The shape of the Julia set of an expanding rational map

Abstract

In memory of Sibe Mardešić, our friend.Sibe Mardešić has enriched algebraic topology developing shape and strong shape theories with important constructions and theorems.This paper relates computational topology to shape theory. We have developed some algorithms and implementations that under some conditions give a shape resolution of some Julia sets.When a semi-flow is induced by a rational map g of degree d defined on the Riemann sphere, one has the associated Julia set J(g). The main objective of this paper is to give a computational procedure to study the shape of the compact metric space J(g).Our main contribution is to provide an inverse system of cubic complexes approaching J(g) by using implemented algorithms based in the notion of spherical multiplier. This inverse system of cubical complexes is used to: (i) obtain nice global visualizations of the fractal structure of the Julia set J(g); (ii) determine the shape of the compact metric space J(g).These techniques also give the possibility of applying overlay theory (introduced by R. Fox and developed among others by S. Mardešić) to study the symmetry properties of the fractal geometry of the Julia set J(g).