Classification of Morse--Smale Diffeomorphisms with a~Finite Set of Heteroclinic Orbits on 3-Manifolds

Ch. Bonatti ; V. Z. Grines ; O. V. Pochinka

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 250 (2005), pp. 5-53

Résumé

A topological classification is obtained for a certain class of Morse–Smale diffeomorphisms defined on a closed smooth orientable three-dimensional manifold $M$. The class $G$ of these diffeomorphisms is determined by the following conditions: the wandering set of each diffeomorphism $f\in G$ contains a finite number of heteroclinic orbits and does not contain heteroclinic curves. For a diffeomorphism $f\in G$, a complete topological invariant (a scheme $S(f)$) is introduced. In particular, this scheme describes the topological structure of the embedding of two-dimensional separatrices of saddle periodic points into an ambient manifold. Moreover, the realization problem is solved: for each abstract invariant (perfect scheme $S$), a representative $f_S$ of a class of topologically conjugate diffeomorphisms is constructed whose scheme is equivalent to the initial one.