Coding and tiling of Julia sets for subhyperbolic rational maps

Let $f:\widehat{C}\to \widehat{C}$ be a subhyperbolic rational map of degree $d$. We construct a set of coding maps $Cod(f)=\{{\pi }_r:\mathrm{\Sigma }\to J{\}}_r$ 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 $\varphi :\tilde{S}\to S$ for the corresponding orbifold, we lift the inverse of $f$ to an iterated function system $I=(g_i{)}_{i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},d}$. 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 ${\varphi }^{-1}(J)$, 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 ${\pi }_{r^{\mathrm{\prime }}}$ to be equal.