Holomorphic functions with dense sets of Plessner points

Abstract

PROOF. First we deal with the case where a = 0, and we begin with the construction of a locally univalent function having exactly one Plessner point. Let G be a simply connected Riemann surface, of finite area, and consisting of the unit disk together with a smooth vermiform appendage that winds its way over the plane in such a way that each point of the plane is a boundary point of G. Let the function f map the disk D conformally onto G, and let it map the origin onto the point w = 0 in the portion of G that covers the unit disk in the w-plane. Then there exists exactly one point ei0 such that the image of each curve in D from 0 to ei0 passes through the full length of the appendage. The cluster set of f relative to each curve in D that terminates at ei0 is the extended plane; therefore ei0 is a Plessner point of f. By adding further appendages to G, each appropriately branched, we obtain a simply connected Riemann surface G* such that every continuous one-to-one mapping from D to G* has a dense set of Plessner points on C. If we observe the precaution of keeping the area of G* finite, then each conformal mappingf* from D to G* has a set of Fatou points of measure 2wr, hence a set of Plessner points of measure 0. The case where a = 2r is trivial, as we see by examining the function