Abstract
Let be an open subset of the complex plane and let be an injective holomorphic self-map of such that the sequence of iterates of is a run-away sequence. We prove that the composition operator with symbol is spherically universal on a suitable function space consisting of sphere-valued functions – in contrast to the known fact that, in general, is not hypercyclic on in case that is multiply connected. Moreover, concrete open sets which support spherically universal functions will explicitly be determined in case that the symbol of the composition operator is given by a finite Blaschke product of degree two that has an attracting fixed point at the origin.
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