Entropy for hyperbolic Riemann surface laminations I

Abstract

We develop a notion of entropy, using hyperbolic time, for laminations by hyperbolic Riemann surfaces. When the lamination is compact and transversally smooth, we show that the entropy is finite and the Poincare metric on leaves is transversally Holder continuous. A notion of metric entropy is also introduced for harmonic measures.

27 pages, Part 1. The article is adapted for the use we need in the second part of our study of hyperbolic entropy for singular foliations