Equilibrium states of endomorphisms of [formula omitted] I: Existence and properties

Abstract

We develop a new method, based on pluripotential theory, to study the transfer (Perron-Frobenius) operator induced on Pk=Pk(C) by a holomorphic endomorphism and a suitable continuous weight. This method allows us to prove the existence and uniqueness of the equilibrium state and conformal measure for very general weights (due to Denker-Przytycki-Urbański in dimension 1 and Urbański-Zdunik in higher dimensions, both in the case of Hölder continuous weights). We establish a number of properties of the equilibrium states, including mixing, K-mixing, mixing of all orders, and an equidistribution of repelling periodic points. Our analytic method replaces all distortion estimates on inverse branches with a unique, global, estimate on dynamical currents, and allows us to reduce the dynamical questions to comparisons between currents and their potentials.