Hypercyclicity of composition operators in Stein manifolds
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Abstract:
We characterize hypercyclic composition operators $C_\varphi :f\mapsto f\circ \varphi$ on the space of holomorphic functions on a connected Stein manifold $\Omega$ with $\varphi$ being a holomorphic self-map of $\Omega$.
In turns out that in the case when all balls with respect to the Carathéodory pseudodistance are relatively compact in $\Omega$, a much simpler characterization may be obtained (many natural classes of domains in $\mathbb {C}^N$ satisfy this condition). Moreover, we show that in such a class of manifolds, as well as in simply connected and infinitely connected planar domains, hypercyclicity of $C_\varphi$ implies its hereditary hypercyclicity with respect to the full sequence of natural numbers.
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Additional Information
- Sylwester Zajac
- Affiliation: Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
- Email: sylwester.zajac@im.uj.edu.pl
- Received by editor(s): May 22, 2015
- Received by editor(s) in revised form: October 14, 2015, and November 21, 2015
- Published electronically: March 17, 2016
- Communicated by: Franc Forstneric
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3991-4000
- MSC (2010): Primary 47B33; Secondary 32H50
- DOI: https://doi.org/10.1090/proc/13046
- MathSciNet review: 3513554