Spherical universality of composition operators

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.

Keywords:

• Universality
• composition operator
• finite Blaschke products

AMS Subject Classifications:

• 30K20
• 37F10
• 30J10

Acknowledgements

The authors thank the referee for his profound report, which helped to improve the presentation.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author has been supported by the Stipendienstiftung Rheinland-Pfalz.