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Examples in the entropy theory of countable group actions

Part of: Ergodic theory

Published online by Cambridge University Press:  25 March 2019

LEWIS BOWEN
LEWIS BOWEN*
Affiliation:
University of Texas at Austin, Mathematics, 1 University Station C1200, Austin, Texas, 78712, USA email [email protected]
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Abstract

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

Keywords

sofic groupentropyOrnstein theoryBenjamini–Schramm convergence

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification
Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 10 , October 2020 , pp. 2593 - 2680
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Copyright
© Cambridge University Press, 2019

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References

Austin, T. and Burton, P.. Uniform mixing and completely positive sofic entropy. J. Anal. Math. , to appear.Google Scholar
Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
Alpeev, A.. The entropy of Gibbs measures on sofic groups. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 436 (2015), 34–48 (Teoriya Predstavlenii, Dinamicheskie Sistemy, Kombinatornye Metody. XXV).Google Scholar
Abramov, L. M. and Rohlin, V. A.. Entropy of a skew product of mappings with invariant measure. Vestnik Leningrad. Univ. 17(7) (1962), 5–13.Google Scholar
Alpeev, A. and Seward, B.. Krieger’s finite generator theorem for ergodic actions of countable groups III. Preprint, 2016.Google Scholar
Austin, T.. Additivity properties of sofic entropy and measures on model spaces. Forum Math. Sigma 4 (2016), e25, 79.CrossRefGoogle Scholar
Austin, T.. Behaviour of entropy under bounded and integrable orbit equivalence. Geom. Funct. Anal. 26(6) (2016), 1483–1525.CrossRefGoogle Scholar
Austin, T.. The geometry of model spaces for probability-preserving actions of sofic groups. Anal. Geom. Metr. Spaces 4 (2016), Art. 6.Google Scholar
Abért, M. and Weiss, B.. Bernoulli actions are weakly contained in any free action. Ergod. Th. & Dynam. Sys. 33(2) (2013), 323–333.CrossRefGoogle Scholar
Ax, J.. The elementary theory of finite fields. Ann. of Math. (2) 88 (1968), 239–271.Google Scholar
Ball, K.. Factors of independent and identically distributed processes with non-amenable group actions. Ergod. Th. & Dynam. Sys. 25(3) (2005), 711–730.Google Scholar
Bekka, B., de la Harpe, P. and Valette, A.. Kazhdan’s Property (T). Cambridge University Press, New York, 2008.Google Scholar
Bowen, L. and Gutman, Y.. A Juzvinskii addition theorem for finitely generated free group actions. Ergod. Th. & Dynam. Sys. 34(1) (2014), 95–109.Google Scholar
Bartholdi, L. and Kielak, D.. Amenability of groups is characterized by Myhill’s Theorem. J. Eur. Math. Soc., to appear.Google Scholar
Bowen, L. and Li, H.. Harmonic models and spanning forests of residually finite groups. J. Funct. Anal. 263(7) (2012), 1769–1808.CrossRefGoogle Scholar
Burger, M. and Mozes, S.. Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Sér. I Math. 324(7) (1997), 747–752.CrossRefGoogle Scholar
Burger, M. and Mozes, S.. Lattices in product of trees. Publ. Math. Inst. Hautes Études Sci. (92) (2000), 151–194, 2001.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) revised edn.. Springer, Berlin, 2008. With a preface by David Ruelle, Edited by Jean-René Chazottes.Google Scholar
Bowen, L.. Free groups in lattices. Geom. Topol. 13(5) (2009), 3021–3054.Google Scholar
Bowen, L.. The ergodic theory of free group actions: entropy and the f-invariant. Groups Geom. Dyn. 4(3) (2010), 419–432.Google Scholar
Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23(1) (2010), 217–245.CrossRefGoogle Scholar
Bowen, L.. Non-abelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula. Ergod. Th. & Dynam. Sys. 30(6) (2010), 1629–1663.Google Scholar
