Morse-Smale systems without heteroclinic submanifolds on codimension one separatrices

TL;DR

This paper classifies certain Morse-Smale systems on closed manifolds with restricted heteroclinic intersections, revealing the topological structure of the underlying manifolds and conditions for heteroclinic connections.

Contribution

It provides a topological classification of manifolds supporting Morse-Smale systems with no heteroclinic submanifolds on codimension one separatrices, including explicit construction methods.

Findings

Manifolds are either spheres or connected sums involving spheres and special manifolds.

Characterization of manifolds with a single saddle as projective-like in specific dimensions.

Derived formulas relate dynamical features to manifold topology and heteroclinic intersection conditions.

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.