no. 2
Dynamical Systems
Frequently dense orbits
Presented by: Gilles Pisier
Grosse-Erdmann, K.-G.1; Peris, Alfredo2
1 Fachbereich Mathematik, FernUniversität Hagen, 58084 Hagen, Germany
2 E.T.S. Arquitectura, Departament de Matemàtica Aplicada and IMPA-UPV, Universitat Politècnica de València, 46022 València, Spain
Comptes Rendus. Mathématique, Volume 341 (2005) no. 2, pp. 123-128

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.05.025

Grosse-Erdmann, K.-G. 1; Peris, Alfredo 2

1 Fachbereich Mathematik, FernUniversität Hagen, 58084 Hagen, Germany
2 E.T.S. Arquitectura, Departament de Matemàtica Aplicada and IMPA-UPV, Universitat Politècnica de València, 46022 València, Spain 
@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 = {https://www.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  - https://www.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  - 
%0 Journal Article
%A Grosse-Erdmann, K.-G.
%A Peris, Alfredo
%T Frequently dense orbits
%J Comptes Rendus. Mathématique
%D 2005
%P 123-128
%V 341
%N 2
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2005.05.025/
%R 10.1016/j.crma.2005.05.025
%G en
%F CRMATH_2005__341_2_123_0
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

[1] Bayart, F.; Grivaux, S. 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] Bernal-González, L.; Grosse-Erdmann, K.-G. The hypercyclicity criterion for sequences of operators, Studia Math., Volume 157 (2003), pp. 17-32

[4] Bès, J.; Peris, A. Hereditarily hypercyclic operators, J. Funct. Anal., Volume 167 (1999), pp. 94-112

[5] A. Bonilla, K.-G. Grosse-Erdmann, Frequently hypercyclic operators, Preprint

[6] Bourdon, P.S.; Feldman, N.S. Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J., Volume 52 (2003), pp. 811-819

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

[8] Grosse-Erdmann, K.-G. Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), Volume 36 (1999), pp. 345-381

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

[10] Stewart, C.L.; Tijdeman, R. On infinite-difference sets, Canad. J. Math., Volume 31 (1979), pp. 897-910

[11] Wengenroth, J. Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc., Volume 131 (2003), pp. 1759-1761

Cited by Sources: