Chaos, Elliptic Curves and All That

Abstract

The iteration of simple functions can rapidly produce chaotic behaviour in the fields of real and complex numbers. In this paper we examine polynomial iteration in the integers modulo N and examine the resulting structures. For each point c in the complex plane the map fc: C→ C is defined by f _c (z) = z ² + c. The Mandelbrot Set is defined to be those points c ∈ C for which the sequence c, f _c (c),f _c (f _c (c)),… does not diverge. See [1] for further details. By considering similar sequences in the integers modulo N, and amending the concept of divergence, we obtain finite analogues of the Mandelbrot Set when N is the product of certain primes. Curiously enough, we are immediately drawn to the application of results concerning elliptic curves defined over finite fields. So far we have only dealt with quadratic iteration. Iteration by higher degree polynomial functions remains an open area, the arithmetic of higher genus curves being much more complex.

The work of the first author was supported by the SERC