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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
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References:
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