Ergodic Theory and Dynamical Systems

Research Article

Gradient-like flows on 3-manifolds

K. A. de Rezendea1

a1 Departamento de Matemática, Universidade Estadual de Campinas, 13081–970 Campinas, Sāo Paulo, Brazil


In this paper, we determine properties that a Lyapunov graph must satisfy for it to be associated with a gradient-like flow on a closed orientable three-manifold. We also address the question of the realization of abstract Lyapunov graphs as gradient-like flows on three-manifolds and as a byproduct we prove a partial converse to the theorem which states the Morse inequalities for closed orientable three-manifolds. We also present cancellation theorems of non-degenerate critical points for flows which arise as realizations of canonical abstract Lyapunov graphs.

(Received January 02 1992)

(Revised July 10 1992)


† Supported in part by the National Science Foundation under Grant DMS 8704877 and by the Conselho Nacional de Desenvolvimento Cientifico e Tecnológico under Grant 300072/90.2.