The specification approach to equilibrium states for parabolic rational maps

Abstract

We develop the specification and orbit-decomposition approach to equilibrium states for parabolic rational maps of the Riemann Sphere. Our result extends the well-known results on uniqueness of equilibrium states in this setting, notably the results of Denker, Przytycki and Urbański. We extend the class of potentials from Hölder to those with the Bowen property on 'good orbits' . We obtain uniqueness of the equilibrium state for potentials satisfying a pressure gap condition which is sharp in the class of potentials we consider. We show that our equilibrium state has the $K$-property, and in particular it has positive entropy. When the potential is Hölder, the theory of equilibrium states is already highly developed. Nevertheless, several interesting results on equilibrium states for Hölder potentials follow readily from our approach. In the family of geometric potentials, we obtain a simple proof of uniqueness of equilibrium states up to the phase transition that occurs at the Hausdorff dimension of the Julia set. For Hölder potentials on parabolic rational maps, we show that hyperbolicity of the potential is equivalent to having a unique equilibrium state which is fully supported. This does not appear to have been stated in the literature before, although it may be considered folklore.