Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

Viatcheslav Grines (1) , Francois Laudenbach (2) , Olga Pochinka (1)

1 Department of mathematics
2 LMJL - Laboratoire de Mathématiques Jean Leray

This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from \cite{GrLaPo}, \cite{GrLaPo1}, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well dynamics of diffeomorphism. The paper is devoted to finding conditions to the existence of such an energy function, that is, a function whose set of critical points coincides with the non-wandering set of the considered diffeomorphism. We show that the necessary and sufficient conditions to the existence of a dynamically ordered energy function reduces to the type of embedding of one-dimensional attractors and repellers of a given Morse-Smale diffeomorphism on a closed 3-manifold.

Mots clés: energy function; Morse-Smale diffeomorphism; Morse-Lyapunov function; energy function.

Domaines: Topologie géométrique [math.GT]

Viatcheslav Grines, Francois Laudenbach, Olga Pochinka. Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds. Proceedings of the Steklov Institute of Mathematics, 2012, 278, pp.1-14. ⟨hal-00561091v2⟩