Bowen, L. P.. A measure-conjugacy invariant for free group actions. Ann. of Math. (2) 171(2) (2010), 1387–1400.CrossRefGoogle Scholar
Bowen, L.. Entropy for expansive algebraic actions of residually finite groups. Ergod. Th. & Dynam. Sys. 31(3) (2011), 703–718.CrossRefGoogle Scholar
Bowen, L.. Weak isomorphisms between Bernoulli shifts. Israel J. Math. 183 (2011), 93–102.Google Scholar
Bowen, L.. Every countably infinite group is almost Ornstein. Dynamical Systems and Group Actions (Contemporary Mathematics, 567) . American Mathematical Society, Providence, RI, 2012, pp. 67–78.Google Scholar
Bowen, L.. Sofic entropy and amenable groups. Ergod. Th. & Dynam. Sys. 32(2) (2012), 427–466.CrossRefGoogle Scholar
Bowen, L.. Entropy theory for sofic groupoids I: the foundations. J. Anal. Math. 124 (2014), 149–233.CrossRefGoogle Scholar
Bowen, L.. Zero entropy is generic. Entropy 18(6) (2016), Paper No. 220, 20.Google Scholar
Bowen, L.. Finitary random interlacements and the Gaboriau–Lyons problem. Preprint, 2017,arXiv:1707.09573.Google Scholar
Burton, R. and Steif, J. E.. Non-uniqueness of measures of maximal entropy for subshifts of finite type. Ergod. Th. & Dynam. Sys. 14(2) (1994), 213–235.CrossRefGoogle Scholar
Burton, R. and Steif, J. E.. New results on measures of maximal entropy. Israel J. Math. 89(1–3) (1995), 275–300.CrossRefGoogle Scholar
Benjamini, I. and Schramm, O.. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23) (2001), 13 (electronic).CrossRefGoogle Scholar
Burton, P.. Naive entropy of dynamical systems. Israel J. Math. 219(2) (2017), 637–659.CrossRefGoogle Scholar
Ciobanu, L., Holt, D. F. and Rees, S.. Sofic groups: graph products and graphs of groups. Pacific J. Math. 271(1) (2014), 53–64.Google Scholar
Chung, N.-P.. Topological pressure and the variational principle for actions of sofic groups. Ergod. Th. & Dynam. Sys. 33(5) (2013), 1363–1390.CrossRefGoogle Scholar
Capraro, V. and Lupini, M.. Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture (Lecture Notes in Mathematics, 2136) . Springer, Cham, 2015. With an appendix by Vladimir Pestov.CrossRefGoogle Scholar
Chung, N.-P. and Li, H.. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. Math. 199(3) (2015), 805–858.CrossRefGoogle Scholar
Cornulier, Y.. A sofic group away from amenable groups. Math. Ann. 2 (2011), 269–275.CrossRefGoogle Scholar
Chung, N.-P. and Zhang, G.. Weak expansiveness for actions of sofic groups. J. Funct. Anal. 268(11) (2015), 3534–3565.Google Scholar
Danilenko, A. I.. Entropy theory from the orbital point of view. Monatsh. Math. 134(2) (2001), 121–141.CrossRefGoogle Scholar
Deninger, C.. Fuglede–Kadison determinants and entropy for actions of discrete amenable groups. J. Amer. Math. Soc. 19(3) (2006), 737–758 (electronic).Google Scholar
Dooley, A. H. and Golodets, V. Ya.. The spectrum of completely positive entropy actions of countable amenable groups. J. Funct. Anal. 196(1) (2002), 1–18.Google Scholar
Dooley, A. H., Golodets, V. Ya., Rudolph, D. J. and Sinel’shchikov, S. D.. Non-Bernoulli systems with completely positive entropy. Ergod. Th. & Dynam. Sys. 28(1) (2008), 87–124.CrossRefGoogle Scholar
Dinaburg, E. I.. A correlation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR 190 (1970), 19–22.Google Scholar
Dykema, K., Kerr, D. and Pichot, M.. Sofic dimension for discrete measured groupoids. Trans. Amer. Math. Soc. 366(2) (2014), 707–748.Google Scholar
Dembo, A. and Montanari, A.. Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24(2) (2010), 137–211.CrossRefGoogle Scholar
Dembo, A. and Montanari, A.. Ising models on locally tree-like graphs. Ann. Appl. Probab. 20(2) (2010), 565–592.Google Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18) . Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Danilenko, A. I. and Park, K. K.. Generators and Bernoullian factors for amenable actions and cocycles on their orbits. Ergod. Th. & Dynam. Sys. 22(6) (2002), 1715–1745.CrossRefGoogle Scholar
