Skip to main content Accessibility help
×
Home

Surjunctivity and topological rigidity of algebraic dynamical systems

  • SIDDHARTHA BHATTACHARYA (a1), TULLIO CECCHERINI-SILBERSTEIN (a2) and MICHEL COORNAERT (a3)

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.

Copyright

References

Hide All
[1] Bhattacharya, S.. Orbit equivalence and topological conjugacy of affine actions on compact abelian groups. Monatsh. Math. 129 (2000), 8996.
[2] Bhattacharya, S. and Ward, T.. Finite entropy characterizes topological rigidity on connected groups. Ergod. Th. & Dynam. Sys. 25 (2005), 365373.
[3] 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. 91133.
[4] Ceccherini-Silberstein, T. and Coornaert, M.. Expansive actions of countable amenable groups, homoclinic pairs, and the Myhill property. Illinois J. Math. 59 (2015), 597621.
[5] Chung, N.-P. and Li, H.. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. Math. 199 (2015), 805858.
[6] Deninger, C.. Fuglede–Kadison determinants and entropy for actions of discrete amenable groups. J. Amer. Math. Soc. 19 (2006), 737758.
[7] Deninger, C. and Schmidt, K.. Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Ergod. Th. & Dynam. Sys. 27 (2007), 769786.
[8] Einsiedler, M. and Rindler, H.. Algebraic actions of the discrete Heisenberg group and other non-abelian groups. Aequationes Math. 62 (2001), 117135.
[9] 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.
[10] 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. 120125.
[11] Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1 (1999), 109197.
[12] Hall, P.. Finiteness conditions for soluble groups. Proc. Lond. Math. Soc. (3) 4 (1954), 419436.
[13] Halmos, P. R.. On automorphisms of compact groups. Bull. Amer. Math. Soc. 49 (1943), 619624.
[14] Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9 (1989), 691735.
[15] Lam, P.-F.. On expansive transformation groups. Trans. Amer. Math. Soc. 150 (1970), 131138.
[16] Lind, D. and Schmidt, K.. Symbolic and algebraic dynamical systems. Handbook of Dynamical Systems. 1A. North-Holland, Amsterdam, 2002, pp. 765812.
[17] Lind, D. and Shmidt, K.. A survey of algebraic actions of the discrete Heisenberg group. Uspekhi Mat. Nauk 70 (2015), 77142.
[18] Linnell, P. A.. Zero divisors and group von Neumann algebras. Pacific J. Math. 149 (1991), 349363.
[19] 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. 209248.
[20] 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.
[21] 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.
[22] Passman, D. S.. The Algebraic Structure of Group Rings (Pure and Applied Mathematics) . Wiley-Interscience [John Wiley & Sons], New York, 1977.
[23] Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128) . Birkhäuser, Basel, 1995.
[24] van Kampen, E.. On almost periodic functions of constant absolute value. J. Lond. Math. Soc. 12 (1937), 36.
[25] Vasconcelos, W. V.. On finitely generated flat modules. Trans. Amer. Math. Soc. 138 (1969), 505512.
[26] Walters, P.. Topological conjugacy of affine transformations of compact abelian groups. Trans. Amer. Math. Soc. 140 (1969), 95107.
[27] 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].
[28] Weiss, B.. Sofic groups and dynamical systems. Sankhyā Ser. A 62 (2000), 350359. Ergodic Theory and Harmonic Analysis (Mumbai, 1999).

Surjunctivity and topological rigidity of algebraic dynamical systems

  • SIDDHARTHA BHATTACHARYA (a1), TULLIO CECCHERINI-SILBERSTEIN (a2) and MICHEL COORNAERT (a3)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed