Complex Ruelle Operator and Hyperbolic Complex Dynamical Systems

Let $R:\overline{\mathbb{C}}arrow\overline{\mathbb{C}} $ be a hyperbolic rational mapping. We assume that all the attractive periodic points of $R $ are fixed points, all the critical points of $R $ are non-degenerate, and that the Julia set of $R, $ $J_{R} $ , is inculed in C. Let $N $ denote the number of attractive fixed points and let $a_{1}, $ $\cdots $ , $a_{N}$ denote the attractive fixed points. Let $A_{k} $ denote the attractive basin of $a_{k} $. Let $C_{R} $ denote the set of critical points of $R $. For $k=1, $ $\cdots $ , $N $ , let $\gamma_{k} $ denote an oriented multicurve in $A_{k} $ , such that $\gamma_{k}=\partial\Omega_{k} $ , where $\Omega_{k} $ is an open set satisfying $R^{-1}(\Omega_{k})\subset\Omega_{k}, $ $\Omega_{k}\cup A_{k}=\overline{\mathbb{C}} $, and $C_{R}\cap\Omega_{k}\cap A_{k}=\phi $. Let $\gamma=\bigcup_{k=1}^{N}\gamma k $ and $\Omega=\mathrm{n}_{k=1}^{N}\Omega_{k} $. For open set $O\subset\overline{\mathbb{C}} $ , let $O_{0}(o) $ denote the space of functions $g:Oarrow \mathbb{C}$ holomorphic in $O $ and has an analytic extention to a neighbourhood of the closure of $O $ , and satisfies $g(\infty)=0 $ if $\infty $ belongs to the closure of $O $. We have the following decomposition of holomorphic functions. The direct sum in the theorem means the uniqueness of the decomposition