We study the notion of frequent hypercyclicity that was recently introduced by Bayart and Grivaux. We show that frequently hypercyclic operators satisfy the Hypercyclicity Criterion, answering a question of Bayart and Grivaux [Trans. Amer. Math. Soc., in press]. We also disprove a conjecture therein concerning frequently hypercyclic weighted shifts, and we prove that vectors which have a somewhere frequently dense orbit are frequently hypercyclic.
On étudie la notion d'hypercyclicité fréquente qui a récemment été introduite par Bayart et Grivaux. Nous démontrons que tout opérateur fréquemment hypercyclique vérifie le Critère d'Hypercyclicité, ce qui répond à une question de Bayart et Grivaux [Trans. Amer. Math. Soc., à paraître]. De plus, nous réfutons une conjecture de Bayart et Grivaux concernant les shifts à poids fréquemment hypercycliques, et nous démontrons que tout vecteur avec une orbite qui est quelque part fréquemment dense est fréquemment hypercyclique.
Accepted:
Published online:
@article{CRMATH_2005__341_2_123_0, author = {Grosse-Erdmann, K.-G. and Peris, Alfredo}, title = {Frequently dense orbits}, journal = {Comptes Rendus. Math\'ematique}, pages = {123--128}, publisher = {Elsevier}, volume = {341}, number = {2}, year = {2005}, doi = {10.1016/j.crma.2005.05.025}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.05.025/} }
TY - JOUR AU - Grosse-Erdmann, K.-G. AU - Peris, Alfredo TI - Frequently dense orbits JO - Comptes Rendus. Mathématique PY - 2005 SP - 123 EP - 128 VL - 341 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.05.025/ DO - 10.1016/j.crma.2005.05.025 LA - en ID - CRMATH_2005__341_2_123_0 ER -
Grosse-Erdmann, K.-G.; Peris, Alfredo. Frequently dense orbits. Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 123-128. doi : 10.1016/j.crma.2005.05.025. http://archive.numdam.org/articles/10.1016/j.crma.2005.05.025/
[1] Hypercyclicité : le rôle du spectre ponctuel unimodulaire, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 703-708
[2] F. Bayart, S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc., in press
[3] The hypercyclicity criterion for sequences of operators, Studia Math., Volume 157 (2003), pp. 17-32
[4] Hereditarily hypercyclic operators, J. Funct. Anal., Volume 167 (1999), pp. 94-112
[5] A. Bonilla, K.-G. Grosse-Erdmann, Frequently hypercyclic operators, Preprint
[6] Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J., Volume 52 (2003), pp. 811-819
[7] On a conjecture of D. Herrero concerning hypercyclic operators, C. R. Acad. Sci. Paris, Sér. I, Volume 330 (2000), pp. 179-182
[8] Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), Volume 36 (1999), pp. 345-381
[9] Multi-hypercyclic operators are hypercyclic, Math. Z., Volume 236 (2001), pp. 779-786
[10] On infinite-difference sets, Canad. J. Math., Volume 31 (1979), pp. 897-910
[11] Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc., Volume 131 (2003), pp. 1759-1761
Cited by Sources: