NEIGHBORHOODS OF CONFORMAL AUTOMORPHISMS
Let G = C be a simply-connected domain preferably with a complicated boundary structure. Following Gaier [1] we denote by Z (G) the metric space of the confonnal automorphisms of G with the topology of uniform chordal convergence. We prove that each neighborhood of an arbitrary element of Z(G) either is an uncountable set or is a singleton. From this we obtain a necessary condition about the boundary structure of G if X(G) is not discrete. AMS-Classification: 30 C 35 1. Let G c C be a simply-connected domain. By Z(G) we denote the set of all conformal automorphisms under view of the metric d(f,g) = sup {x(f(z). g(z)) I z e G} where % is the chordal distance. The space Z(G) has been introduced and studied by D. Gaier [1]. This concept has a recent s^plication in the problem of characterization of Jordan domains (s.[7]) which itself raised from the theory of iterations of rational functions (s.[4],[3]).