Intrinsic ergodicity, generators, and symbolic representations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 58 (2024) no. 1, pp. 50-83.

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct natural symbolic representations of intrinsically ergodic, but not necessarily expansive, principal algebraic actions of countably infinite amenable groups and use these representations to find explicit generating partitions (up to null-sets) for such actions.
Mots-clés : principal algebraic actions, summable homoclinic points.
Keywords: symbolic representations, generating partitions, intrinsic ergodicity
@article{FAA_2024_58_1_a4,
     author = {Hanfeng Li and Klaus Schmidt},
     title = {Intrinsic ergodicity, generators, and symbolic representations},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {50--83},
     publisher = {mathdoc},
     volume = {58},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2024_58_1_a4/}
}
TY  - JOUR
AU  - Hanfeng Li
AU  - Klaus Schmidt
TI  - Intrinsic ergodicity, generators, and symbolic representations
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2024
SP  - 50
EP  - 83
VL  - 58
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2024_58_1_a4/
LA  - ru
ID  - FAA_2024_58_1_a4
ER  - 
%0 Journal Article
%A Hanfeng Li
%A Klaus Schmidt
%T Intrinsic ergodicity, generators, and symbolic representations
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2024
%P 50-83
%V 58
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2024_58_1_a4/
%G ru
%F FAA_2024_58_1_a4
Hanfeng Li; Klaus Schmidt. Intrinsic ergodicity, generators, and symbolic representations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 58 (2024) no. 1, pp. 50-83. http://geodesic.mathdoc.fr/item/FAA_2024_58_1_a4/

[1] L. Bowen, H. Li, “Harmonic models and spanning forests of residually finite groups”, J. Funct. Anal., 263:7 (2012), 1769–1808 | DOI | MR | Zbl

[2] R. Bowen, “Markov partitions for Axiom ${\rm A}$ diffeomorphisms”, Amer. J. Math., 92 (1970), 725–747 | DOI | MR | Zbl

[3] C.-S. Tullio, C. Miel, H. Li, Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity, 2021 http://math.buffalo.edu/~hfli/surjunctive-first.pdf

[4] N.-P. Chung, H. Li, “Homoclinic groups, IE groups, and expansive algebraic actions”, Invent. Math., 199:3 (2015), 805–858 | DOI | MR | Zbl

[5] A. I. Danilenko, “Entropy theory from the orbital point of view”, Monatsh. Math., 134:2 (2001), 121–141 | DOI | MR | Zbl

[6] C. Deninger, “Fuglede–Kadison determinants and entropy for actions of discrete amenable groups”, J. Amer. Math. Soc., 19:3 (2006), 737–758 | DOI | MR | Zbl

[7] C. Deninger, “Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents”, J. Reine Angew. Math., 651 (2011), 165–185 | MR | Zbl

[8] C. Deninger, K. Schmidt, “Expansive algebraic actions of discrete residually finite amenable groups and their entropy”, Ergodic Theory Dynam. Systems, 27:3 (2007), 769–786 | DOI | MR | Zbl

[9] M. Einsiedler, H. Rindler, “Algebraic actions of the discrete Heisenberg group and other non-abelian groups”, Aequationes Math., 62:1–2 (2001), 117–135 | DOI | MR | Zbl

[10] M. Einsiedler, K. Schmidt, “Markov partitions and homoclinic points of algebraic $\mathbb Z^d$-actions”, Dinamicheskie sistemy i smezhnye voprosy, Sbornik statei. K 60-letiyu so dnya rozhdeniya akademika Dmitriya Viktorovicha Anosova, Trudy MIAN, 216, Nauka, M., 1997, 265–284 | MR

[11] A. Furman, “Random walks on groups and random transformations”, Handbook of dynamical systems, v. 1A, North-Holland, Amsterdam, 2002, 931–1014 | MR | Zbl

[12] M. Göll, Principal algebraic actions of the discrete Heisenberg group, PhD Thesis, University of Leiden, 2015, 167 pp. | Zbl

[13] M. Göll, K. Schmidt, E. Verbitskiy, “Algebraic actions of the discrete Heisenberg group: expansiveness and homoclinic points”, Indag. Math. (N.S.), 25:4 (2014), 713–744 | DOI | MR | Zbl

[14] B. Hayes, “Fuglede–Kadison determinants and sofic entropy”, Geom. Funct. Anal., 26:2 (2016), 520–606 | DOI | MR | Zbl

