A survey of algebraic actions of the discrete Heisenberg group

D. Lind   ; K. Schmidt

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 70 (2015) no. 4, pp. 657-714

Résumé

The study of actions of countable groups by automorphisms of compact Abelian groups has recently undergone intensive development, revealing deep connections with operator algebras and other areas. The discrete Heisenberg group is the simplest non-commutative example, where dynamical phenomena related to its non-commutativity already illustrate many of these connections. The explicit structure of this group means that these phenomena have concrete descriptions, which are not only instances of the general theory but are also testing grounds for further work. This paper surveys what is known about such actions of the discrete Heisenberg group, providing numerous examples and emphasizing many of the open problems that remain. Bibliography: 71 titles.

Détail

Keywords: Heisenberg group, expansiveness, entropy.
Mots-clés : algebraic action

Bibliographie

[1] R. L. Adler, A. G. Konheim, M. H. McAndrew, “Topological entropy”, Trans. Amer. Math. Soc., 114:2 (1965), 309–319 | DOI | MR | Zbl

[2] G. R. Allan, “Ideals of vector-valued functions”, Proc. London Math. Soc. (3), 18:2 (1968), 193–216 | DOI | MR | Zbl

[3] J. Anderson, W. Paschke, “The rotation algebra”, Houston J. Math., 15:1 (1989), 1–26 | MR | Zbl

[4] G. Atkinson, “A class of transitive cylinder transformations”, J. London Math. Soc. (2), 17:2 (1978), 263–270 | DOI | MR | Zbl

[5] C. Béguin, A. Valette, A. Zuk, “On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator”, J. Geom. Phys., 21:4 (1997), 337–356 | DOI | MR | Zbl

[6] A. S. Besicovitch, “A problem on topological transformations of the plane. II”, Proc. Cambridge Philos. Soc., 47 (1951), 38–45 | DOI | MR | Zbl

[7] L. Bowen, “Entropy for expansive algebraic actions of residually finite groups”, Ergodic Theory Dynam. Systems, 31:3 (2011), 703–718 | DOI | MR | Zbl

[8] R. Bowen, “Entropy for group endomorphisms and homogeneous spaces”, Trans. Amer. Math. Soc., 153 (1971), 401–414 | DOI | MR | Zbl

[9] D. W. Boyd, “Kronecker's theorem and Lehmer's problem for polynomials in several variables”, J. Number Theory, 13:1 (1981), 116–121 | DOI | MR | Zbl

[10] D. W. Boyd, “Uniform approximation to Mahler's measure in several variables”, Canad. Math. Bull., 41:1 (1998), 125–128 | DOI | MR | Zbl

[11] N.-P. Chung, H. Li, “Homoclinic groups, IE groups, and expansive algebraic actions”, Invent. Math., 199:3 (2015), 805–858 ; (2011 (v3 – 2014)), 49 pp., arXiv: 1103.1567 | DOI | MR | Zbl

[12] P. J. Cohen, “A note on constructive methods in Banach algebras”, Proc. Amer. Math. Soc., 12 (1961), 159–163 | DOI | MR | Zbl

[13] J. B. Conway, A course in functional analysis, Grad. Texts in Math., 96, 2nd ed., Springer-Verlag, New York, 1990, xvi+399 pp. | MR | Zbl

[14] P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, IL, 2000, vi+310 pp. | MR | Zbl

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

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

[17] Ch. 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

[18] 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

[19] 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, Tr. MIAN, 216, Nauka, M., 1997, 265–284 | MR | Zbl

[20] K. Fra̧czek, M. Lemańczyk, “On symmetric logarithm and some old examples in smooth ergodic theory”, Fund. Math., 180:3 (2003), 241–255 | DOI | MR | Zbl

[21] B. Fuglede, R. V. Kadison, “Determinant theory in finite factors”, Ann. of Math. (2), 55:3 (1952), 520–530 | DOI | MR | Zbl

[22] H. Furstenberg, “Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation”, Math. Systems Theory, 1 (1967), 1–49 | DOI | MR | Zbl

