A Ruelle operator for a real Julia set

Let R be an expanding rational function with a real bounded Julia set, and let {Mathematical expression} be a Ruelle operator acting in a space of functions analytic in a neighbourhood of the Julia set. We obtain explicit expressions for the resolvent function {Mathematical expression} and, in particular, for the Fredholm determinant D(λ)=det(I-λL). It gives us an equation for calculating the escape rate. We relate our results to orthogonal polynomials with respect to the balanced measure of R. Two examples are considered.