Compositional hypercyclicity equals supercyclicity

Abstract

In this note it is proved that the sequence of composition operators generated by automorphisms of a simply connected domain strictly contained in the complex plane is hypercyclic –that is, possesses some dense orbit– if and only if it is supercyclic –i.e., possesses some dense projective orbit–. When the domain is the full complex plane, a result in this direction is also obtained. In addition, a number of statements about the corresponding cyclicity properties of single composition operators are either proved directly or extracted as a consequence