Complex projective structures: Lyapunov exponent, degree and harmonic measure

TL;DR

This paper introduces and relates new invariants for holomorphic projective structures on Riemann surfaces, including the Lyapunov exponent, degree, and harmonic measures, revealing their interconnections and implications for the structure space.

Contribution

It establishes a simple formula linking the Lyapunov exponent and degree, and provides estimates for harmonic measure dimensions, advancing understanding of projective structures in complex dynamics.

Findings

Lyapunov exponent and degree are related by a simple formula.

Estimates for Hausdorff dimension of harmonic measures are provided.

The structure space resembles that of polynomials in holomorphic dynamics.

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.