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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: etienne.matheron@euler.univ-artois.fr)
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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

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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