Existence of the Mandelbrot Set in the Parameter Planes of Certain Rational Functions

Abstract

In complex dynamics we compose a complex valued function with itself repeatedly and observe the orbits of values of that function. Particular interest is in the orbit of critical points of that function (critical orbits). One famous, studied example is the quadratic polynomial Pc(z) = z^2 +c and how changing the value of c makes a difference to the orbit of the critical point z = 0. The set of c values for which the critical orbit is bounded is called the Mandelbrot set. This paper studies rational functions of the form Rn;a;c(z) = z^n + a/z^n + c and their critical orbits. It turns out that for certain fixed values of n, a, and c, Rn;a;c locally behaves like Pc. On those regions Rn;a;c is said to be a degree two polynomial-like map. We then consider fixing a while allowing c to vary and study the a-parameter plane of Rn;a;c and then vice-versa. We show that homeomorphic copies of the Mandelbrot set exist in both the a and c-parameter planes. Finally we observe peculiar behaviors of the parameter planes when multiple Mandelbrot copies exist and are in close proximity to each other.