Complex projective structures: Lyapunov exponent, degree, and harmonic measure. (English) Zbl 1381.57013

Deroin, Bertrand; Dujardin, Romain

Duke Math. J. 166, No. 14, 2643-2695 (2017).

Let \(X\) be a Riemann surface, endowed with a projective structure \(\sigma\). The paper under review is devoted to associating to this projective structure some invariants, and to studying their relationships.
The first is a Lyapunov exponent \(\chi (\sigma)\) that the authors constructed in [Commun. Math. Phys. 340, No. 2, 433–469 (2015; Zbl 1367.37046)]; the second is a degree deg\((\sigma)\), and the third is a family of harmonic measures. The Lyapunov exponent depends only on the holonomy of the structure \(\sigma\) and the Riemann surface structure induced on \(X\) by \(\sigma\). The degree deg\((\sigma)\) is the normalized asymptotic covering degree of the developing map from the universal cover of \(X\) in \(\mathbb{CP}^1\). It is normalized in the sense that deg\((\sigma) = \text{{Vol}}(X) \delta(\sigma)\). Finally, the harmonic measures \(\{\nu_x\}_{x \in \tilde{X}}\) on \(\mathbb{CP}^1\) generalize the harmonic measures on the limit sets of Kleinian groups.
The main results in the paper are Theorems A and B. Theorem A relates \(\chi (\sigma)\), \(\delta (\sigma)\) and deg\((\sigma)\) by means of the very simple expression \(\chi (\sigma) = \frac{1}{2} + 2\pi \delta (\sigma)\). Its proof runs along Sections 2, 3 and 4, and is based on the ergodic theory of holomorphic foliations. The most delicate point appears when the surface is not compact, and then the projective structure is assumed to be parabolic.
Theorem B relates the Hausdorff dimension of the harmonic measures and the Lyapunov exponent \(\chi (\sigma)\), and states that \(\text{{dim}}_H(\nu_x) \leq \frac{1}{2\chi} \leq 1\). This result is proved in Section 5.
This very interesting and carefully written article ends with some applications to Teichmüller theory, and an appendix on branched projective structures.

Reviewer: José Javier Etayo (Madrid)

MSC:

57M50	General geometric structures on low-dimensional manifolds
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F50	Klein surfaces
37H15	Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30C85	Capacity and harmonic measure in the complex plane

Keywords:

complex projective structures; Kleinian groups; Lyapunov exponent; covering degree; harmonic measure; Brownian motion

Citations:

Zbl 1367.37046