On an extension of a theorem by Ruelle to long-range potentials

Ruelle's transfer operator plays an important role in understanding thermodynamic and probabilistic properties of dynamical systems. In this work, we develop a method of finding eigenfunctions of transfer operators based on comparing Gibbs measures on the half-line $\mathbb Z_+$ and the whole line $\Z$. For a rather broad class of potentials, including both the ferromagnetic and antiferromagnetic long-range Dyson potentials, we are able to establish the existence of integrable, but not necessarily continuous, eigenfunctions. For a subset thereof we prove that the eigenfunction is actually continuous.