Deninger, C. and Schmidt, K.. Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Ergod. Th. & Dynam. Sys. 27(3) (2007), 769–786.CrossRefGoogle Scholar
Dye, H. A.. On groups of measure preserving transformation. I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
Dye, H. A.. On groups of measure preserving transformations. II. Amer. J. Math. 85 (1963), 551–576.CrossRefGoogle Scholar
Epstein, I.. Orbit inequivalent actions of non-amenable groups. Preprint, 2008, arXiv:0707.4215.Google Scholar
Elek, G. and Szabó, E.. Sofic groups and direct finiteness. J. Algebra 280(2) (2004), 426–434.Google Scholar
Elek, G. and Szabó, E.. Hyperlinearity, essentially free actions and L 2 -invariants. The sofic property. Math. Ann. 332(2) (2005), 421–441.CrossRefGoogle Scholar
Elek, G. and Szabó, E.. On sofic groups. J. Group Theory 9(2) (2006), 161–171.CrossRefGoogle Scholar
Elek, G. and Szabó, E.. Sofic representations of amenable groups. Proc. Amer. Math. Soc. 139(12) (2011), 4285–4291.Google Scholar
Elek, G. and Szegedy, B.. A measure-theoretic approach to the theory of dense hypergraphs. Adv. Math. 231(3–4) (2012), 1731–1772.CrossRefGoogle Scholar
Friedman, N. A. and Ornstein, D. S.. On isomorphism of weak Bernoulli transformations. Adv. Math. 5(1970) (1970), 365–394.Google Scholar
Foreman, M. and Weiss, B.. An anti-classification theorem for ergodic measure preserving transformations. J. Eur. Math. Soc. (JEMS) 6(3) (2004), 277–292.CrossRefGoogle Scholar
Gaboriau, D.. Entropie sofique. Astérisque 390 (2017), 101–138 , Exp. No. 1108, Séminaire Bourbaki 2015/2016. Exposés 1104–1119.Google Scholar
Georgii, H.-O.. Gibbs Measures and Phase Transitions (de Gruyter Studies in Mathematics, 9) , 2nd edn. Walter de Gruyter & Co, Berlin, 2011.CrossRefGoogle Scholar
Gaboriau, D. and Lyons, R.. A measurable-group-theoretic solution to von Neumann’s problem. Invent. Math. 177(3) (2009), 533–540.Google Scholar
Glasner, E.. Ergodic Theory Via Joinings (Mathematical Surveys and Monographs, 101) . American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Wayne Goodwyn, L.. Topological entropy bounds measure-theoretic entropy. Proc. Amer. Math. Soc. 23 (1969), 679–688.CrossRefGoogle Scholar
Goodman, T. N. T.. Relating topological entropy and measure entropy. Bull. Lond. Math. Soc. 3 (1971), 176–180.CrossRefGoogle Scholar
Wayne Goodwyn, L.. Comparing topological entropy with measure-theoretic entropy. Amer. J. Math. 94 (1972), 366–388.CrossRefGoogle Scholar
Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1(2) (1999), 109–197.CrossRefGoogle Scholar
Golodets, V. Ya. and Sinel’shchikov, S. D.. Complete positivity of entropy and non-Bernoullicity for transformation groups. Colloq. Math. 84/85(part 2) (2000), 421–429. Dedicated to the memory of Anzelm Iwanik.CrossRefGoogle Scholar
Greschonig, G. and Schmidt, K.. Ergodic decomposition of quasi-invariant probability measures. Colloq. Math. 84/85(part 2) (2000), 495–514. Dedicated to the memory of Anzelm Iwanik.CrossRefGoogle Scholar
Gaboriau, D. and Seward, B.. Cost, ℓ 2 -Betti numbers, and the sofic entropy of some algebraic actions. J. Anal. Math. Preprint, 2015, arXiv:1509.02482.Google Scholar
Glasner, E., Thouvenot, J.-P. and Weiss, B.. Entropy theory without a past. Ergod. Th. & Dynam. Sys. 20(5) (2000), 1355–1370.CrossRefGoogle Scholar
Hayes, B.. Mixing and spectral gap relative to Pinsker factors for sofic groups. Proc. 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F. R. Jones’ 60th Birthday (Proc. Centre Math. Appl. Austral. Nat. Univ., 46) . Australian National University, Canberra, 2017, pp. 193–221.Google Scholar
