308.5853, 2013.
[24]	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.
[25]	B. Seward, Krieger's finite generator theorem for ergodic actions of countable groups Ⅱ, J. Mod. Dyn., 15 (2019), 1-39. doi: 10.3934/jmd.2019012.
[26]	Ya. G. Sinai, On the concept of entropy for a dynamical system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.
[27]	Ya. G. Sinai, A weak isomorphism of transformations with invariant measure, Dokl. Akad. Nauk SSSR, 147 (1962), 797-800.
[28]	A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.