On Morse--Smale Diffeomorphisms without Heteroclinic Intersections on Three-Manifolds

Ch. Bonatti ; V. Z. Grines ; V. S. Medvedev ; E. Peku

Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 66-78

Résumé

A class of Morse–Smale diffeomorphisms is considered that do not admit heteroclinic intersections and are defined on three-manifolds. To each diffeomorphism $f$, we associate an enriched graph $G(f)$ and, for each sink $\omega$, we define a scheme $S(\omega )$ which is a link of tori, the Klein bottle, and simple closed curves embedded in $S^2\times S^1$. We show that diffeomorphisms $f_1$ and $f_2$ are topologically conjugate if and only if (1) the corresponding graphs $G(f_1)$ and $G(f_2)$ are isomorphic and the permutations induced by the dynamics $f_1$ and $f_2$ on the vertices and edges of the graphs are conjugate; (2) two sinks corresponding to isomorphic vertices have equivalent schemes; and (3) for any two saddles corresponding to isomorphic vertices and having one-dimensional unstable manifolds, the corresponding pairs of curves in $S^2\times S^1$ associated with the one-dimensional separatrices are concordantly embedded.