Positive entropy actions of countable groups factor onto Bernoulli shifts

Seward, Brandon

doi:10.1090/jams/931

Positive entropy actions of countable groups factor onto Bernoulli shifts
HTML articles powered by AMS MathViewer

by Brandon Seward;

J. Amer. Math. Soc. 33 (2020), 57-101

DOI: https://doi.org/10.1090/jams/931

Published electronically: September 27, 2019

HTML | PDF | Request permission

Abstract:

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy), then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory.

References

Miklós Abért, Yair Glasner, and Bálint Virág, Kesten’s theorem for invariant random subgroups, Duke Math. J. 163 (2014), no. 3, 465–488. MR 3165420, DOI 10.1215/00127094-2410064
A. Alpeev and B. Seward, Krieger’s finite generator theorem for ergodic actions of countable groups III, preprint. https://arxiv.org/abs/1705.09707.
Lewis Phylip Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math. (2) 171 (2010), no. 2, 1387–1400. MR 2630067, DOI 10.4007/annals.2010.171.1387
Lewis Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc. 23 (2010), no. 1, 217–245. MR 2552252, DOI 10.1090/S0894-0347-09-00637-7
Lewis Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 427–466. MR 2901354, DOI 10.1017/S0143385711000253
Lewis Bowen, Every countably infinite group is almost Ornstein, Dynamical systems and group actions, Contemp. Math., vol. 567, Amer. Math. Soc., Providence, RI, 2012, pp. 67–78. MR 2931910, DOI 10.1090/conm/567/11234
Lewis Bowen, Finitary random interlacements and the Gaboriau-Lyons problem, Geom. Funct. Anal. 29 (2019), no. 3, 659–689. MR 3962876, DOI 10.1007/s00039-019-00494-4
Lewis Bowen, Daniel Hoff, and Adrian Ioana, von Neumann’s problem and extensions of non-amenable equivalence relations, Groups Geom. Dyn. 12 (2018), no. 2, 399–448. MR 3813200, DOI 10.4171/GGD/456
R. M. Burton, M. S. Keane, and Jacek Serafin, Residuality of dynamical morphisms. part 2, Colloq. Math. 84/85 (2000), no. part 2, 307–317. Dedicated to the memory of Anzelm Iwanik. MR 1784199, DOI 10.4064/cm-84/85-2-307-317
Alexandre I. Danilenko and Kyewon K. Park, Generators and Bernoullian factors for amenable actions and cocycles on their orbits, Ergodic Theory Dynam. Systems 22 (2002), no. 6, 1715–1745. MR 1944401, DOI 10.1017/S014338570200072X
A. H. Dooley and V. Ya. Golodets, The spectrum of completely positive entropy actions of countable amenable groups, J. Funct. Anal. 196 (2002), no. 1, 1–18. MR 1941988, DOI 10.1006/jfan.2002.3966
R. Dougherty, S. Jackson, and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. 341 (1994), no. 1, 193–225. MR 1149121, DOI 10.1090/S0002-9947-1994-1149121-0
Tomasz Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2809170, DOI 10.1017/CBO9780511976155
R. H. Farrell, Representation of invariant measures, Illinois J. Math. 6 (1962), 447–467. MR 150264, DOI 10.1215/ijm/1255632504
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math. 5 (1970), 365–394 (1970). MR 274718, DOI 10.1016/0001-8708(70)90010-1
D. Gaboriau, Arbres, groupes, quotients, Habilitation á diriger des recherches, 2002. http://perso.ens-lyon.fr/gaboriau/Travaux-Publi/Habilitation/Habilitation.html.
Damien Gaboriau and Russell Lyons, A measurable-group-theoretic solution to von Neumann’s problem, Invent. Math. 177 (2009), no. 3, 533–540. MR 2534099, DOI 10.1007/s00222-009-0187-5
Giovanni Gallavotti and Donald S. Ornstein, Billiards and Bernoulli schemes, Comm. Math. Phys. 38 (1974), 83–101. MR 355003, DOI 10.1007/BF01651505
Ben Hayes, Fuglede-Kadison determinants and sofic entropy, Geom. Funct. Anal. 26 (2016), no. 2, 520–606. MR 3513879, DOI 10.1007/s00039-016-0370-y
Ben Hayes, Polish models and sofic entropy, J. Inst. Math. Jussieu 17 (2018), no. 2, 241–275. MR 3773269, DOI 10.1017/S1474748015000468
Ben Hayes, Mixing and spectral gap relative to Pinsker factors for sofic groups, 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., vol. 46, Austral. Nat. Univ., Canberra, 2017, pp. 193–221. MR 3635672
Ben Hayes, Sofic entropy of Gaussian actions, Ergodic Theory Dynam. Systems 37 (2017), no. 7, 2187–2222. MR 3693125, DOI 10.1017/etds.2016.6
Ben Hayes, Independence tuples and Deninger’s problem, Groups Geom. Dyn. 11 (2017), no. 1, 245–289. MR 3641841, DOI 10.4171/GGD/396
S. Jackson, A. S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Log. 2 (2002), no. 1, 1–80. MR 1900547, DOI 10.1142/S0219061302000138
