Titre :

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

Auteur(s) :

Bianchi, Fabrizio [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Centre National de la Recherche Scientifique [CNRS]
Dinh, Tien-Cuong [Auteur]
National University of Singapore [NUS]

Discipline(s) HAL :

Mathématiques [math]/Systèmes dynamiques [math.DS]
Mathématiques [math]/Variables complexes [math.CV]

Résumé en anglais : [en]

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. ...
Lire la suite >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.Lire moins >

Langue :

Anglais

Projet ANR :

ULNE
Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Commentaire :

Part 1 of the previous version. Part 2 to be submitted separately.

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

Date de dépôt :

2025-01-24T15:54:59Z