# American Institute of Mathematical Sciences

2019, 15: 1-39. doi: 10.3934/jmd.2019012

## Krieger's finite generator theorem for actions of countable groups Ⅱ

 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

Received  May 23, 2018 Revised  November 2018 Published  February 2019

We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért–Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (ⅰ) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ⅱ) Bernoulli shifts have completely positive Rokhlin entropy; and (ⅲ) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.

Citation: 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
##### References:
 [1] M. Abért and B. Weiss, Bernoulli actions are weakly contained in any free action, Ergodic Theory and Dynamical Systems, 23 (2013), 323-333.  doi: 10.1017/S0143385711000988.  Google Scholar [2] A. Alpeev and B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅲ, preprint, arXiv: 1705.09707. Google Scholar [3] P. Ara, K. C. O'Meara and F. Perera, Stable finiteness of group rings in arbitrary characteristic, Advances in Math., 170 (2002), 224-238.  doi: 10.1006/aima.2002.2075.  Google Scholar [4] L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, Journal of the American Mathematical Society, 23 (2010), 217-245.  doi: 10.1090/S0894-0347-09-00637-7.  Google Scholar [5] L. Bowen, Sofic entropy and amenable groups, Ergod. Th. & Dynam. Sys., 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.  Google Scholar [6] 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.  Google Scholar [7] M. Burger and A. Valette, Idempotents in complex group rings: Theorems of Zalesskii and Bass revisited, Journal of Lie Theory, 8 (1998), 219-228.   Google Scholar [8] V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, With an appendix by Vladimir Pestov, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. doi: 10.1007/978-3-319-19333-5.  Google Scholar [9] N.-P. Chung, Topological pressure and the variational principle for actions of sofic groups, Ergodic Theory and Dynamical Systems, 33 (2013), 1363-1390.  doi: 10.1017/S0143385712000429.  Google Scholar [10] T. Downarowicz, Entropy in Dynamical Systems, Cambridge University Press, New York, 2011.  doi: 10.1017/CBO9780511976155.  Google Scholar [11] G. Elek and E. Szabó, Sofic groups and direct finiteness, Journal of Algebra, 280 (2004), 426-434.  doi: 10.1016/j.jalgebra.2004.06.023.  Google Scholar [12] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar [13] W. Gottschalk, Some general dynamical notions, in Recent Advances in Topological Dynamics, Lecture Notes in Mathematics, 318, Springer, Berlin, 1973, 120–125.  Google Scholar [14] C. Grillenberger and U. Krengel, On marginal distributions and isomorphisms of stationary processes, Math. Z., 149 (1976), 131-154.  doi: 10.1007/BF01301571.  Google Scholar [15] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. European Math. Soc., 1 (1999), 109-197.  doi: 10.1007/PL00011162.  Google Scholar [16] I. Kaplansky, Fields and Rings, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, IL, 1972.   Google Scholar [17] A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.  Google Scholar [18] A. Kechris, S. Solecki and S. Todorcevic, Borel chromatic numbers, Adv. in Math., 141 (1999), 1-44.  doi: 10.1006/aima.1998.1771.  Google Scholar [19] D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632.  doi: 10.4171/GGD/200.  Google Scholar [20] D. Kerr, Bernoulli actions of sofic groups have completely positive entropy, Israel Journal of Math., 202 (2014), 461-474.  doi: 10.1007/s11856-014-1077-0.  Google Scholar [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.  Google Scholar [22] D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, American Journal of Mathematics, 135 (2013), 721-761.  doi: 10.1353/ajm.2013.0024.  Google Scholar [23] D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672.  doi: 10.4171/GGD/142.  Google Scholar [24] A. N. Kolmogorov, New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces, Doklady of Russian Academy of Sciences, 119 (1958), 861-864.   Google Scholar [25] A. N. Kolmogorov, Entropy per unit time as a metric invariant for automorphisms, Doklady of Russian Academy of Sciences, 124 (1959), 754-755.   Google Scholar [26] D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4 (1970), 337-352.  doi: 10.1016/0001-8708(70)90029-0.  Google Scholar [27] 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.  Google Scholar [28] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d'Analyse Mathématique, 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar [29] D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Annals of Mathematics, 151 (2000), 1119-1150.  doi: 10.2307/121130.  Google Scholar [30] Ya. G. Sinam${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, On the concept of entropy for a dynamical system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.   Google Scholar [31] B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅰ, Inventiones Mathematicae, 215 (2019), 265-310.  doi: 10.1007/s00222-018-0826-9.  Google Scholar [32] B. Seward, Weak containment, Pinsker algebras, and Rokhlin entropy, preprint, arXiv: 1602.06680. Google Scholar [33] B. Seward and R. D. Tucker-Drob, Borel structurability on the 2-shift of a countable group, Ann. Pure Appl. Logic, 167 (2016), 1-21.  doi: 10.1016/j.apal.2015.07.005.  Google Scholar [34] A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.   Google Scholar [35] B. Weiss, Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350–359.  Google Scholar

