A complete topological classification of Morse-Smale diffeomorphisms on surfaces: a kind of kneading theory in dimension two

TL;DR

This paper provides a comprehensive topological classification of Morse-Smale diffeomorphisms on surfaces, introducing a finite data set that uniquely determines their conjugacy classes, thus extending and completing previous classification methods.

Contribution

It introduces a new finite data framework that fully classifies Morse-Smale surface diffeomorphisms topologically, establishing a one-to-one correspondence with conjugacy classes.

Findings

Complete classification via finite data set

Existence of unique conjugacy class for each data set

New proof that nearby MS diffeomorphisms are topologically conjugate

Abstract

In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a scheme. Here we will assign to any MS surface diffeomorphism a finite amount of data which completely determines its topological conjugacy class. Moreover, we show that associated to any abstract version of this data, there exists a unique conjugacy class of MS orientation preserving diffeomorphisms (on some orientation preserving surface). As a corollary we obtain a different proof that nearby MS diffeomorphisms are topologically conjugate.