Self-overlays and symmetries of Julia sets of expanding maps

Abstract

When a semi-flow is induced by a d-fold branched covering $$ f:M \rightarrow M $$ defined on a Riemannian manifold M, the associated Julia set J(f) is a compact invariant subset of M and, therefore, there exists an induced restriction $$ f | _ {J (f)} :J (f) \rightarrow J (f) $$ . In order to construct an inverse system of regular sub-complexes whose inverse limit is J(f) we use computational techniques to iterate subdivision processes for a regular CW-structure given in M. The invariants of this inverse system can be used to study some topology and shape properties of J(f) . In particular, for the case of an expanding rational map we have constructed a resolution using global multipliers. The advantage of this resolution is that we can develop many algorithms that give an explicit description of the complexes of this resolution and implemented versions of this procedure can be used to give nice visualizations of the Julia set or to compute its shape invariants. If J(f) does not contain critical points of f, the restriction $$f | _ {J (f)} $$ inherits a d-fold overlay structure which is the limit of d-fold coverings and the classification of this overlay structure can be given in terms of representations of the fundamental pro-groupoid of J(f) in the symmetric group $$\varSigma _d$$ .