Equilibrium states of endomorphisms of ℙ^k I: existence and properties

2020/07/09 by Bianchi, Fabrizio, Dinh, Tien-Cuong
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.04595

Abstract

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-Urbański in dimension 1 and Urbański-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.

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