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##### References:
 [1] M. Abért and B. Weiss, Bernoulli actions are weakly contained in any free action, Ergodic Theory and Dynamical Systems, 23 (2013), 323-333.  doi: 10.1017/S0143385711000988.  Google Scholar [2] A. Alpeev and B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅲ, preprint, arXiv: 1705.09707. Google Scholar [3] P. Ara, K. C. O'Meara and F. Perera, Stable finiteness of group rings in arbitrary characteristic, Advances in Math., 170 (2002), 224-238.  doi: 10.1006/aima.2002.2075.  Google Scholar [4] L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, Journal of the American Mathematical Society, 23 (2010), 217-245.  doi: 10.1090/S0894-0347-09-00637-7.  Google Scholar [5] L. Bowen, Sofic entropy and amenable groups, Ergod. Th. & Dynam. Sys., 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.  Google Scholar [6] 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.  Google Scholar [7] M. Burger and A. Valette, Idempotents in complex group rings: Theorems of Zalesskii and Bass revisited, Journal of Lie Theory, 8 (1998), 219-228.   Google Scholar [8] V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, With an appendix by Vladimir Pestov, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. doi: 10.1007/978-3-319-19333-5.  Google Scholar [9] N.-P. Chung, Topological pressure and the variational principle for actions of sofic groups, Ergodic Theory and Dynamical Systems, 33 (2013), 1363-1390.  doi: 10.1017/S0143385712000429.  Google Scholar [10] T. Downarowicz, Entropy in Dynamical Systems, Cambridge University Press, New York, 2011.  doi: 10.1017/CBO9780511976155.  Google Scholar [11] G. Elek and E. Szabó, Sofic groups and direct finiteness, Journal of Algebra, 280 (2004), 426-434.  doi: 10.1016/j.jalgebra.2004.06.023.  Google Scholar [12] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar [13] W. Gottschalk, Some general dynamical notions, in Recent Advances in Topological Dynamics, Lecture Notes in Mathematics, 318, Springer, Berlin, 1973, 120–125.  Google Scholar [14] C. Grillenberger and U. Krengel, On marginal distributions and isomorphisms of stationary processes, Math. Z., 149 (1976), 131-154.  doi: 10.1007/BF01301571.  Google Scholar [15] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. European Math. Soc., 1 (1999), 109-197.  doi: 10.1007/PL00011162.  Google Scholar [16] I. Kaplansky, Fields and Rings, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, IL, 1972.   Google Scholar [17] A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.  Google Scholar [18] A. Kechris, S. Solecki and S. Todorcevic, Borel chromatic numbers, Adv. in Math., 141 (1999), 1-44.  doi: 10.1006/aima.1998.1771.  Google Scholar [19] D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632.  doi: 10.4171/GGD/200.  Google Scholar [20] D. Kerr, Bernoulli actions of sofic groups have completely positive entropy, Israel Journal of Math., 202 (2014), 461-474.  doi: 10.1007/s11856-014-1077-0.  Google Scholar [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.  Google Scholar [22] D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, American Journal of Mathematics, 135 (2013), 721-761.  doi: 10.1353/ajm.2013.0024.  Google Scholar [23] D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672.  doi: 10.4171/GGD/142.  Google Scholar [24] A. N. Kolmogorov, New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces, Doklady of Russian Academy of Sciences, 119 (1958), 861-864.   Google Scholar [25] A. N. Kolmogorov, Entropy per unit time as a metric invariant for automorphisms, Doklady of Russian Academy of Sciences, 124 (1959), 754-755.   Google Scholar [26] D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4 (1970), 337-352.  doi: 10.1016/0001-8708(70)90029-0.  Google Scholar [27] 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.  Google Scholar [28] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d'Analyse Mathématique, 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar [29] D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Annals of Mathematics, 151 (2000), 1119-1150.  doi: 10.2307/121130.  Google Scholar [30] Ya. G. Sinam${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, On the concept of entropy for a dynamical system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.   Google Scholar [31] B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅰ, Inventiones Mathematicae, 215 (2019), 265-310.  doi: 10.1007/s00222-018-0826-9.  Google Scholar [32] B. Seward, Weak containment, Pinsker algebras, and Rokhlin entropy, preprint, arXiv: 1602.06680. Google Scholar [33] B. Seward and R. D. Tucker-Drob, Borel structurability on the 2-shift of a countable group, Ann. Pure Appl. Logic, 167 (2016), 1-21.  doi: 10.1016/j.apal.2015.07.005.  Google Scholar [34] A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302.   Google Scholar [35] B. Weiss, Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350–359.  Google Scholar
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