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

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.

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 |