  We develop a new method, based on pluripotential theory, to study the
transfer (Perron-Frobenius) operator induced on $\mathbb P^k = \mathbb P^k
(\mathbb C)$ 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-Urba\'nski in dimension 1 and Urba\'nski-Zdunik in higher
dimensions, both in the case of H\''older 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.
