Hypercyclic semigroups and somewhere dense orbits
[Semigroupes hypercycliques et orbites quelque part denses]
Costakis, George1 ; Peris, Alfredo2
1 Department of Mathematics, University of Maryland, College Park, MA 20742, USA
2 Departamento de Matemática Aplicada, E.T.S. Arquitectura, Universidad Politécnica de Valencia, 46071 Valencia, Spain
Comptes Rendus. Mathématique, Tome 335 (2002) no. 11, pp. 895-898.

Nous étudions l'hypercyclicité des semigroupes linéaires et fortement continus. En ce qui concerne l'iteration d'un opérateur, Bourdon et Feldman ont montré que l'existence des orbites quelque part denses implique hypercyclicité. Nous démontrons le resultat correspondant pour des semigroupes. Une conséquence est la generalisation d'une conjecture de Herrero à des semigroupes.

We study hypercyclicity of linear strongly continuous semigroups. In the case of iterations of a single operator Bourdon and Feldman have recently proved that the existence of somewhere dense orbits implies hypercyclicity. We show the corresponding result for semigroups. As a consequence, a conjecture of Herrero concerning iterations of a single operator also holds for strongly continuous semigroups.

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Accepté le :
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DOI : 10.1016/S1631-073X(02)02572-4
Costakis, George 1 ; Peris, Alfredo 2

1 Department of Mathematics, University of Maryland, College Park, MA 20742, USA
2 Departamento de Matemática Aplicada, E.T.S. Arquitectura, Universidad Politécnica de Valencia, 46071 Valencia, Spain 
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Costakis, George; Peris, Alfredo. Hypercyclic semigroups and somewhere dense orbits. Comptes Rendus. Mathématique, Tome 335 (2002) no. 11, pp. 895-898. doi : 10.1016/S1631-073X(02)02572-4. http://www.numdam.org/articles/10.1016/S1631-073X(02)02572-4/

[1] Bourdon, P.S. Invariant manifolds of hypercyclic vestors, Proc. Amer. Math. Soc, Volume 118 (1993), pp. 845-847

[2] P.S. Bourdon, N.S. Feldman, Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J., to appear

[3] Costakis, G. On a conjecture of D. Herrero concerning hypercyclic operators, C. R. Acad. Sci. Paris, Serie I, Volume 330 (2000), pp. 179-182

[4] Desch, W.; Schappacher, W.; Webb, G.F. Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynamical Systems, Volume 17 (1997), pp. 793-819

[5] Herrero, D.A. Hypercyclic operators and chaos, J. Operator Theory, Volume 28 (1992), pp. 93-103

[6] Miller, V.G. Remarks on finitely hypercyclic and finitely supercyclic operators, Integral Equations Operator Theory, Volume 29 (1997), pp. 110-115

[7] Peris, A. Multi-hypercyclic operators are hypercyclic, Math. Z, Volume 236 (2001), pp. 779-786

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