Thermodynamics, dimension and the Weil-Petersson metric (Q938287)

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.