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Self-overlays and symmetries of Julia sets of expanding maps
- Source :
- Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 112:829-848
- Publication Year :
- 2018
- Publisher :
- Springer Science and Business Media LLC, 2018.
-
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 $$\varSigma _d$$ .
- Subjects :
- Discrete mathematics
Algebra and Number Theory
Inverse system
Applied Mathematics
010102 general mathematics
010103 numerical & computational mathematics
Riemannian manifold
01 natural sciences
Julia set
Computational Mathematics
Symmetric group
Homogeneous space
Geometry and Topology
Inverse limit
Branched covering
0101 mathematics
Invariant (mathematics)
Analysis
Mathematics
Subjects
Details
- ISSN :
- 15791505 and 15787303
- Volume :
- 112
- Database :
- OpenAIRE
- Journal :
- Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
- Accession number :
- edsair.doi...........fdfdb221f907da38ba9189a8fa7fade4