CHAOS AND FREQUENT HYPERCYCLICITY FOR WEIGHTED SHIFTS

Stéphane Charpentier (1) , Karl Grosse-Erdmann (2) , Quentin Menet (2)

1 I2M - Institut de Mathématiques de Marseille
2 UMONS - University of Mons [Belgium]

Bayart and Ruzsa [Ergodic Theory Dynam. Systems 35 (2015)] have recently shown that every frequently hypercyclic weighted shift on p is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Trans. Amer. Math. Soc. 358 (2006)] who constructed a non-chaotic frequently hypercyclic weighted shift on c 0. We first generalize the Bayart-Ruzsa theorem to all Banach sequence spaces in which the unit sequences are a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on H(D) is chaotic, while H(C) admits a non-chaotic frequently hypercyclic weighted shift.

Domains: Mathematics [math]; Functional Analysis [math.FA]

Stéphane Charpentier, Karl Grosse-Erdmann, Quentin Menet. CHAOS AND FREQUENT HYPERCYCLICITY FOR WEIGHTED SHIFTS. Ergodic Theory and Dynamical Systems, 2020, 41 (12), pp.3634-3670. ⟨10.1017/etds.2020.122⟩. ⟨hal-03466290⟩