Gibbs state for some class of meromorphic functions

This paper is a continuation of our earlier works [1,2] on the fractal structure of expanding and subexpanding meromorphic functions of the form F = H o exp o Q, where H and Q are non-constant rational maps. Under some assumptions on the forward trajectories of asymptotic values ofF we define a class of summable potentials for the maps f of the punctured cylinder induced by F. We prove the existence and uniqueness of Gibbs states for these potentials.