Entropy for hyperbolic Riemann surface laminations I

TL;DR

This paper introduces a new entropy concept for hyperbolic Riemann surface laminations, demonstrating finiteness and regularity properties in compact, transversally smooth cases, and also defining a metric entropy for harmonic measures.

Contribution

It develops a novel entropy framework for hyperbolic Riemann surface laminations and establishes key regularity and finiteness results.

Findings

Entropy is finite for compact, transversally smooth laminations.

Poincare metric on leaves is transversally Holder continuous.

A metric entropy for harmonic measures is introduced.

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.