Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T03:04:02.641Z Has data issue: false hasContentIssue false
  • English
  • Français

Surjunctivity and topological rigidity of algebraic dynamical systems

Published online by Cambridge University Press:  20 June 2017

SIDDHARTHA BHATTACHARYA ,
TULLIO CECCHERINI-SILBERSTEIN  and
MICHEL COORNAERT
SIDDHARTHA BHATTACHARYA
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India email [email protected]
TULLIO CECCHERINI-SILBERSTEIN
Affiliation:
Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 107, 82100 Benevento, Italy email [email protected]
MICHEL COORNAERT
Affiliation:
Université de Strasbourg, CNRS, IRMA UMR 7501, F-67000 Strasbourg, France email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $X$ be a compact metrizable group and let $\unicode[STIX]{x1D6E4}$ be a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\unicode[STIX]{x1D6E4})$ is surjunctive, i.e. every injective continuous map $\unicode[STIX]{x1D70F}:X\rightarrow X$ commuting with the action of $\unicode[STIX]{x1D6E4}$ is surjective.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 604 - 619
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bhattacharya, S.. Orbit equivalence and topological conjugacy of affine actions on compact abelian groups. Monatsh. Math. 129 (2000), 89–96.Google Scholar
Bhattacharya, S. and Ward, T.. Finite entropy characterizes topological rigidity on connected groups. Ergod. Th. & Dynam. Sys. 25 (2005), 365–373.Google Scholar
Ceccherini-Silberstein, T. and Coornaert, M.. Surjunctivity and reversibility of cellular automata over concrete categories. Trends in Harmonic Analysis (Springer INdAM Series, 3) . Springer, Milan, 2013, pp. 91–133.Google Scholar
Ceccherini-Silberstein, T. and Coornaert, M.. Expansive actions of countable amenable groups, homoclinic pairs, and the Myhill property. Illinois J. Math. 59 (2015), 597–621.Google Scholar
Chung, N.-P. and Li, H.. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. Math. 199 (2015), 805–858.Google Scholar
Deninger, C.. Fuglede–Kadison determinants and entropy for actions of discrete amenable groups. J. Amer. Math. Soc. 19 (2006), 737–758.Google Scholar
Deninger, C. and Schmidt, K.. Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Ergod. Th. & Dynam. Sys. 27 (2007), 769–786.Google Scholar
Einsiedler, M. and Rindler, H.. Algebraic actions of the discrete Heisenberg group and other non-abelian groups. Aequationes Math. 62 (2001), 117–135.Google Scholar
Godement, R.. Integration and spectral theory, harmonic analysis, the garden of modular delights. Analysis IV (Universitext) . Springer, Cham, 2015, translated from the French edition by Urmie Ray.Google Scholar
Gottschalk, W.. Some general dynamical notions. Recent Advances in Topological Dynamics (Proc. Conf. Topological Dynamics, Yale Univiversity, New Haven, CN, 1972; in honor of Gustav Arnold Hedlund) (Lecture Notes in Mathematics, 318) . Springer, Berlin, 1973, pp. 120–125.Google Scholar
Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197.Google Scholar
Hall, P.. Finiteness conditions for soluble groups. Proc. Lond. Math. Soc. (3) 4 (1954), 419–436.Google Scholar
Halmos, P. R.. On automorphisms of compact groups. Bull. Amer. Math. Soc. 49 (1943), 619–624.Google Scholar
Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9 (1989), 691–735.Google Scholar
Lam, P.-F.. On expansive transformation groups. Trans. Amer. Math. Soc. 150 (1970), 131–138.Google Scholar
Lind, D. and Schmidt, K.. Symbolic and algebraic dynamical systems. Handbook of Dynamical Systems. 1A. North-Holland, Amsterdam, 2002, pp. 765–812.Google Scholar
Lind, D. and Shmidt, K.. A survey of algebraic actions of the discrete Heisenberg group. Uspekhi Mat. Nauk 70 (2015), 77–142.Google Scholar
Linnell, P. A.. Zero divisors and group von Neumann algebras. Pacific J. Math. 149 (1991), 349–363.Google Scholar
Linnell, P. A.. Analytic versions of the zero divisor conjecture. Geometry and Cohomology in Group Theory (Durham, 1994) (London Mathematical Society Lecture Note Series, 252) . Cambridge University Press, Cambridge, 1998, pp. 209–248.Google Scholar
Matsumura, H.. Commutative Ring Theory (Cambridge Studies in Advanced Mathematics, 8) . 2nd edn. Cambridge University Press, Cambridge, 1989, translated from the Japanese by M. Reid.Google Scholar
Morris, S. A.. Pontryagin Duality and the Structure of Locally Compact Abelian Groups (London Mathematical Society Lecture Note Series, 29) . Cambridge University Press, Cambridge, 1977.Google Scholar
Passman, D. S.. The Algebraic Structure of Group Rings (Pure and Applied Mathematics) . Wiley-Interscience [John Wiley & Sons], New York, 1977.Google Scholar
Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128) . Birkhäuser, Basel, 1995.Google Scholar
van Kampen, E.. On almost periodic functions of constant absolute value. J. Lond. Math. Soc. 12 (1937), 3–6.Google Scholar
Vasconcelos, W. V.. On finitely generated flat modules. Trans. Amer. Math. Soc. 138 (1969), 505–512.Google Scholar
Walters, P.. Topological conjugacy of affine transformations of compact abelian groups. Trans. Amer. Math. Soc. 140 (1969), 95–107.Google Scholar
Weil, A.. L’intégration dans les groupes topologiques et ses applications (Actual Sci. Ind., 869) . Hermann et Cie, Paris, 1940 [this book has been republished by the author at Princeton, NJ, 1941].Google Scholar
Weiss, B.. Sofic groups and dynamical systems. Sankhyā Ser. A 62 (2000), 350–359. Ergodic Theory and Harmonic Analysis (Mumbai, 1999).Google Scholar