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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hypercyclicity of composition operators in Stein manifolds
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by Sylwester Zajac PDF
Proc. Amer. Math. Soc. 144 (2016), 3991-4000 Request permission


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:
  • 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:
  • MathSciNet review: 3513554