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Abstract and Applied Analysis
Volume 2003 (2003), Issue 4, Pages 217-260

Fixed points and periodic points of semiflows of holomorphic maps

Politecnico di Torino, Dipartimento di Matematica, Corso Ducadegli Abruzzi 24, Torino 10129, Italy

Received 16 September 2001

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let ϕ be a semiflow of holomorphic maps of a bounded domain D in a complex Banach space. The general question arises under which conditions the existence of a periodic orbit of ϕ implies that ϕ itself is periodic. An answer is provided, in the first part of this paper, in the case in which D is the open unit ball of a J-algebra and ϕ acts isometrically. More precise results are provided when the J-algebra is a Cartan factor of type one or a spin factor. The second part of this paper deals essentially with the discrete semiflow ϕ generated by the iterates of a holomorphic map. It investigates how the existence of fixed points determines the asymptotic behaviour of the semiflow. Some of these results are extended to continuous semiflows.