Fixed points and periodic points of semiflows of holomorphic maps

Abstract

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 $J^{\ast }$-algebra and $\phi$ acts isometrically. More precise results are provided when the $J^{\ast }$-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.