Abstract
We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of \(p\)-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is \(1\)-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on \(X\), it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in \(X\) exactly when the action has completely positive entropy.
Similar content being viewed by others
References
Anderson, F.W., Fuller, K.R.: Rings and categories of modules. Graduate Texts in Mathematics, 2nd edn. Springer, New York (1992)
Arveson, W.: An invitation to \(C^*\)-algebras. Graduate Texts in Mathematics. Springer, New York (1976)
Berg, K.R.: Convolution of invariant measures, maximal entropy. Math. Syst. Theory 3, 146–150 (1969)
Bergelson, V., Gorodnik, A.: Ergodicity and mixing of non-commuting epimorphisms. Proc. Lond. Math. Soc. (3) 95(2), 329–359 (2007)
Bhattacharya, S.: Expansiveness of algebraic actions on connected groups. Trans. Am. Math. Soc. 356(12), 4687–4700 (2004)
Blanchard, F.: A disjointness theorem involving topological entropy. Bull. Soc. Math. Fr. 121(4), 465–478 (1993)
Blanchard, F., Glasner, E., Host, B.: A variation on the variational principle and applications to entropy pairs. Ergod. Theory Dyn. Syst. 17(1), 29–43 (1997)
Blanchard, F., Glasner, E., Kolyada, S., Maass, A.: On Li–Yorke pairs. J. Reine Angew. Math. 547, 51–68 (2002)
Blanchard, F., Host, B., Maass, A., Martinez, S., Rudolph, D.J.: Entropy pairs for a measure. Ergod. Theory Dyn. Syst. 15(4), 621–632 (1995)
Blanchard, F., Host, B., Ruette, S.: Asymptotic pairs in positive-entropy systems. Ergod. Theory Dyn. Syst. 22(3), 671–686 (2002)
Bowen, L.: Entropy for expansive algebraic actions of residually finite groups. Ergod. Theory Dyn. Syst. 31(3), 703–718 (2011)
Bryant, B.F.: On expansive homeomorphisms. Pac. J. Math. 10, 1163–1167 (1960)
Chou, C.: Elementary amenable groups. Ill. J. Math. 24(3), 396–407 (1980)
Danilenko, A.I.: Entropy theory from the orbital point of view. Mon. Math. 134(2), 121–141 (2001)
Deninger, C.: Fuglede–Kadison determinants and entropy for actions of discrete amenable groups. J. Am. Math. Soc. 19, 737–758 (2006)
Deninger, C.: Mahler measures and Fuglede–Kadison determinants. Münster J. Math. 2, 45–63 (2009)
Deninger, C., Schmidt, K.: Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Ergod. Theory Dyn. Syst. 27, 769–786 (2007)
Einsiedler, M., Rindler, H.: Algebraic actions of the discrete Heisenberg group and other non-abelian groups. Aequ. Math. 62(1–2), 117–135 (2001)
Einsiedler, M., Schmidt, K.: The adjoint action of an expansive algebraic \({\mathbb{Z}}^d\)-action. Mon. Math. 135(3), 203–220 (2002)
Einsiedler, M., Schmidt, K.: Irreducibility, homoclinic points and adjoint actions of algebraic Z\(^{d}\)-actions of rank one. In: Dynamics and Randomness (Santiago, 2000), pp. 95–124. Kluwer Academic Publications, Dordrecht (2002)
Einsiedler, M., Ward, T.: Entropy geometry and disjointness for zero-dimensional algebraic actions. J. Reine Angew. Math. 584, 195–214 (2005)
Elek, G.: On the analytic zero divisor conjecture of Linnell. Bull. Lond. Math. Soc. 35(2), 236–238 (2003)
Fuglede, B., Kadison, R.V.: Determinant theory in finite factors. Ann. Math. (2) 55, 520–530 (1952)
Glasner, E.: Ergodic theory via joinings. American Mathematical Society, Providence (2003)
Glasner, E., Thouvenot, J.-P., Weiss, B.: Entropy theory without a past. Ergod. Theory Dyn. Syst. 20(5), 1355–1370 (2000)
Glasner, E., Ye, X.: Local entropy theory. Ergod. Theory Dyn. Syst. 29(2), 321–356 (2009)
Gottschalk, W.H., Hedlund, G.A.: Topological dynamics. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (1955)
