Complex projective structures: Lyapunov exponent, degree and harmonic measure
Abstract
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developing map; and a family of harmonic measures on the Riemann sphere, previously introduced by Hussenot. We show that the degree and the Lyapunov exponent are related by a simple formula and give estimates for the Hausdorff dimension of the harmonic measures in terms of the Lyapunov exponent. In accordance with the famous "Sullivan dictionary", this leads to a description of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics.
Cite
@article{arxiv.1308.0541,
title = {Complex projective structures: Lyapunov exponent, degree and harmonic measure},
author = {Bertrand Deroin and Romain Dujardin},
journal= {arXiv preprint arXiv:1308.0541},
year = {2017}
}