(Non-)weakly mixing operators and hypercyclicity sets
[Opérateurs (non) faiblement mélangeants et ensembles d’hypercyclicité]
Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 1-35.

On étudie la fréquence d’hypercyclicité des opérateurs hypercycliques non faiblement mélangeants. On montre en particulier qu’il est possible de construire sur l’espace 1 des opérateurs non faiblement mélangeants de fréquence d’hypercyclicité arbitrairement grande. On obtient un résultat analogue (mais plus faible) sur c 0 ou p , 1<p<. Certains de nos résultats font intervenir des propriétés de lacunarité de type “Sidon” pour les suites d’entiers.

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space 1 (), any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for c 0 () or p (), 1<p<. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

DOI : https://doi.org/10.5802/aif.2425
Classification : 47A16,  37B99,  11B99
Mots clés : opérateurs hypercycliques, opérateurs faiblement mélangeants, ensembles d’hypercyclicité, suites de Sidon
@article{AIF_2009__59_1_1_0,
     author = {Bayart, Fr\'ed\'eric and Matheron, \'Etienne},
     title = {(Non-)weakly mixing operators and hypercyclicity sets},
     journal = {Annales de l'Institut Fourier},
     pages = {1--35},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2425},
     mrnumber = {2514860},
     zbl = {1178.47003},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2425/}
}
Bayart, Frédéric; Matheron, Étienne. (Non-)weakly mixing operators and hypercyclicity sets. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 1-35. doi : 10.5802/aif.2425. http://archive.numdam.org/articles/10.5802/aif.2425/

[1] Ansari, S. Hypercyclic and cyclic vectors, J. Funct. Anal., Volume 128 (1995) no. 2, pp. 374-383 | Article | MR 1319961 | Zbl 0853.47013

[2] Bayart, F.; Grivaux, S. Frequently hypercyclic operators, Trans. Amer. Math. Soc., Volume 358 (2006) no. 11, pp. 5083-5117 | Article | MR 2231886 | Zbl 1115.47005

[3] Bayart, F.; Matheron, É. Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces, J. Funct. Anal., Volume 250 (2007), pp. 426-441 | Article | MR 2352487 | Zbl 1131.47006

[4] Bès, J.; Peris, A. Hereditarily hypercyclic operators, J. Funct. Anal., Volume 167 (1999) no. 1, pp. 94-112 | Article | MR 1710637 | Zbl 0941.47002

[5] Bonilla, A.; Grosse-Erdmann, K.-G. Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems, Volume 27 (2007), pp. 383-404 | Article | MR 2308137 | Zbl 1119.47011

[6] Bourdon, P. S.; Feldman, N. S. Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J., Volume 52 (2003) no. 3, pp. 811-819 | Article | MR 1986898 | Zbl 1049.47002

[7] Costakis, G.; Sambarino, M. Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc., Volume 132 (2004) no. 2, pp. 385-389 | Article | MR 2022360 | Zbl 1054.47006

[8] De La Rosa, M.; Read, C. J. A hypercyclic operator whose direct sum is not hypercyclic (Journal of Operator Theory, to appear)

[9] Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, 1981 | MR 603625 | Zbl 0459.28023

[10] Glasner, E. Ergodic theory via joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, 2003 | MR 1958753 | Zbl 1038.37002

[11] Glasner, E.; Weiss, B. On the interplay between mesurable and topological dynamics, Handbook of dynamical systems, 1B, Elsevier B. V., 2006 (597–648) | MR 2186250 | Zbl 1130.37303

[12] Grivaux, S. Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory, Volume 54 (2005) no. 1, pp. 147-168 | MR 2168865 | Zbl 1104.47010

[13] Grosse-Erdmann, K.-G.; Peris, A. Frequently dense orbits, C. R. Acad. Sci. Paris, Volume 341 (2005), pp. 123-128 | MR 2153969 | Zbl 1068.47012

[14] Halberstam, H.; Roth, K. F. Sequences, Springer-Verlag, 1983 | MR 687978 | Zbl 0498.10001

[15] Peris, A.; Saldivia, L. Syndetically hypercyclic operators, Integral Equations Operator Theory, Volume 51 (2005) no. 2, pp. 275-281 | Article | MR 2120081 | Zbl 1082.47004

[16] Rusza, I. Z. An infinite Sidon sequence, J. Number Theory, Volume 68 (1998), pp. 63-71 | Article | MR 1492889 | Zbl 0927.11005