Ergodic Theory and Dynamical Systems



Frequently hypercyclic operators and vectors


A. BONILLA a1 and K.-G. GROSSE-ERDMANN a2
a1 Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain (e-mail: abonilla@ull.es)
a2 Fakultät für Mathematik und Informatik, FernUniversität Hagen, 58084 Hagen, Germany (e-mail: kg.grosse-erdmann@fernuni-hagen.de)

Article author query
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Abstract

We study frequently hypercyclic operators, a natural new concept in hypercyclicity that was recently introduced by F. Bayart and S. Grivaux. We derive a strengthened version of their Frequent Hypercyclicity Criterion, which allows us to obtain examples of frequently hypercyclic operators in a straightforward way. Moreover, Bayart and Grivaux have noted that the frequent hypercyclicity setting differs from general hypercyclicity in that the set of frequently hypercyclic vectors need not be residual. We show here that, under weak assumptions, this set is only of first category. Motivated by this we study the question of whether one may write every vector in the underlying space as the sum of two frequently hypercyclic vectors. This investigation leads us to the introduction of a new notion, that of Runge transitivity.

(Published Online March 9 2007)
(Received February 15 2005)
(Revised September 7 2006)