On the Peano curves associated with some conformal maps
On the Peano curves associated with some conformal maps
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关于与一些共角贴图相关的皮亚诺曲线
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发表时间：
1955
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通讯作者：
G. R. Maclane
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作者：
G. R. Maclane
1. Salem和Zygmund [5]利用适当的带间隙Taylor级数，证明了存在函数f（z），它在Izl <1时全纯，在Izi <1时连续，且曲线w =f（eit），0 <t <27 r，填充某个正方形。使用不同类型的系列，Piranian，Titus和Young [4]给出了一个非常简单的例子，也表明曲线w =f（eit）可以是正方形。圆周j z = 1是塞勒姆和齐格蒙特函数的自然边界，而皮兰尼安、提图斯和杨函数的奇性I z i = 1可能为零。Schaeffer [6]用间隙级数证明了f（eit）可以取f（z）在Jz < 1时所取的任何值。所有这些证明基本上都是算术的，人们不禁要问，这样一个令人吃惊的映射w =f（z）的几何结构究竟是什么。本文的目的是通过构造一个合适的Riemann曲面证明一个类似的定理，其中IzI <1由w =f（z）映射到该Riemann曲面上.在这个结构中，很容易精确地看到相应的皮亚诺曲线是什么。由此产生的曲线不同于通常的皮亚诺曲线的例子，因为它在某种意义上是由线性段组成的。Ohtsuka [7]用这种方法首先构造一个合适的Riemann曲面来获得所需的函数，最近证明了以下定理：1存在一个函数w= F（z），在j z j <1中有界且解析，使得F（ei'）= lim，_1 F（reit）对几乎所有eit在IzI =1上都有模1，并且对于每个c，|c I 1，则方程F（eit）= c在Izj =1的eil的不可数集合上满足. Ohtsuka定理与下面的定理1和定理2（对于D是单位圆的情况）之间的主要区别在于对F（eit）填充域的条件的选择。也就是说，或者对于几乎所有的t，IF（ei）I =1，或者F（z）在IzI <1时是连续的。Ohtsuka函数的不可数性质类似于Salem和Zygmund函数的性质。
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.