Polygons and Chaos

Abstract

An infinite number of periodic trajectories are derived for the map giving rise to the Mandelbrot set at a value of the parameter corresponding to the extreme point on the real axis of the Mandelbrot set. Beginning with the edge of a family of star n-gons as the seed, the trajectory of the logistic map cycles through a sequence of edges of other star n-gons. Each n-gon for n odd is shown to have its own characteristic cycle length. The logistic map is shown to be the first of two infinite families of maps, all exhibiting periodic trajectories, derived from two families of polynomials, the Chebyshev polynomials and another related to the Lucas sequence. Using the family of Lucas polynomials, the Mandelbrot set is generalized to an infinite family of sets with similar properties.