On the work of Jorge Lewowicz on expansive systems

TL;DR

This paper reviews Jorge Lewowicz's significant classification of expansive homeomorphisms on surfaces, highlighting his innovative tools and the interplay between topology and dynamics that underpin his groundbreaking results.

Contribution

It presents Lewowicz's conceptual framework and approach, emphasizing the use of Lyapunov functions and persistence in classifying expansive surface homeomorphisms.

Findings

Classification of expansive homeomorphisms on surfaces

Introduction of Lyapunov functions and persistence concepts

Deep interaction between topology and dynamics

Abstract

We will try to give an overview of one of the landmark results of Jorge Lewowicz: his classification of expansive homeomorphisms of surfaces. The goal will be to present the main ideas with the hope of giving evidence of the deep and beautiful contributions he made to dynamical systems. We will avoid being technical and try to concentrate on the tools introduced by Lewowicz to obtain these classification results such as Lyapunov functions and the concept of persistence for dynamical systems. The main contribution that we will try to focus on is his conceptual framework and approach to mathematics reflected by the previously mentioned tools and fundamentally by the delicate interaction between topology and dynamics of expansive homeomorphisms of surfaces he discovered in order to establish his result.