Hayes, B.. Local and doubly empirical convergence and the entropy of algebraic actions of sofic groups. Ergod. Th. & Dynam. Sys. , to appear. Preprint, 2016, arXiv:1603.06450.Google Scholar
Hayes, B.. Fuglede–Kadison determinants and sofic entropy. Geom. Funct. Anal. 26(2) (2016), 520–606.CrossRefGoogle Scholar
Hayes, B.. Relative entropy and the Pinsker product formula for sofic groups. Preprint, 2016, arXiv: 1605.01747.Google Scholar
Hayes, B.. Independence tuples and Deninger’s problem. Groups Geom. Dyn. 11(1) (2017), 245–289.Google Scholar
Hayes, B.. Sofic entropy of Gaussian actions. Ergod. Th. & Dynam. Sys. 37(7) (2017), 2187–2222.CrossRefGoogle Scholar
Hayes, B.. Polish models and sofic entropy. J. Inst. Math. Jussieu 17(2) (2018), 241–275.CrossRefGoogle Scholar
Hatami, H., Lovász, L. and Szegedy, B.. Limits of locally-globally convergent graph sequences. Geom. Funct. Anal. 24(1) (2014), 269–296.CrossRefGoogle Scholar
Hoffman, C.. Subshifts of finite type which have completely positive entropy. Discrete Contin. Dyn. Syst. 29(4) (2011), 1497–1516.CrossRefGoogle Scholar
Hayes, B. and Sale, A.. The wreath product of two sofic groups is sofic. Preprint, 2016, arXiv:1601.03286.Google Scholar
Hayes, B. and Sale, A. W.. Metric approximations of wreath products. Ann. Inst. Fourier (Grenoble) 68(1) (2018), 423–455.CrossRefGoogle Scholar
Ioana, A., Kechris, A. S. and Tsankov, T.. Subequivalence relations and positive-definite functions. Groups Geom. Dyn. 3(4) (2009), 579–625.CrossRefGoogle Scholar
Juzvinskiĭ, S. A.. Metric properties of automorphisms of locally compact commutative groups. Sibirsk. Mat. Ž. 6 (1965), 244–247.Google Scholar
Juzvinskiĭ, S. A.. Metric properties of the endomorphisms of compact groups. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 1295–1328.Google Scholar
Kechris, A. S.. Global Aspects of Ergodic Group Actions (Mathematical Surveys and Monographs, 160) . American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
Keller, G.. Equilibrium States in Ergodic Theory (London Mathematical Society Student Texts, 42) . Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
Kerr, D.. Sofic measure entropy via finite partitions. Groups Geom. Dyn. 7(3) (2013), 617–632.CrossRefGoogle Scholar
Kerr, D.. Bernoulli actions of sofic groups have completely positive entropy. Israel J. Math. 202(1) (2014), 461–474.CrossRefGoogle Scholar
Keynes, H. B.. Lifting of topological entropy. Proc. Amer. Math. Soc. 24 (1970), 440–445.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995, with a supplementary chapter by Katok and Leonardo Mendoza.CrossRefGoogle Scholar
Kida, Y.. Orbit equivalence rigidity for ergodic actions of the mapping class group. Geom. Dedicata 131 (2008), 99–109.CrossRefGoogle Scholar
Kieffer, J. C.. A generalized Shannon–McMillan theorem for the action of an amenable group on a probability space. Ann. Probab. 3(6) (1975), 1031–1037.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Bernoulli actions and infinite entropy. Groups Geom. Dyn. 5(3) (2011), 663–672.Google Scholar
Kerr, D. and Li, H.. Entropy and the variational principle for actions of sofic groups. Invent. Math. 186(3) (2011), 501–558.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Combinatorial independence and sofic entropy. Commun. Math. Stat. 1(2) (2013), 213–257.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Soficity, amenability, and dynamical entropy. Amer. J. Math. 135(3) (2013), 721–761.CrossRefGoogle Scholar
Kerr, D. and Li, H.. Ergodic Theory (Springer Monographs in Mathematics) . Springer, Cham, 2016, Independence and dichotomies.CrossRefGoogle Scholar
Kolmogorov, A. N.. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861–864.Google Scholar
Kolmogorov, A. N.. Entropy per unit time as a metric invariant of automorphisms. Dokl. Akad. Nauk SSSR 124 (1959), 754–755.Google Scholar
