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

Equilibrium states of endomorphisms of $\mathbb{P}^k$: spectral stability and limit theorems

created by bianchi on 15 Jul 2022
modified on 04 Jul 2024


Accepted Paper

Inserted: 15 jul 2022
Last Updated: 4 jul 2024

Journal: Geometric and Functional Analysis (GAFA)
Year: 2022

ArXiv: 2204.02856 PDF


We establish the existence of a spectral gap for the transfer operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. The main issue to overcome is the rigidity of the complex objects, since the transfer operator is a non-holomorphic perturbation of the operator $f_*$. The system is moreover non-uniformly hyperbolic and one may have critical points on the Julia set. The construction of our norm requires the introduction and study of several intermediate new norms, and a careful combination of ideas from pluripotential and interpolation theory. As far as we know, this is the first time that pluripotential methods have been applied to solve a mixed real-complex problem. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure. Moreover, we obtain a full list of statistical properties for the equilibrium states: exponential mixing, CLT, Berry-Esseen theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

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