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Γ-supercyclicity

We characterize the subsets $Γ$ of $\C$ 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 $x\in X$ such that $\overline{\text{Orb}}(Γx, T)=X$. In addition we characterize the sets $Γ\subset \C$ for which, for every operator $T$ on $X$, $T$ is hypercyclic if and only if there exists a vector $x\in X$ such that the set $\text{Orb}(Γx, T)$ 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 $Γ\subset \C$ for which $Γ$-supercyclicity is equivalent to supercyclicity.