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

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.

On the realisability of Smale orders

“…From our point of view, not only does the answer to this question and the techniques developed in order to acquire it complement the ongoing classification of stable diffeomorphisms in dimension 2 (see [BL] for the periodic attractors case, [GPS2] for the Morse-Smale case, [GPS1], [MMP]), but they also establish a bridge between the combinatorics of the dynamics and the topology of the underlying space. A similar bridge has been constructed for gradient-like diffeomorphisms in dimension 3 in [BP], but has yet to be established for general Morse-Smale diffeomorphisms in dimension 3, whose classification was recently announced in [BGP].…”