Constructing subdivision rules from rational maps

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Abstract
This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n n the iterate f ∘ n 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.