Predictability, topological entropy, and invariant random orders

Alpeev, Andrei; Meyerovitch, Tom; Ryu, Sieye

doi:10.1090/proc/15158

Predictability, topological entropy, and invariant random orders
HTML articles powered by AMS MathViewer

by Andrei Alpeev, Tom Meyerovitch and Sieye Ryu PDF

Proc. Amer. Math. Soc. 149 (2021), 1443-1457 Request permission

Abstract:

We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. We investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.

References

A. Alpeev, A random ordering formula for sofic and Rokhlin entropy of Gibbs measures, preprint, arXiv:1705.08559, 2017.
A. Alpeev and B. Seward, Krieger’s finite generator theorem for actions of countable groups III, arXiv:1705.09707, (2017).
Tim Austin and Moumanti Podder, Gibbs measures over locally tree-like graphs and percolative entropy over infinite regular trees, J. Stat. Phys. 170 (2018), no. 5, 932–951. MR 3767001, DOI 10.1007/s10955-018-1959-3
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 P. Bowen, A brief introduction of sofic entropy theory, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1847–1866. MR 3966832
L. Bowen, Examples in the entropy theory of countable group actions, arXiv:1704.06349, (2017).
Andrés del Junco, On minimal self-joinings in topological dynamics, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 211–227. MR 896794, DOI 10.1017/S0143385700003965
B. Deroin, A. Navas, and C. Rivas, Groups, orders, and dynamics, 2014, arXiv:1408.5805.
Edward Formanek, Extending partial right orders on nilpotent groups, J. London Math. Soc. (2) 7 (1973), 131–134. MR 327605, DOI 10.1112/jlms/s2-7.1.131
Eli Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, Providence, RI, 2003. MR 1958753, DOI 10.1090/surv/101
E. Glasner and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 597–648. MR 2186250, DOI 10.1016/S1874-575X(06)80035-4
B. Hayes, Relative entropy and the Pinsker product formula for sofic groups, arXiv:1605.01747, (2016).
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
Michael Hochman, On notions of determinism in topological dynamics, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 119–140. MR 2873161, DOI 10.1017/S0143385710000738
Wen Huang, Lei Jin, and Xiangdong Ye, On extensions over semigroups and applications, Entropy 18 (2016), no. 6, Paper No. 230, 6. MR 3530053, DOI 10.3390/e18060230
Wen Huang, Leiye Xu, and Yingfei Yi, Asymptotic pairs, stable sets and chaos in positive entropy systems, J. Funct. Anal. 268 (2015), no. 4, 824–846. MR 3296581, DOI 10.1016/j.jfa.2014.10.025
Brunon Kamiński, Artur Siemaszko, and Jerzy Szymański, The determinism and the Kolmogorov property in topological dynamics, Bull. Polish Acad. Sci. Math. 51 (2003), no. 4, 401–417. MR 2025310
Brunon Kamiński, Artur Siemaszko, and Jerzy Szymański, Extreme relations for topological flows, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 1, 17–24. MR 2162751, DOI 10.4064/ba53-1-3
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, Ergodic theory, Springer Monographs in Mathematics, Springer, Cham, 2016. Independence and dichotomies. MR 3616077, DOI 10.1007/978-3-319-49847-8
J. C. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Probability 3 (1975), no. 6, 1031–1037. MR 393422, DOI 10.1214/aop/1176996230
Jonathan L. King, A map with topological minimal self-joinings in the sense of del Junco, Ergodic Theory Dynam. Systems 10 (1990), no. 4, 745–761. MR 1091424, DOI 10.1017/S0143385700005873
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 0103254
Peter Linnell and Dave Witte Morris, Amenable groups with a locally invariant order are locally indicable, Groups Geom. Dyn. 8 (2014), no. 2, 467–478. MR 3231224, DOI 10.4171/GGD/234
Dave Witte Morris, Amenable groups that act on the line, Algebr. Geom. Topol. 6 (2006), 2509–2518. MR 2286034, DOI 10.2140/agt.2006.6.2509
Jean Moulin Ollagnier and Didier Pinchon, Groupes pavables et principe variationnel, Z. Wahrsch. Verw. Gebiete 48 (1979), no. 1, 71–79 (French). MR 533007, DOI 10.1007/BF00534883
Jean Moulin Ollagnier and Didier Pinchon, The variational principle, Studia Math. 72 (1982), no. 2, 151–159. MR 665415, DOI 10.4064/sm-72-2-151-159
Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
A. H. Rhemtulla, Right-ordered groups, Canadian J. Math. 24 (1972), 891–895. MR 311538, DOI 10.4153/CJM-1972-088-x
Brandon Seward, Krieger’s finite generator theorem for actions of countable groups II, J. Mod. Dyn. 15 (2019), 1–39. MR 3959054, DOI 10.3934/jmd.2019012
B. Seward, Weak containment and Rokhlin entropy, arXiv:1602.06680, (2016).
Brandon Seward, Positive entropy actions of countable groups factor onto Bernoulli shifts, J. Amer. Math. Soc. 33 (2020), no. 1, 57–101. MR 4066472, DOI 10.1090/jams/931
M. Shub and B. Weiss, Can one always lower topological entropy?, Ergodic Theory Dynam. Systems 11 (1991), no. 3, 535–546. MR 1125888, DOI 10.1017/S0143385700006325
A. M. Stepin and A. T. Tagi-Zade, Variational characterization of topological pressure of the amenable groups of transformations, Dokl. Akad. Nauk SSSR 254 (1980), no. 3, 545–549 (Russian). MR 590147
Thomas Ward and Qing Zhang, The Abramov-Rokhlin entropy addition formula for amenable group actions, Monatsh. Math. 114 (1992), no. 3-4, 317–329. MR 1203977, DOI 10.1007/BF01299386
B. Weiss, Multiple recurrence and doubly minimal systems, Topological dynamics and applications (Minneapolis, MN, 1995) Contemp. Math., vol. 215, Amer. Math. Soc., Providence, RI, 1998, pp. 189–196. MR 1603185, DOI 10.1090/conm/215/02940