Fixed points of the Ruelle-Thurston operator and the Cauchy transform

We give necessary and sufficient conditions for a function in a naturally appearing functional space to be a fixed point of the Ruelle-Thurston operator associated to a rational function, see Lemma 2.1. The proof uses essentially a recent [13]. As an immediate consequence, we revisit Theorem 1 and Lemma 5.2 of [11], see Theorem 1 and Lemma 2.2 below.