Complex Projective Structures: Lyapunov Exponent, Degree and Harmonic Measure

Abstract : We study several new invariants associated to a holomorphic projective struc- ture 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 descrip- tion of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics.
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Bertrand Deroin, Romain Dujardin. Complex Projective Structures: Lyapunov Exponent, Degree and Harmonic Measure. Duke Mathematical Journal, Duke University Press, 2017, 166 (14 ). ⟨hal-01068575⟩

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