Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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F. Bianchi - T. C. Dinh

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

created by bianchi on 11 Dec 2020
modified on 29 Mar 2024


Published Paper

Inserted: 11 dec 2020
Last Updated: 29 mar 2024

Journal: Journal de Mathématiques Pures et Appliquées
Volume: 172
Pages: 164-201
Year: 2023

ArXiv: 2007.04595 PDF


We study the transfer (Perron-Frobenius) operator induced on $\mathbb{P}^k = \mathbb{P}^k (\mathbb{C})$ by a generic holomorphic endomorphism and a suitable continuous weight. We prove the existence and uniqueness of the equilibrium state and conformal measure and the existence of a spectral gap for the transfer operator and its perturbations on various functional spaces. Moreover, we establish an equidistribution property for the backward orbits of points, with exponential speed of convergence, towards the conformal measure. Several statistical properties of the equilibrium state, such as the K-mixing, mixing of all orders, exponential mixing, ASIP, LIL, CLT, LDP, local CLT, almost sure CLT are also obtained. Our study in particular applies to the case of H{\"o}lder continuous weights and observables, and many results are already new in the case of zero weight function and even in dimension $k = 1$. Our approach is based on pluripotential theory and on the introduction of new invariant functional spaces in this mixed real-complex setting.

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