[23] A. O. Gelfond, Transcendental and algebraic numbers, Dover Publications, Inc., New York, 1960, vii+190 pp. | MR | MR | Zbl | Zbl

[24] 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

[25] M. Göll, K. Schmidt, E. Verbitskiy, A Wiener lemma for the discrete Heisenberg group: invertibility criteria and applications to algebraic dynamics, preprint

[26] M. Göll, E. Verbitskiy, Homoclinic points of principal algebraic actions, preprint

[27] W. H. Gottschalk, G. A. Hedlund, Topological dynamics, Amer. Math. Soc. Colloq. Publ., 36, Amer. Math. Soc., Providence, RI, 1955, vii+151 pp. | MR | Zbl

[28] D. Gretete, “Random walk on a discrete Heisenberg group”, Rend. Circ. Mat. Palermo (2), 60:3 (2011), 329–335 | DOI | MR | Zbl

[29] P. R. Halmos, “On automorphisms of compact groups”, Bull. Amer. Math. Soc., 49 (1943), 619–624 | DOI | MR | Zbl

[30] B. Hayes, Ergodicity of nilpotent group actions, Gauss's lemma and mixing in the Heisenberg group, Senior thesis, Univ. of Washington, Seattle, 2009, 21 pp.

[31] B. Hayes, Mixing principal algebraic actions of torsion-free nilpotent groups, preprint

[32] J. W. Jenkins, “An amenable group with a nonsymmetric group algebra”, Bull. Amer. Math. Soc., 75 (1969), 357–360 | DOI | MR | Zbl

[33] I. Kaplansky, “Groups with representations of bounded degree”, Canadian J. Math., 1 (1949), 105–112 | DOI | MR | Zbl

[34] I. Kaplansky, Fields and rings, Univ. of Chicago Press, Chicago, IL–London, 1969, ix+198 pp. | MR | Zbl

[35] 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

[36] B. Kitchens, K. Schmidt, “Automorphisms of compact groups”, Ergodic Theory Dynam. Systems, 9:4 (1989), 691–735 | DOI | MR | Zbl

[37] F. Ledrappier, “Un champ markovien peut être d'entropie nulle et mélangeant”, C. R. Acad. Sci. Paris Sér. A-B, 287:7 (1978), A561–A563 | MR | Zbl

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

[39] H. Li, J. Peterson, K. Schmidt, “Ergodicity of principal algebraic group actions”, Recent trends in ergodic theory and dynamical systems, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015, 201–210 | DOI

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

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

[42] D. Lind, K. Schmidt, E. Verbitskiy, “Entropy and growth rate of periodic points of algebraic $\mathbb{Z}^d$-actions”, Dynamical numbers – interplay between dynamical systems and number theory, Contemp. Math., 532, Amer. Math. Soc., Providence, RI, 2010, 195–211 | DOI | MR | Zbl

[43] 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

[44] D. Lind, K. Schmidt, T. Ward, “Mahler measure and entropy for commuting automorphisms of compact groups”, Invent. Math., 101:3 (1990), 593–629 | DOI | MR | Zbl

[45] D. A. Lind, T. Ward, “Automorphisms of solenoids and $p$-adic entropy”, Ergodic Theory Dynam. Systems, 8:3 (1988), 411–419 | DOI | MR | Zbl

[46] E. Lindenstrauss, B. Weiss, “Mean topological dimension”, Israel J. Math., 115 (2000), 1–24 | DOI | MR | Zbl

[47] W. Lück, $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergeb. Math. Grenzgeb. (3), 44, Springer-Verlag, Berlin, 2002, xvi+595 pp. | DOI | MR | Zbl

[48] K. W. MacKenzie, “Prime ideals in skew Laurent polynomial rings”, Proc. Edinburgh Math. Soc. (2), 36:2 (1993), 299–317 | DOI | MR | Zbl

[49] K. Mahler, “An application of Jensen's formula to polynomials”, Mathematika, 7:2 (1960), 98–100 | DOI | MR | Zbl

[50] K. Mahler, “On some inequalities for polynomials in several variables”, J. London Math. Soc., 37:1 (1962), 341–344 | DOI | MR | Zbl

