The shape of the Julia set of an expanding rational map
In this paper the authors establish a bridge between computational topology and shape/strong shape theory. The techniques developed permit to construct resolutions of CW complexes and to study the shape of the Julia set of a rational map (using topological, geometrical, and computational tools). The original contribution of this article is the construction of the shape resolutions using the spherical multiplier of an analytic function defined on the Riemann sphere. The authors implement an algorithm for computing a finite number of cubic complexes of these resolutions. This can be used to obtain global visualizations of the Julia set of an expanding rational map or to compute homotopical/homological shape invariants. A global graphic representation of the Julia set on the unit 2-sphere is obtained. The 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) \(J(g)\). The main advantage of this algorithm based on spherical multipliers is that it is very fast. As further work the authors intend to implement in Julia language (high level/high performance dynamic programming language) the algorithms presented in this paper with possible main steps (iv)--(viii) presented in the final section. The paper also contains examples.
scientific article; zbMATH DE number 6853365
scientific article