Existence and properties of equilibrium states of holomorphic endomorphisms of $\mathbb{P}^k$

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 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, as well as the equidistribution of repelling periodic points with respect to the equilibrium state. 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 Hölder 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.

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

Fabrizio Bianchi, Tien-Cuong Dinh. Existence and properties of equilibrium states of holomorphic endomorphisms of $\mathbb{P}^k$. 2021. ⟨hal-02894047v3⟩