Global attractor and repeller of Morse--Smale diffeomorphisms

V. Z. Grines ; E. V. Zhuzhoma ; V. S. Medvedev ; O. V. Pochinka

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. II, Tome 271 (2010), pp. 111-133

Résumé

Let $f$ be an orientation-preserving Morse–Smale diffeomorphism of an $n$-dimensional ($n\ge3$) closed orientable manifold $M^n$. We show the possibility of representing the dynamics of $f$ in a “source–sink” form. The roles of the “source” and “sink” are played by invariant closed sets one of which, $A_f$, is an attractor, and the other, $R_f$, is a repeller. Such a representation reveals new topological invariants that describe the embedding (possibly, wild) of stable and unstable manifolds of saddle periodic points in the ambient manifold. These invariants have allowed us to obtain a classification of substantial classes of Morse–Smale diffeomorphisms on 3-manifolds. In this paper, for any $n\ge3$, we describe the topological structure of the sets $A_f$ and $R_f$ and of the space of orbits that belong to the set $M^n\setminus(A_f\cup R_f)$.