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Constructing subdivision rules from rational maps. (English) Zbl 1138.37023
Summary: This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if $$f$$ is a critically finite rational map with no periodic critical points, then for any sufficiently large integer $$n$$ the iterate $$f^{\circ n}$$ is the subdivision map of a finite subdivision rule. This enables one to give essentially combinatorial models for the dynamics of such iterates.

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 57M12 Low-dimensional topology of special (e.g., branched) coverings 37F20 Combinatorics and topology in relation with holomorphic dynamical systems
##### Keywords:
finite subdivision rule; conformality
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##### References:
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