Common A-hypercyclicity

In the last decades the notion of hypercyclicity has been subject to many investigations in linear dynamics. An operator T on a Fréchet space X is called hypercyclic if there exists some vector x ∈ X such that the set {T n (x) | n ≥ 0} is dense in X. In this case x is called hypercyclic for T. In other terms an operator is hypercyclic if there exists some vector that visits under this operator each non-empty open set. A fundamental theorem in linear dynamics is the transitivity theorem of Birkhoff. This ensures that the set of hypercyclic vectors of an operator is a dense G δ -set and this provides an equivalent formulation of hypercyclicity. Later this result has been generalized for families of operators. Instead of studying the behaviour of an operator one then care about a family of operators. In this context it is rather natural to study the existence of vectors which are hypercyclic for each operator of the family. This yields the notion of common hypercyclicity. Most of the common hypercyclicity criteria are based on the generalization of the transitivity theorem of Birkhoff.

Recently Bés, Menet, Peris and Puig have defined a notion which generalizes hypercyclicity: the A-hypercyclicity. This notion also includes upper frequent hypercyclicity. Intuitively an operator is upper frequently hypercyclic if it possesses a vector that visits 'very often' each non-empty open set. Imposing some conditions on the family A, Bonilla and Grosse-Erdmann have found an analogue of the theorem of Birkhoff for A-hypercyclicity.

In this thesis we study common A-hypercyclicity. Relying on the arguments used in common hypercyclicity we establish a generalization of the theorem of Bonilla and Grosse-Erdmann for families of operators. In particular this allows us to obtain a common upper frequent hypercyclicity criterion. In addition to these positive results we develop an approach ensuring the nonexistence of common A-hypercyclic vectors.