Coding and tiling of Julia sets for subhyperbolic rational maps

Let be a subhyperbolic rational map of degree d. We construct a set of “proper” coding maps Cod^°(f)={π_r:Σ→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 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 φ^-1(J), and the multiplicity of π_r is well-defined. Moreover, we see that the equivalence relation induced by π_r is described by a finite directed graph, and give a necessary and sufficient condition for two coding maps π_r and π_r^′ to be equal.