Asymptotic behavior of light interior functions defined in the unit disc

Abstract

Let D be the unit disk, C the unit circle. Let f be a light interior function from D into the Riemann sphere W, i.e. f is a continuous open map which does not take any continuum into a single point. It is known that f has a factorization f = g o h where h is a homeomorphism of the unit disk onto either the unit disk or the finite complex plane and g is a nonconstant meromorphic function. We will be concerned with the case when the range of h is the unit disk. A simple continuous curve r: z(t) (O _ t . The end of a boundary path r is the set rC C. A boundary path r is an asymptotic path of f for the value c provided f(z(t))->c as t->1. The set A (f) is defined to be the set of all ei0 for which there exists an asymptotic path of f in D with end E and ei'CE. The set Ap(f) is defined to be the set of all ei0 for which there exists an asymptotic path of f in D with end the point e40. For a homeomorphism h of D onto D the set B (h) is defined to be the set of all e40 for which there exists an asymptotic path of h in D with end E and e40eint E, where int E is the interior of E. The cluster set, C(f, 0), of f at ei0 is defined to be the set of all points wC W for which there exists a sequence {z2 } of points in D with Zn->e40 and f(Zn)->w. If S is a subset of D and ei0C [S3nD], then the cluster set, Cs(f, 0), of f at ei0 relative to S is defined to be the set of all points wC W for which there exists a sequence { zn} of points in S with zn-->e4O and f (Zn) ->W.