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.
https://hal-upec-upem.archives-ouvertes.fr/hal-01068575 Contributor : Romain DujardinConnect in order to contact the contributor Submitted on : Friday, September 26, 2014 - 4:46:03 PM Last modification on : Thursday, March 17, 2022 - 10:08:17 AM Long-term archiving on: : Friday, April 14, 2017 - 2:02:17 PM