Hall, P.: Finiteness conditions for soluble groups. Proc. Lond. Math. Soc. (3) 4, 419–436 (1954)
Herz, C.: The theory of \(p\)-spaces with an application to convolution operators. Trans. Am. Math. Soc. 154, 69–82 (1971)
Herz, C.: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23(3), 91–123 (1973)
Huang, W., Li, S.M., Shao, S., Ye, X.: Null systems and sequence entropy pairs. Ergod. Theory Dyn. Syst. 23(5), 1505–1523 (2003)
Huang, W., Maass, A., Romagnoli, P.P., Ye, X.: Entropy pairs and a local Abramov formula for a measure theoretical entropy of open covers. Ergod. Theory Dyn. Syst. 24(4), 1127–1153 (2004)
Huang, W., Ye, X.: A local variational relation and applications. Isr. J. Math. 151, 237–279 (2006)
Huang, W., Ye, X.: Combinatorial lemmas and applications to dynamics. Adv. Math. 220(6), 1689–1716 (2009)
Huang, W., Ye, X., Zhang, G.: Local entropy theory for a countable discrete amenable group action. J. Funct. Anal. 261(4), 1028–1082 (2011)
Katok, A., Katok, S., Schmidt, K.: Rigidity of measurable structure for \({\mathbb{Z}}^d\)-actions by automorphisms of a torus. Comment. Math. Helv. 77(4), 718–745 (2002)
Katok, A., Spatzier, R.J.: Invariant measures for higher-rank hyperbolic abelian actions. Ergod. Theory Dyn. Syst. 16(4), 751–778 (1996)
Kechris, A.S.: Classical descriptive set theory. Graduate Texts in Mathematics. Springer, New York (1995)
Kerr, D., Li, H.: Independence in topological and \(C^*\)-dynamics. Math. Ann. 338(4), 869–926 (2007)
Kerr, D., Li, H.: Combinatorial independence in measurable dynamics. J. Funct. Anal. 256(5), 1341–1386 (2009)
Kerr, D., Li, H.: Entropy and the variational principle for actions of sofic groups. Invent. Math. 186(3), 501–558 (2011)
Kitchens, B., Schmidt, K.: Isomorphism rigidity of irreducible algebraic \({\mathbb{Z}}^d\)-actions. Invent. Math. 142(3), 559–577 (2000)
Lam, T.Y.: Lectures on modules and rings. Graduate Texts in Mathematics. Springer, New York (1999)
Lang, S.: Complex analysis. Graduate Texts in Mathematics, 4th edn. Springer, New York (1999)
Lang, S.: Algebra. Revised third edition. Graduate Texts in Mathematics. Springer, New York (2002)
Ledrappier, F.: Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A B 287(7), A561–A563 (1978)
Li, H.: Compact group automorphisms, addition formulas and Fuglede–Kadison determinants. Ann. Math. (2) 176(1), 303–347 (2012)
Lind, D.: The structure of skew products with ergodic group automorphisms. Isr. J. Math. 28(3), 205–248 (1977)
Lind, D., Schmidt, K.: Homoclinic points of algebraic \({\mathbb{Z}}^d\)-actions. J. Am. Math. Soc. 12(4), 953–980 (1999)
Lind, D., Schmidt, K.: Symbolic and algebraic dynamical systems. In: Handbook of dynamical systems, vol. 1A, pp. 765–812. North-Holland, Amsterdam (2002)
Lind, D., Schmidt, K., Verbitskiy, E.: Entropy and growth rate of periodic points of algebraic \({\mathbb{Z}}^d\)-actions. In: Dynamical numbers: interplay between dynamical systems and number theory, pp. 195–211, Contemporary Mathematics, vol. 532. American Mathematical Society, Providence, RI (2010)
Lind, D., Schmidt, K., Ward, T.: Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101, 593–629 (1990)
Lindenstrauss, E., Weiss, B.: Mean topological dimension. Isr. J. Math. 115, 1–24 (2000)
Linnell, P.A.: Zero divisors and group von Neumann algebras. Pac. J. Math. 149(2), 349–363 (1991)
Linnell, P.A.: Zero divisors and \(L^2(G)\). C. R. Acad. Sci. Paris Sér. I Math. 315(1), 49–53 (1992)
Linnell, P.A.: Division rings and group von Neumann algebras. Forum Math. 5(6), 561–576 (1993)
Linnell, P.A.: Analytic versions of the zero divisor conjecture. In: Geometry and cohomology in group theory (Durham, 1994), Lecture Notes Series, vol. 252, pp. 209–248. London Mathematical Society, Cambridge University Press, Cambridge (1998)
