The Structure of Open Continuous Mappings Having Two Valences

Abstract

Abstract.The authors study open continuous functions which map the unit disc to compact Riemann surfaces and which assume each value in the range space (with a finite number of exceptions) either p or q times for some positive integers p, q . Although the questions here originated in efforts to understand mapping properties of locally univalent analytic functions, the authors remove analyticity assumptions and show that the underlying issues are topological and combinatoric in nature. The mappings are studied by embedding their image surfaces in compact covering spaces, a setting which allows the consideration of fairly general ranges and which accommodates branch and exceptional points. Known results are generalized and extended; several open questions are posed, particularly regarding the higher dimensional analogues of the results.

A Generalisation of the Clunie–sheil-Small Theorem

“…The authors thank the referee for their useful comments and remarks. In particular, for suggesting the results from [13,14] and the results described in Remark 3.3.…”

Covering properties of open continuous mappings having two valences between Riemann surfaces

“…A detailed proof of this result can be found in [5]. However, for later reference we indicate that "V can be chosen (as in the proof in [5]) as if S is not open, and as <S' = S f) "V otherwise.…”

“…It is immediate that the n-depth function r n (y) is non-negative, Lipschitz in y and monotone decreasing in n. This function was introduced in [5] to establish the following result which is fundamental to the work of this paper.…”

“…The object of this section is to introduce the notation, concepts and preliminary results that will be used throughout the paper. Our two main sources will be the book of Ahlfors and Sario[l] and the paper of Lyzzaik and Stephenson [5].…”

“…Subsequently, the idea was extended independently by Brannan and Lyzzaik [3] and Carne, Ortel and Smith [4] to locally-univalent (p,g)-maps between the unit disc and the Riemann sphere. With the same desire, Lyzzaik and Stephenson [5] considered the broader class of (p, <7)-maps from the unit disc to closed Riemann surfaces, allowing finite sets of branch points and exceptional points, thus establishing all the earlier results on (p, g)-maps of the unit disc. In the latter paper the authors observed that their so-called embedding theorem may extend to more general (p,g r )-maps (see example 5 of [5]), and they asked whether the theorem can generalize to (p^-maps of multiplyconnected domains.…”