Ergodic theory for Riemann surface laminations: a survey

TL;DR

This survey reviews recent advances in ergodic theory applied to hyperbolic Riemann surface laminations, highlighting similarities and differences with ergodic theory of maps, especially in the context of singular holomorphic foliations.

Contribution

It compiles and discusses recent developments in ergodic theory for Riemann surface laminations, emphasizing the role of singular holomorphic foliations and their unique features.

Findings

Shows strong analogy between map ergodic theory and Riemann surface laminations.

Highlights fundamental differences between these two ergodic theories.

Provides insights into the ergodic properties of singular holomorphic foliations.

Abstract

We survey some recent developments in the ergodic theory for hyperbolic Riemann surface laminations. The emphasis is on singular holomorphic foliations. These results not only illustrate the strong similarity between the ergodic theory of maps and that of Riemann surface laminations, but also indicate the fundamental differences between these two theories.