Hypercyclicity of composition operators in pseudoconvex domains

Details Hypercyclicity of composition operators in pseudoconvex domains Hypercyclicity of composition operators in pseudoconvex domains

2012-02-29

arxiv.org/abs/1202.6638v1

Mathematics

Complex Variables

Scientific paper

We characterize hypercyclic composition operators $C_\varphi:f\mapsto f\circ\varphi$ on the space of functions holomorphic on $\Omega$, where $\Omega\subset\mathbb{C}^N$ is a pseudoconvex domain and $\varphi$ is a holomorphic self-mapping of $\Omega$. In the case when all the balls with respect to the Carath\'{e}odory pseudodistance are relatively compact in $\Omega$, we show that much simpler characterisation is possible (many natural classes of domains satisfy this conditioon, i.e. strictly pseudoconvex domains, bounded convex domains, etc.). Moreover, we show that in such a class of domains, and in simply connected or infinitely connected planar domains, hypercyclicity of $C_\varphi$ implies its hereditary hypercyclicity.

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