Equilibrium states of endomorphisms of $\mathbb{P}^k$ I: existence and properties

Fabrizio Bianchi (1, 2) , Tien-Cuong Dinh (3)

1 CNRS - Centre National de la Recherche Scientifique
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 NUS - National University of Singapore

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.

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Fabrizio Bianchi, Tien-Cuong Dinh. Equilibrium states of endomorphisms of $\mathbb{P}^k$ I: existence and properties. 2022. ⟨hal-02894047v4⟩