Linnell, P.A., Puls, M.J.: Zero divisors and \(L^p(G)\), II. N. Y. J. Math. 7, 49–58 (2001). (electronic)
Lück, W.: \(L^2\)-Invariants: theory and applications to geometry and \(K\)-theory. Springer, Berlin (2002)
Miles, G., Thomas, R.K.: Generalized torus automorphisms are Bernoullian. In: Studies in probability and ergodic theory, Advances in Math. Suppl. Stud., vol. 2, pp. 231–249 Academic Press, New York (1978)
Miles, R.: Expansive algebraic actions of countable abelian groups. Mon. Math. 147(2), 155–164 (2006)
Miles, R., Ward, T.: Orbit-counting for nilpotent group shifts. Proc. Am. Math. Soc. 137(4), 1499–1507 (2009)
Moulin Ollagnier, J.: Ergodic theory and statistical mechanics. Lecture Notes in Mathematics. Springer, Berlin (1985)
Ornstein, D.S., Weiss, B.: Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48, 1–141 (1987)
Passman, D.S.: The algebraic structure of group rings. Pure and Applied Mathematics. Wiley, New York (1977)
Peters, J.: Entropy on discrete abelian groups. Adv. Math. 33(1), 1–13 (1979)
Pisier, G.: The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989)
Puls, M.J.: Zero divisors and \(L^p(G)\). Proc. Am. Math. Soc. 126(3), 721–728 (1998)
Rosenthal, H.P.: A characterization of Banach spaces containing \(l_1\). Proc. Natl. Acad. Sci. USA 71, 2411–2413 (1974)
Rudin, W.: Real and complex analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)
Rudolph, D.J., Schmidt, K.: Almost block independence and Bernoullicity of \(Z^d\)-actions by automorphisms of compact abelian groups. Invent. Math. 120(3), 455–488 (1995)
Ruelle, D.: Statistical mechanics on a compact set with \(Z^{v}\) action satisfying expansiveness and specification. Trans. Am. Math. Soc. 187, 237–251 (1973)
Schmidt, K.: Automorphisms of compact abelian groups and affine varieties. Proc. Lond. Math. Soc. (3) 61(3), 480–496 (1990)
Schmidt, K.: Dynamical systems of algebraic origin. Progress in Mathematics. Birkhäuser, Basel (1995)
Schmidt, K.: The cohomology of higher-dimensional shifts of finite type. Pac. J. Math. 170(1), 237–269 (1995)
Schmidt, K.: The dynamics of algebraic \({\mathbb{Z}}^d\)-actions. In: European Congress of Mathematics, vol. I (Barcelona, 2000), pp. 543–553, Progr. Math., 201, Birkhäuser, Basel (2001)
Schmidt, K., Verbitskiy, E.: Abelian sandpiles and the harmonic model. Commun. Math. Phys. 292(3), 721–759 (2009)
Schmidt, K., Ward, T.: Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent. Math. 111(1), 69–76 (1993)
Segal, D.: Polycyclic groups. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1983)
Walters, A.: An introduction to ergodic theory. Graduate Texts in Mathematics. Springer, New York (1982)
Wang, X.: Volumes of generalized unit balls. Math. Mag. 78(5), 390–395 (2005)
Yuzvinskiĭ, S.A.: Metric properties of the endomorphisms of compact groups. (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 29, 1295–1328 (1965) [Translated in Amer. Math. Soc. Transl. (2) 66, 63–98 (1968)]
Yuzvinskiĭ, S.A.: Computing the entropy of a group of endomorphisms. (Russian) Sibirsk. Mat. Ẑ. 8, 230–239 (1967) [Translated in Siberian Math. J. 8, 172–178 (1967)]
Acknowledgments
The second named author was partially supported by NSF grants DMS-0701414 and DMS-1001625. He is grateful to Doug Lind and Klaus Schmidt for very interesting discussions. We thank David Kerr for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chung, NP., Li, H. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. math. 199, 805–858 (2015). https://doi.org/10.1007/s00222-014-0524-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-014-0524-1