Contractive Curves

We discuss the dynamics of the correspondences associated to those plane curves whose local sections contract the Poincaré metric in a hyperbolic planar domain. 1. Introduction. We consider certain 1-dimensional, holomorphic correspondences of hyperbolic type, which we call " contractive curves. " These are curves whose local sections contract the Poincaré metric of a hyperbolic planar domain. The model for our analysis is given by the hyperbolic Julia sets. This work has been motivated by the work in Ochs' recent paper [5]. In the first part, we adapt to our purposes the Schwarz lemma for correspondences that appears in Minda's paper [3]. In the second part, we discuss the dynamics of a contractive curve, and the properties of the associated attractor. Such curves usually do not have global sections that contract the hyperbolic metric. Nevertheless, the associated dynamical systems have much in common with the iterated function systems of hyperbolic type. The basis for our discussion is the paper [2] by Barnsley and Demko.