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Fixed points and periodic points of semiflows of holomorphic maps. (English) Zbl 1106.47303

Summary: Let \(\phi\) 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 \(\phi\) implies that \(\phi\) 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 \(\phi\) 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 \(\phi\) 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.

MSC:

47H10 Fixed-point theorems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
46G20 Infinite-dimensional holomorphy
58C10 Holomorphic maps on manifolds
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