Thermodynamics, dimension and the Weil-Petersson metric. (English) Zbl 1156.30035
The author considers 1-parameter families of compact Riemann surfaces of genus \(g>1\). He reconstructs the Petersson-Weil metric on the corresponding Teichmüller space from the Hausdorff dimension of limit sets associated to such a family. He, moreover, establishes parallel results in complex dynamics for proper holomorphic selfmaps of the unit disk of degree \(>1\), leading, among other things, to a definition of a natural hermitian metric on the moduli space of such maps.
Reviewer: Ingo Lieb (Bonn)
MSC:
| 30F60 | Teichmüller theory for Riemann surfaces |
| 30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 37F30 | Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) |
| 30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |
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