Julia集为Cantor集的一族有理函数

A Class of Rational Functions with Cantor Julia Sets

摘要: 参数空间的研究是复解析动力系统研究的一个重要部分，著名的Mandelbrot集是含有单参数的多项式p(z)=z²+c(c为复常数)的参数空间，它是一个复杂的分形图。人们猜测，对同样含有单参数的函数族，应该也有像M集一样复杂的参数空间。为此，我们研究了含有单参数的二次有理函数族R_λ(z)=λz/(1−z)²(λ为复常数)。运用复解析动力系统中临界点与Fatou分支的关系，我们得到当参数λ∈(−1,0)∪(0,1)时，其Julia集为广义cantor集。此时由于该函数族的所有临界点都在Fatou集的一个吸性分支里面，所以该函数族中的函数全为双曲有理函数。

Abstract: Parameter space is an important part of complex analytic dynamics. Mandelbrot set, which is famous in the world, is the parameter space of p(z)=z²+c (c is a complex constant). And p(z) has one parameter. Many people conjecture that functions with one parameter also have the intricate parameter space as Mandelbrot set. Similarly, we study a class of rational functions with one parameter. They are R_λ(z)=λz/(1−z)² (λ is a complex constant). By using the relation between critical points and the component of Fatou set, we have found that they have cantor Julia sets when λ∈(−1,0)∪(0,1). Moreover, they are all hyperbolic rational functions since all critical points of them stay in an attract component of Fatou set.