Notes on Chain Recurrence and Lyapunonv Functions

TL;DR

This paper introduces chain recurrence in topological dynamics and proves the existence of complete Lyapunov functions for homeomorphisms on compact metric spaces, serving as educational supplementary material.

Contribution

It provides an accessible introduction to chain recurrence and offers a proof of the existence of Lyapunov functions, enhancing understanding in introductory dynamics.

Findings

Proof of existence of complete Lyapunov functions for homeomorphisms

Introduction to chain recurrence in topological dynamics

Educational supplementary material for dynamics courses

Abstract

This short expository note provides an introduction to the concept of chain recurrence in topological dynamics and a proof of the existence complete Lyapunov functions for homeomorphisms of compact metric spaces due to Charles Conley. I have used it as supplementary material in introductory dynamics courses.