Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T09:49:38.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Subsemigroups of transitive semigroups

Published online by Cambridge University Press:  24 May 2011

ÉTIENNE MATHERON
ÉTIENNE MATHERON*
Affiliation:
Faculté des Sciences Jean Perrin, Laboratoire de Mathématiques de Lens, Université d’Artois, Rue Jean Souvraz S. P. 18, 62307 LENS, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Γ be a topological semigroup acting on a topological space X, and let Γ0 be a subsemigroup of Γ. We give general conditions ensuring that Γ and Γ0 have the same transitive points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 1043 - 1071
Copyright
Copyright © Cambridge University Press 2011

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

[1]Ansari, S.. Hypercyclic and cyclic vectors. J. Funct. Anal. 128 (1995), 374–383.CrossRefGoogle Scholar
[2]Bayart, F.. Dynamics of holomorphic groups. Semigroup Forum 82 (2011), 229–241.CrossRefGoogle Scholar
[3]Bayart, F., Costakis, G. and Hadjiloucas, D.. Topologically transitive skew-products of operators. Ergod. Th. & Dynam. Sys. 30 (2010), 33–49.CrossRefGoogle Scholar
[4]Bayart, F. and Matheron, É.. Dynamics of Linear Operators (Cambridge Tracts in Mathematics, 179). Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
[5]Bernal-González, L. and Grosse-Erdmann, K.-G.. Existence and non-existence of hypercyclic semigroups. Proc. Amer. Math. Soc. 135 (2007), 755–766.CrossRefGoogle Scholar
[6]Bourbaki, N.. Topologie Générale. Hermann, Paris, 1958.Google Scholar
[7]Conejero, J. A., Müller, V. and Peris, A.. Hypercyclic behaviour of operators in a hypercyclic C 0-semigroup. J. Funct. Anal. 244 (2007), 342–348.CrossRefGoogle Scholar
[8]Costakis, G. and Peris, A.. Hypercyclic semigroups and somewhere dense orbits. C. R. Acad. Sci. Paris 335 (2002), 895–898.Google Scholar
[9]Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477–515.CrossRefGoogle Scholar
[10]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
[11]Glasner, E. and Weiss, B.. On the Interplay Between Measurable and Topological Dynamics (Handbook of Dynamical Systems, 1B). Elsevier B.V., Amsterdam, 2006, pp. 597–648.Google Scholar
[12]Grosse-Erdmann, K.-G. and Peris, A.. Frequently dense orbits. C. R. Acad. Sci. Paris 341 (2005), 123–128.Google Scholar
[13]Grosse-Erdmann, K.-G. and Peris, A.. Linear Chaos (Universitext). Springer, Berlin, to appear.CrossRefGoogle Scholar
[14]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, Vol. 1. Springer, Berlin, 1963.Google Scholar
[15]Keynes, H. B. and Robertson, J. B.. Eigenvalue theorems in topological transformation groups. Trans. Amer. Math. Soc. 139 (1969), 359–369.CrossRefGoogle Scholar
[16]León Saavedra, F. and Müller, V.. Rotations of hypercyclic and supercyclic operators. Integral Equations Operator Theory 50 (2004), 385–391.CrossRefGoogle Scholar
[17]Oxtoby, J. and Ulam, S.. Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42 (1941), 874–920.CrossRefGoogle Scholar
[18]Parry, W.. Compact abelian group extensions of discrete dynamical systems. Probability Theory Related Fields (Z. Wahrscheinlichkeitstheorie verw. GeL) 13 (1969), 95–113.Google Scholar
[19]Petersen, K.. Disjointnness and weak mixing of minimal sets. Probab. Theory Related Fields 24 (1970), 278–280.Google Scholar
[20]De La Rosa, M. and Read, C.. A hypercyclic operator whose direct sum is not hypercyclic. J. Operator Theory 61 (2009), 369–380.Google Scholar
[21]Shkarin, S.. Universal elements for non-linear operators and their applications. J. Math. Anal. Appl. 348 (2008), 193–210.CrossRefGoogle Scholar
[22]Shkarin, S.. On supercyclicity of operators from a supercyclic semigroup. J. Math. Anal. Appl. to appear. Available at www.sciencedirect.com.Google Scholar
[23]Shkarin, S.. Hypercyclic and mixing operator semigroups. Proc. Edinb. Math. Soc. to appear.Google Scholar
[24]Wengenroth, J.. Hypercyclic operators on non-locally convex spaces. Proc. Amer. Math. Soc. 131 (2003), 1759–1761.CrossRefGoogle Scholar