[15] R. Kenyon, A. Vershik, “Arithmetic construction of sofic partitions of hyperbolic toral automorphisms”, Ergodic Theory Dynam. Systems, 18:2 (1998), 357–372 | DOI | MR | Zbl

[16] D. Kerr, H. Li, Ergodic theory: Independence and Dichotomies, Springer Monographs in Mathematics, Springer, Cham, 2016 | DOI | MR | Zbl

[17] S. Le Borgne, “Un codage sofique des automorphismes hyperboliques du tore”, C. R. Acad. Sci. Paris Sér. I Math., 323:10 (1996), 1123–1128 | MR | Zbl

[18] H. Li, “Compact group automorphisms, addition formulas and Fuglede–Kadison determinants”, Ann. of Math. (2), 176:1 (2012), 303–347 | DOI | MR | Zbl

[19] H. Li, A. Thom, “Entropy, determinants, and $L^2$-torsion”, J. Amer. Math. Soc., 27:1 (2014), 239–292 | MR | Zbl

[20] D. A. Lind, “Dynamical properties of quasihyperbolic toral automorphisms”, Ergodic Theory Dynam. Systems, 2:1 (1982), 49–68 | DOI | MR | Zbl

[21] D. Lind, K. Schmidt, “Homoclinic points of algebraic $Z^d$-actions”, J. Amer. Math. Soc., 12:4 (1999), 953–980 | DOI | MR | Zbl

[22] D. Lind, K. Shmidt, “Obzor algebraicheskikh deistvii diskretnoi gruppy Geizenberga”, UMN, 70:4(424) (2015), 77–142 | DOI | MR | Zbl

[23] D. Lind, K. Schmidt, “New examples of Bernoulli algebraic actions”, Ergodic Theory Dynam. Systems, 42:9 (2022), 2923–2934 | DOI | MR | Zbl

[24] D. Lind, K. Schmidt, E. Verbitskiy, “Homoclinic points, atoral polynomials, and periodic points of algebraic $\mathbb{Z}^d$-actions”, Ergodic Theory Dynam. Systems, 33:4 (2013), 1060–1081 | DOI | MR | Zbl

[25] A. Pajor, Sous-espaces $l^n_1$ des espaces de Banach, With an Introduction by Gilles Pisier, Travaux en Cours, 16, Hermann, Paris, 1985 | MR

[26] K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, 3, Academic Press, Inc., New York–London, 1967 | MR

[27] D. S. Passman, The algebraic structure of group rings, Pure Appl. Math., Wiley-Interscience [John Wiley Sons], 1977 | MR | Zbl

[28] N. Sauer, “On the density of families of sets”, J. Combinatorial Theory Ser. A, 13 (1972), 145–147 | DOI | MR | Zbl

[29] K. Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995 | DOI | MR | Zbl

[30] K. Schmidt, “Representations of toral automorphisms”, Topology Appl., 205 (2016), 88–116 | DOI | MR | Zbl

[31] B. Seward, “Krieger's finite generator theorem for actions of countable groups I”, Invent. Math., 215:1 (2019), 265–310 | DOI | MR | Zbl

[32] B. Seward, “Krieger's finite generator theorem for actions of countable groups II”, J. Mod. Dyn., 15 (2019), 1–39 | MR | Zbl

[33] S. Shelah, “A combinatorial problem; stability and order for models and theories in infinitary languages”, Pacific J. Math., 41 (1972), 247–261 | DOI | MR | Zbl

[34] Ya. G. Sinai, “Postroenie markovskikh razbienii”, Funkts. analiz i ego pril., 2:3 (1968), 70–80 | MR | Zbl

[35] N. Th. Varopoulos, “Long range estimates for Markov chains”, Bull. Sci. Math. (2), 109:3 (1985), 225–252 | MR | Zbl

[36] A. M. Vershik, “Arifmeticheskii izomorfizm giperbolicheskikh avtomorfizmov tora i soficheskikh sdvigov”, Funkts. analiz i ego pril., 26:3 (1992), 22–27 | MR | Zbl

[37] B. Weiss, “Intrinsically ergodic systems”, Bull. Amer. Math. Soc., 76 (1970), 1266–1269 | DOI | MR | Zbl

[38] Sauer–Shelah lemma, Wikipedia, The Free Encyclopedia, accessed 21, October 2021

[39] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138, Cambridge University Press, Cambridge, 2000 | MR | Zbl