What are Lyapunov exponents, and why are they interesting?

TL;DR

This paper introduces Lyapunov exponents, explaining their significance across ergodic theory, Teichmüller theory, and spectral theory, highlighting their role in understanding complex dynamical systems.

Contribution

It provides an accessible exposition of Lyapunov exponents and discusses their applications in three advanced mathematical fields, inspired by Artur Avila's influential work.

Findings

Lyapunov exponents are crucial in understanding stability in dynamical systems.

The paper connects Lyapunov exponents to recent advances in Teichmüller and spectral theories.

It emphasizes the importance of Lyapunov exponents in various mathematical contexts.

Abstract

This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichm\"uller theory, and the spectral theory of one-frequency Schr\"odinger operators. The inspiration for this paper is the work of 2014 Fields Medalist Artur Avila, and his work in these areas is given special attention.