Fixed Points and Periodic Points of Semiflows of Holomorphic Maps

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.