Recently, I read a theorem of existence of conformal measure for the rational map.

I did not understand two places in the proof. The author claims that there exists an open set $V\subset \hat{C}\setminus J(R)$, such that each inverse branch $R_{j}^{-n}$ of $R^{n}$ is a single valued function. And also the inverse orbits $$R_{j_1}^{-1}(V), R_{j_2}^{-1}R_{j_1}^{-1}(V),\dots R_{j_k}^{-1}R_{j_(k-1)}^{-1}(V)\dots R_{j_1}^{-1}(V)$$ are disjoint and the orbits will uniformly converge to $J(R)$, if $V$ is a subset in Siegel disk and Herman ring. there is at most one exceptional inverse orbit.

I was confused with this argument for a very long time, I did know how to give a complete proof. any reference and comments will be appreaciated.

EDIT(Thanks for professor Eremenko's advice.): this arguement is form theorem 3 (page 740) in Sullivan's paper. http://download.springer.com/static/pdf/268/chp%253A10.1007%252FBFb0061443.pdf?auth66=1398441124_3dd347091ebaae9a952145910da7220b&ext=.pdf