Γ-supercyclicity

We characterize the subsets Γ of CC for which the notion of Γ-supercyclicity coincides with the notion of hypercyclicity, where an operator T on a Banach space X   is said to be Γ-supercyclic if there exists x∈Xx∈X such that Orb‾(Γx,T)=X. In addition we characterize the sets Γ⊂CΓ⊂C for which, for every operator T on X, T   is hypercyclic if and only if there exists a vector x∈Xx∈X such that the set Orb(Γx,T)Orb(Γx,T) is somewhere dense in X  . This extends results by León–Müller and Bourdon–Feldman respectively. We are also interested in the description of those sets Γ⊂CΓ⊂C for which Γ-supercyclicity is equivalent to supercyclicity.