Reversing the Escape Algorithm for the Plane

Abstract

The objective of this research is to develop a method of finding a function which produces a given escape diagram. An escape diagram is a graphical representation of the set of inputs in the domain of a function which remain bounded under iteration of the function. This research considers functions which have domains that are subsets of the complex numbers, although the functions are not necessarily complex analytic. The findings of this research include two methods: the fixed point method, and the conjugate method. The fixed point method provides a set of conditions which are contingent on the boundary of the given escape diagram consisting of repulsive fixed points (the boundary must be given as two functions defined on the interval (y1,y2)). If a function satisfies the conditions, the function will produce the given escape diagram. The conjugate method uses conjugacy to modify functions which have already known escape diagrams. The resulting function produces an escape diagram which is a distortion of the escape diagram of the known function. These methods were tested numerically, using computer programs to determine whether the functions that they result in produce the correct escape diagrams for multiple cases. Both of these methods were found to be successful in the cases tested.