Krieger, W.. On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc. 149 (1970), 453–464.Google Scholar
Kun, G.. On sofic approximations of property (T) groups. Preprint, 2016, arXiv:1606.04471.Google Scholar
Li, H.. Compact group automorphisms, addition formulas and Fuglede–Kadison determinants. Ann. of Math. (2) 176(1) (2012), 303–347.Google Scholar
Li, H. and Liang, B.. Sofic mean length. Adv. Math. , to appear.Google Scholar
Lind, D., Schmidt, K. and Ward, T.. Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101(3) (1990), 593–629.CrossRefGoogle Scholar
Li, H. and Thom, A.. Entropy, determinants, and L 2 -torsion. J. Amer. Math. Soc. 27(1) (2014), 239–292.Google Scholar
Malcev, A.. On isomorphic matrix representations of infinite groups. Rec. Math. [Mat. Sbornik] N.S. 8(50) (1940), 405–422.Google Scholar
Miles, R. and Björklund, M.. Entropy range problems and actions of locally normal groups. Discrete Contin. Dyn. Syst. 25(3) (2009), 981–989.CrossRefGoogle Scholar
Meyerovitch, T.. Positive sofic entropy implies finite stabilizer. Entropy 18(7) (2016), 14, Paper No. 263.CrossRefGoogle Scholar
Miles, R.. The entropy of algebraic actions of countable torsion-free abelian groups. Fund. Math. 201(3) (2008), 261–282.CrossRefGoogle Scholar
Misiurewicz, M.. A short proof of the variational principle for a Z + N action on a compact space. Astérisque 40 (1976), 147–157.Google Scholar
Moulin Ollagnier, J.. Ergodic Theory and Statistical Mechanics (Lecture Notes in Mathematics, 1115) . Springer, Berlin, 1985.Google Scholar
Ol’shanskiĭ, A. Yu.. Geometry of Defining Relations in Groups (Mathematics and its Applications (Soviet Series), 70) . Kluwer Academic Publishers Group, Dordrecht, 1991, translated from the 1989 Russian original by Yu. A. Bakhturin.Google Scholar
Ozawa, N. and Popa, S.. On a class of II1 factors with at most one Cartan subalgebra. Ann. of Math. (2) 172(1) (2010), 713–749.CrossRefGoogle Scholar
Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4(1970) (1970), 337–352.CrossRefGoogle Scholar
Ornstein, D.. Factors of Bernoulli shifts are Bernoulli shifts. Adv. Math. 5(1970) (1970), 349–364.CrossRefGoogle Scholar
Ornstein, D.. Two Bernoulli shifts with infinite entropy are isomorphic. Adv. Math. 5(1970) (1970), 339–348.CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B.. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) 2(1) (1980), 161–164.CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1–141.CrossRefGoogle Scholar
Ornstein, D. and Weiss, B.. Entropy is the only finitely observable invariant. J. Mod. Dyn. 1(1) (2007), 93–105.Google Scholar
Păunescu, L.. On sofic actions and equivalence relations. J. Funct. Anal. 261(9) (2011), 2461–2485.CrossRefGoogle Scholar
Pestov, V. G.. Hyperlinear and sofic groups: a brief guide. Bull. Symbolic Logic 14(4) (2008), 449–480.Google Scholar
Petersen, K.. Ergodic Theory (Cambridge Studies in Advanced Mathematics, 2) . Cambridge University Press, Cambridge, 1989, corrected reprint of the 1983 original.Google Scholar
Pestov, V. G. and Kwiatkowska, A.. An introduction to hyperlinear and sofic groups. Appalachian Set Theory (London Mathematical Society Lecture Notes Series) , to appear. Preprint, 2009, arXiv:0911.4266v2.Google Scholar
Popa, S.. Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions. J. Inst. Math. Jussieu 5(2) (2006), 309–332.CrossRefGoogle Scholar
Popa, S.. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. II. Invent. Math. 165(2) (2006), 409–451.Google Scholar
Popa, S.. Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170(2) (2007), 243–295.CrossRefGoogle Scholar
Popa, S.. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21(4) (2008), 981–1000.CrossRefGoogle Scholar
Popa, S. and Sasyk, R.. On the cohomology of Bernoulli actions. Ergod. Th. & Dynam. Sys. 27(1) (2007), 241–251.CrossRefGoogle Scholar
