On the Peano curves associated with some conformal maps

Abstract

1. Salem and Zygmund [5], by use of an appropriate Taylor series with gaps, proved that there exists a function, f(z), which is holomorphic in Izl <1, continuous in Izi <1, and such that the curve w =f(eit), 0 <t <27r, fills some square. Using a different type of series, Piranian, Titus, and Young [4] gave a very simple example which shows also that the curve w =f(eit) can be exactly a square. The circumference j z = 1 is a natural boundary for the functions of Salem and Zygmund, whereas the singularities on I zI = 1 of the function of Piranian, Titus, and Young may have measure zero. Using gap series, Schaeffer [6 has shown that f(eit) can assume every value assumed by f(z) in J z < 1. All of these proofs are essentially arithmetical, and one is led to wonder just what may be the geometry of such a startling map w =f(z). The purpose of the present note is to prove a similar theorem by constructing a suitable Riemann surface onto which IzI <1 is mapped by w =f(z). In this construction it is easy to see precisely what the corresponding Peano curve is. The resulting curve differs from the usual examples of Peano curves in that it is, in a sense, made up of linear segments. Using this method of obtaining the desired function by first constructing a suitable Riemann surface, Ohtsuka [7] has recently proved the following theorem.1 There exists a function w= F(z), bounded and analytic in j z j <1, such that F(ei') = lim,_1 F(reit) has modulus 1 for almost all eit on IzI =1, and such that, for each c with |c I 1, the equation F(eit) = c is satisfied on an uncountable set of eil on I z j =1. The primary difference between Ohtsuka's theorem and Theorems 1 and 2 below (for the case where D is the unit circle) is in the choice of the condition added to the condition that F(eit) fill a domain. Namely, either I F(ei) I =1 for almost all t, or F(z) is continuous in I zI <1. The noncountable aspect of Ohtsuka's function is similar to properties of the function of Salem and Zygmund.