Bers and H\'enon, Painlev\'e and Schroedinger

TL;DR

This paper explores the dynamics of mapping class groups on character varieties, revealing similarities with Hénon mappings, and applies these insights to problems in geometry, Painlevé equations, and Schrödinger operators.

Contribution

It establishes a novel analogy between pseudo-Anosov dynamics and Hénon mappings, advancing understanding in geometric and integrable systems.

Findings

Pseudo-Anosov dynamics resemble Hénon mappings

Applied analogy to open questions in geometry and differential equations

Provided new insights into discrete Schrödinger operators

Abstract

We study the dynamics of mapping class groups on 2-dimensional character varieties. We shall show that the dynamics of pseudo-Anosov mapping classes resembles in many ways the dynamics of H\'enon mappings, and then apply this idea to answer open questions concerning the geometry of discrete and faithful representations, Painlev\'e sixth equation, and discrete Schroedinger operators.