Rohlin, V. A.. Lectures on the entropy theory of transformations with invariant measure. Uspekhi Mat. Nauk 22(5 (137)) (1967), 3–56.Google Scholar
Rudolph, D. J.. If a finite extension of a Bernoulli shift has no finite rotation factors, it is Bernoulli. Israel J. Math. 30(3) (1978), 193–206.CrossRefGoogle Scholar
Rudolph, D. J.. Fundamentals of Measurable Dynamics (Oxford Science Publications) . The Clarendon Press, Oxford University Press, New York, 1990, Ergodic theory on Lebesgue spaces.Google Scholar
Rudolph, D. J. and Weiss, B.. Entropy and mixing for amenable group actions. Ann. of Math. (2) 151(3) (2000), 1119–1150.Google Scholar
Schmidt, K.. Dynamical Systems of Algebraic Origin (Modern Birkhäuser Classics) . Birkhäuser/Springer Basel AG, Basel, 1995, 2011 reprint of the 1995 original.Google Scholar
Seward, B.. Every action of a nonamenable group is the factor of a small action. J. Mod. Dyn. 8(2) (2014), 251–270.CrossRefGoogle Scholar
Seward, B.. A subgroup formula for f-invariant entropy. Ergod. Th. & Dynam. Sys. 34(1) (2014), 263–298.CrossRefGoogle Scholar
Seward, B.. Ergodic actions of countable groups and finite generating partitions. Groups Geom. Dyn. 9(3) (2015), 793–810.Google Scholar
Seward, B.. Krieger’s finite generator theorem for ergodic actions of countable groups II. J. Mod. Dynam. , to appear.Google Scholar
Seward, B.. Finite entropy actions of free groups, rigidity of stabilizers, and a Howe–Moore type phenomenon. J. Anal. Math. 129 (2016), 309–340.CrossRefGoogle Scholar
Seward, B.. Weak containment and Rokhlin entropy. Preprint, 2016, arXiv:1602.06680.Google Scholar
Seward, B.. Positive entropy actions of countable groups factor onto Bernoulli shifts. J. Amer. Math. Soc. , to appear.Google Scholar
Seward, B.. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310.Google Scholar
Sinaĭ, Ja. G.. On a weak isomorphism of transformations with invariant measure. Mat. Sb. (N.S.) 63(105) (1964), 23–42.Google Scholar
Shields, P. and Thouvenot, J.-P.. Entropy zero × Bernoulli processes are closed in the d̄-metric. Ann. Probab. 3(4) (1975), 732–736.CrossRefGoogle Scholar
Seward, B. and Tucker-Drob, R. D.. Borel structurability on the 2-shift of a countable group. Ann. Pure Appl. Logic 167(1) (2016), 1–21.Google Scholar
Stepin, A. M.. Bernoulli shifts on groups. Dokl. Akad. Nauk SSSR 223(2) (1975), 300–302.Google Scholar
Thomas, R. K.. The addition theorem for the entropy of transformations of G-spaces. Trans. Amer. Math. Soc. 160 (1971), 119–130.Google Scholar
Thouvenot, J.-P.. 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(2–3) (1975), 177–207. Conference on Ergodic Theory and Topological Dynamics (Kibbutz, Lavi, 1974).CrossRefGoogle Scholar
Thom, A.. Examples of hyperlinear groups without factorization property. Groups Geom. Dyn. 4(1) (2010), 195–208.CrossRefGoogle Scholar
Weiss, B.. Sofic groups and dynamical systems. Sankhyā Ser. A 62(3) (2000), 350–359. Conference on Ergodic Theory and Harmonic Analysis (Mumbai, 1999).Google Scholar
Weiss, B.. Actions of amenable groups. Topics in Dynamics and Ergodic Theory (London Mathematical Society Lecture Note Series, 310) . Cambridge University Press, Cambridge, 2003, pp. 226–262.CrossRefGoogle Scholar
Weiss, B.. Entropy and actions of sofic groups. Discrete Contin. Dyn. Syst. Ser. B 20(10) (2015), 3375–3383.Google Scholar
Ward, T. and Zhang, Q.. The Abramov–Rokhlin entropy addition formula for amenable group actions. Monatsh. Math. 114(3–4) (1992), 317–329.CrossRefGoogle Scholar
Zhang, G.. Local variational principle concerning entropy of sofic group action. J. Funct. Anal. 262(4) (2012), 1954–1985.Google Scholar
Zimmer, R. J.. Ergodic Theory and Semisimple Groups (Monographs in Mathematics, 81) . Birkhäuser, Basel, 1984.CrossRefGoogle Scholar