Complex projective structures: Lyapunov exponent, degree and harmonic measure

Bertrand Deroin; Romain Dujardin

doi:10.1215/00127094-2017-0012

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arXiv · 1308.0541

Complex projective structures: Lyapunov exponent, degree and harmonic measure

Bertrand Deroin·Romain Dujardin

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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.

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BibTeX RIS

Bertrand Deroin, Romain Dujardin. 2013-08-02. Complex projective structures: Lyapunov exponent, degree and harmonic measure. https://doi.org/10.1215/00127094-2017-0012

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