Morse-Smale systems without heteroclinic submanifolds on codimension one separatrices

Abstract

We study a topological structure of a closed $n$-manifold $M^n$ ($n\geq 3$) which admits a Morse-Smale diffeomorphism such that codimension one separatrices of saddles periodic points have no heteroclinic intersections different from heteroclinic points. Also we consider gradient like flow on $M^n$ such that codimension one separatices of saddle singularities have no intersection at all. We show that $M^n$ is either an $n$-sphere $S^n$, or the connected sum of a finite number of copies of $S^{n-1}\otimes S^1$ and a finite number of special manifolds $N^n_i$ admitting polar Morse-Smale systems. Moreover, if some $N^n_i$ contains a single saddle, then $N^n_i$ is projective-like (in particular, $n\in\{4,8,16\}$, and $N^n_i$ is a simply-connected and orientable manifold). Given input dynamical data, one constructs a supporting manifold $M^n$. We give a formula relating the number of sinks, sources and saddle periodic points to the connected sum for $M^n$. As a consequence, we obtain conditions for the existence of heteroclinic intersections for Morse-Smale diffeomorphisms and a periodic trajectory for Morse-Smale flows.