1 October 2017 Complex projective structures: Lyapunov exponent, degree, and harmonic measure
Bertrand Deroin, Romain Dujardin
Duke Math. J. 166(14): 2643-2695 (1 October 2017). DOI: 10.1215/00127094-2017-0012


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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Bertrand Deroin. Romain Dujardin. "Complex projective structures: Lyapunov exponent, degree, and harmonic measure." Duke Math. J. 166 (14) 2643 - 2695, 1 October 2017. https://doi.org/10.1215/00127094-2017-0012


Received: 21 July 2015; Revised: 17 February 2017; Published: 1 October 2017
First available in Project Euclid: 26 August 2017

zbMATH: 1381.57013
MathSciNet: MR3707286
Digital Object Identifier: 10.1215/00127094-2017-0012

Primary: 57M50
Secondary: 30C85 , 30D35 , 30F40 , 30F50 , 37H15

Keywords: Brownian motion , complex projective structures , harmonic measure , Kleinian groups , Lyapunov exponent

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 14 • 1 October 2017
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