A New Proof for the Description of Holomorphic Flows on Multiply Connected Domains

We provide a new proof for the description of holomorphic and biholomorphic flows on multiply connected domains in the complex plane. In contrast to the original proof of Heins (1941) we do this by the means of operator theory and by utilizing the techniques of universal coverings of the underlying domains of holomorphic flows and their liftings on the corresponding universal coverings. 1. Introduction Holomorphic flows on simply connected domains in the complex plane constitute a rich and well-studied class of maps (cf. [1]). The basic description of holomorphic flows on multiply connected domains is due to Heins [2]. In this paper we provide a new proof of Heins' results and show that, with the notable exception of the punctured disc, every holomorphic flow on a multiply connected domain in the plane is biholomorphic. We do this by the means of operator theory and by utilizing the techniques of universal coverings of the underlying domains of holomorphic flows and their liftings on the universal coverings. Let be an interval and let , be sets. As customary, a mapping can be interpreted as a one-parameter family, , of self-mappings , where runs in and vice versa. Definition 1. Let be a topological space. A family of mappings , is called a (semigroup) flow on , if , , and the composition semigroup rule holds for every , . A flow is called trivial if all its mappings are equal to the identity on . Flows can be viewed as semigroups of composition operators on . The set is called the underlying domain of the flow. If are defined and property (1) holds for every real , then the flow is called a group flow. It is easy to see that if is a topological space, is a homeomorphism, and the family is a flow on , then the family is a flow on . A subset is called an invariant space of a flow on if all its functions map into itself. In this case the restrictions form a flow on . A singleton invariant space of a flow is called a fixed point of the flow. Namely, is a fixed point of a flow if for every . By holomorphic (resp., biholomorphic) flow on a domain in this paper we will understand, as usual, a flow on such that all are holomorphic (resp., biholomorphic) functions from into . 2. Holomorphic Flows on Multiply Connected Domains in Holomorphic flows on simply connected domains in the complex plane are well studied. As shown in [1], any holomorphic flow on is biholomorphic. Theorem 2 (see [1]). If is a holomorphic flow on , then one of the following holds:(i) does not have fixed points in and for some .(ii) has one fixed point in and . Recall that any 2-connected