Abstract
We unify and extend several results on the dynamic behaviour of composition operators on the space of holomorphic functions on a simply connected plane domain and endowed with the compact open topology. In particular, we show that a composition operator is weakly supercyclic if and only if the algebra it generates consists entirely, except for the null one, of operators that are topologically mixing, Devaney-chaotic, and have a frequently hypercyclic subspace. We also show that all non-zero scalar multiples of powers of a given hypercyclic composition operator share a dense set of common hypercyclic vectors.
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