{\Gamma}-supercyclicity

We characterize the subsets $\mathrm{\Gamma }$ of $\text{C}$ for which the notion of $\mathrm{\Gamma }$-supercyclicity coincides with the notion of hypercyclicity, where an operator $T$ on a Banach space $X$ is said to be $\mathrm{\Gamma }$-supercyclic if there exists $x\in X$ such that $\overline{\text{Orb}}(\mathrm{\Gamma }x,T)=X$. In addition we characterize the sets $\mathrm{\Gamma }\subset \text{C}$ 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 $\text{Orb}(\mathrm{\Gamma }x,T)$ is somewhere dense in $X$. This extends results by Le'on-M"uller and Bourdon-Feldman respectively. We are also interested in the description of those sets $\mathrm{\Gamma }\subset \text{C}$ for which $\mathrm{\Gamma }$-supercyclicity is equivalent to supercyclicity.