Yitzhak Katznelson and Benjamin Weiss, Commuting measure-preserving transformations, Israel J. Math. 12 (1972), 161–173. MR 316680, DOI 10.1007/BF02764660
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Alexander S. Kechris, Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010. MR 2583950, DOI 10.1090/surv/160
A. S. Kechris, S. Solecki, and S. Todorcevic, Borel chromatic numbers, Adv. Math. 141 (1999), no. 1, 1–44. MR 1667145, DOI 10.1006/aima.1998.1771
David Kerr and Hanfeng Li, Entropy and the variational principle for actions of sofic groups, Invent. Math. 186 (2011), no. 3, 501–558. MR 2854085, DOI 10.1007/s00222-011-0324-9
David Kerr and Hanfeng Li, Soficity, amenability, and dynamical entropy, Amer. J. Math. 135 (2013), no. 3, 721–761. MR 3068400, DOI 10.1353/ajm.2013.0024
David Kerr and Hanfeng Li, Combinatorial independence and sofic entropy, Commun. Math. Stat. 1 (2013), no. 2, 213–257. MR 3197860, DOI 10.1007/s40304-013-0001-y
David Kerr and Hanfeng Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn. 5 (2011), no. 3, 663–672. MR 2813530, DOI 10.4171/GGD/142
John C. Kieffer and Maurice Rahe, Selecting universal partitions in ergodic theory, Ann. Probab. 9 (1981), no. 4, 705–709. MR 624699
A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861–864 (Russian). MR 103254
A. N. Kolmogorov, Entropy per unit time as a metric invariant of automorphisms, Dokl. Akad. Nauk SSSR 124 (1959), 754–755 (Russian). MR 103255
D. A. Lind, The structure of skew products with ergodic group automorphisms, Israel J. Math. 28 (1977), no. 3, 205–248. MR 460593, DOI 10.1007/BF02759810
L. D. Mešalkin, A case of isomorphism of Bernoulli schemes, Dokl. Akad. Nauk SSSR 128 (1959), 41–44 (Russian). MR 110782
G. Miles and R. K. Thomas, The breakdown of automorphisms of compact topological groups, Studies in probability and ergodic theory, Adv. Math. Suppl. Stud., vol. 2, Academic Press, New York-London, 1978, pp. 207–218. MR 517262
Donald Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337–352. MR 257322, DOI 10.1016/0001-8708(70)90029-0
Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339–348 (1970). MR 274716, DOI 10.1016/0001-8708(70)90008-3
Donald Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math. 5 (1970), 349–364 (1970). MR 274717, DOI 10.1016/0001-8708(70)90009-5
Donald S. Ornstein, Ergodic theory, randomness, and dynamical systems, Yale Mathematical Monographs, No. 5, Yale University Press, New Haven, Conn.-London, 1974. James K. Whittemore Lectures in Mathematics given at Yale University. MR 447525
Donald S. Ornstein and Benjamin Weiss, Geodesic flows are Bernoullian, Israel J. Math. 14 (1973), 184–198. MR 325926, DOI 10.1007/BF02762673
Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141. MR 910005, DOI 10.1007/BF02790325
M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math. 17 (1974), 380–391. MR 374387, DOI 10.1007/BF02757140
Brandon Seward, Krieger’s finite generator theorem for actions of countable groups I, Invent. Math. 215 (2019), no. 1, 265–310. MR 3904452, DOI 10.1007/s00222-018-0826-9
B. Seward, Krieger’s finite generator theorem for ergodic actions of countable groups II, preprint. http://arxiv.org/abs/1501.03367.
B. Seward, Weak containment and Rokhlin entropy, preprint. https://arxiv.org/abs/1602.06680.
B. Seward, The Koopman representation and positive Rokhlin entropy, preprint. https://arxiv.org/abs/1804.05270.
B. Seward, Bernoulli shifts with bases of equal entropy are isomorphic, preprint. https://arxiv.org/abs/1805.08279.
Brandon Seward and Robin D. Tucker-Drob, Borel structurability on the 2-shift of a countable group, Ann. Pure Appl. Logic 167 (2016), no. 1, 1–21. MR 3413491, DOI 10.1016/j.apal.2015.07.005
Ja. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768–771 (Russian). MR 103256
Ja. G. Sinaĭ, A weak isomorphism of transformations with invariant measure, Dokl. Akad. Nauk SSSR 147 (1962), 797–800 (Russian). MR 161960
M. Takesaki, Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition; Operator Algebras and Non-commutative Geometry, 5. MR 1873025
Jean-Paul 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), no. 2-3, 177–207 (French, with English summary). MR 399419, DOI 10.1007/BF02760797
V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191–220. MR 159923, DOI 10.1090/S0002-9947-1963-0159923-5
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4