[51] M. S. Montgomery, “Left and right inverses in group algebras”, Bull. Amer. Math. Soc., 75:3 (1969), 539–540 | DOI | MR | Zbl

[52] J. Moulin Ollagnier, Ergodic theory and statistical mechanics, Lecture Notes in Math., 1115, Springer-Verlag, Berlin, 1985, vi+147 pp. | MR | Zbl

[53] D. S. Ornstein, B. Weiss, “Entropy and isomorphism theorems for actions of amenable groups”, J. Analyse Math., 48 (1987), 1–141 | DOI | MR | Zbl

[54] V. I. Oseledets, “A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems”, Trans. Moscow Math. Soc., 19 (1968), 197–231 | MR | Zbl

[55] D. S. Passman, “Idempotents in group rings”, Proc. Amer. Math. Soc., 28:2 (1971), 371–374 | DOI | MR | Zbl

[56] D. S. Passman, The algebraic structure of group rings, Pure Appl. Math., Wiley-Interscience [John Wiley Sons], New York–London–Sydney, 1977, xiv+720 pp. | MR | Zbl

[57] K. Purbhoo, “A Nullstellensatz for amoebas”, Duke Math. J., 141:3 (2008), 407–445 | DOI | MR | Zbl

[58] D. J. Rudolph, K. Schmidt, “Almost block independence and Bernoullicity of $\mathbb{Z}^d$-actions by automorphisms of compact abelian groups”, Invent. Math., 120:3 (1995), 455–488 | DOI | MR | Zbl

[59] K. Schmidt, “Automorphisms of compact abelian groups and affine varieties”, Proc. London Math. Soc. (3), 61:3 (1990), 480–496 | DOI | MR | Zbl

[60] K. Schmidt, Dynamical systems of algebraic origin, Progr. Math., 128, Birkhäuser Verlag, Basel, 1995, xviii+310 pp. | MR | Zbl

[61] K. Schmidt, “Algebraic coding of expansive group automorphisms and two-sided beta-shifts”, Monatsh. Math., 129:1 (2000), 37–61 | DOI | MR | Zbl

[62] N. A. Sidorov, “Bijective and general arithmetic codings for Pisot toral automorphisms”, J. Dynam. Control Systems, 7:4 (2001), 447–472 | DOI | MR | Zbl

[63] N. Sidorov, A. Vershik, “Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus, and binary quadratic forms”, J. Dynam. Control Systems, 4:3 (1998), 365–399 | DOI | MR | Zbl

[64] C. J. Smyth, “A Kronecker-type theorem for complex polynomials in several variables”, Canad. Math. Bull., 24:4 (1981), 447–452 | DOI | MR | Zbl

[65] C. J. Smyth, “On measures of polynomials in several variables”, Bull. Austral. Math. Soc., 23:1 (1981), 49–63 | DOI | MR | Zbl

[66] R. K. Thomas, “The addition theorem for the entropy of transformations of $G$-spaces”, Trans. Amer. Math. Soc., 160 (1971), 119–130 | DOI | MR | Zbl

[67] A. M. Vershik, “Arithmetic isomorphism of hyperbolic toral automorphisms and sofic shifts”, Funct. Anal. Appl., 26:3 (1992), 170–173 | DOI | MR | Zbl

[68] T. B. Ward, Q. Zhang, “The Abramov–Rokhlin entropy addition formula for amenable group actions”, Monatsh. Math., 114:3-4 (1992), 317–329 | DOI | MR | Zbl

[69] B. Weiss, “Monotileable amenable groups”, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, 2001, 257–262 | MR | Zbl

[70] S. A. Yuzvinskii, “Metric properties of endomorphisms of compact groups”, Amer. Math. Soc. Transl. Ser. 2, 66, Amer. Math. Soc., Providence, RI, 1968, 63–98 | MR | Zbl

[71] S. A. Yuzvinskii, “Computing the entropy of a group of endomorphisms”, Siberian Math. J., 8 (1967), 172–178 | DOI | MR | Zbl