We call a rational map f graph critical if any critical point either belongs to an invariant finite graph G, or has minimal limit set, or is nonrecurrent and has limit set disjoint from G. We prove that, for any conformal measure, either for almost every point of the Julia set J(f) its limit set coincides with J(f), or for almost every point of J(f) its limit set coincides with the limit set of...

We consider the stochastic fixed point equation Xn d = Xn−D +B that appears in several applications, for example in the Bolthausen-Sznitman coalescent or in the analysis or recursive algorithms. In this paper we provide a systematic study on these kinds of stochastic fixed point equations. It turns out that depending on the singularity structure of the the moment generating function of D the li...