Coding and tiling of Julia sets for subhyperbolic rational maps

Abstract

Let $f:\hat{C}\to\hat{C}$ be a subhyperbolic rational map of degree $d$. We construct a set of coding maps $Cod(f)=\{\pi_r:\Sigma\to J\}_r$ of the Julia set $J$ by geometric coding trees, where the parameter $r$ ranges over mappings from a certain tree to the Riemann sphere. Using the universal covering space $\phi:\tilde S\to S$ for the corresponding orbifold, we lift the inverse of $f$ to an iterated function system $I=(g_i)_{i=1,2,...,d}$. For the purpose of studying the structure of $Cod(f)$, we generalize Kenyon and Lagarias-Wang's results : If the attractor $K$ of $I$ has positive measure, then $K$ tiles $\phi^{-1}(J)$, and the multiplicity of $\pi_r$ is well-defined. Moreover, we see that the equivalence relation induced by $\pi_r$ is described by a finite directed graph, and give a necessary and sufficient condition for two coding maps $\pi_r$ and $\pi_{r'}$ to be equal.