On the Conformal Mapping of a Region Into a Part of Itself

Abstract

1. This note will extend to domains of any degree of connectivity a theorem proved for simply connected domains by G. Julia in the preliminaries to his prize memoir on the iteration of rational functions.* Consider, in the plane of the variable z, a closed and bounded domain A, consisting of a two-dimensional continuum plus its boundary. Let a function f(z) be analytic in A and assume throughout A values which correspond to inner points of A. Then f(z) maps A conformally on a Riemann surface of one or more sheets, every interior and boundary point of which lies withzn A. If we project the points of this Riemann surface upon the z plane, we obtain a closed domain A1. To every value assumed by f(z) in A, no matter how many times, there corresponds one and only one point of A1. The domain A1 consists of inner points and boundary points, each boundary point being a limit point of inner points. Every value which f(z) takes at an inner point of A gives an inner point of A1; the boundary points of A1 correspond only to values which f(z) takes at boundary points, but not at inner points, of A; in certain cases, however, values which f(z) takes only on the boundary of A may give inner points of A1. It is easv to show that the inner points of A1 form a continuum, but we shall not have occasion to use this fact. It is evident that f(z) transforms A1 into a closed domain A.2 whose points are all inner points of A1, A.. into a smaller domain A3, etc. The theorem we are to prove states that: (a) In the transformation of A into A1, one and only one point of A stays fixed. (b) At this fixed point a, we have f'(a) < 1. (c) The domains A, A1, A, .,, * A** ... converge to the fixed point. The application of this theorem to doubly connected regions shows that it is impossible to shrink a ring conformally into a ring situated in its interior; that is, if the given ring is bounded by certain curves C1 and C2 (C2 interior to C1), it is impossible to map the ring conformally in a one-to-one manner upon a ring bounded by curves r1 and r., With r1 interior to C1 and C2 interior to r2. For it is clear that the points of