• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Article

Morse-Smale systems with few non-wandering points

Topology and its Applications. 2013. Vol. 160. No. 3. P. 498-507.

 

Let MS^{t}(M^n,k)$ and $MS(M^n,k)$ be Morse-Smale flows and diffeomorphisms respectively the non-wandering set of those consists of $k$ fixed points on a closed $n$-manifold $M^n$. For $k=3$, we show that the only values of $n$ possible are $n\in\{2,4,8,16\}$, and $M^2$ is the projective plane. For $n\geq 4$, $M^n$ is simply connected and orientable. We prove that the closure of any separatrix of $f^t\in MS^{t}(M^n,3)$ is a locally flat n/2-sphere while there is $f^t\in MS^{t}(M^n,4)$ such that the closure of separatrix of $f^t$ is a wildly embedded codimension two sphere. This allows us to classify flows from $MS^{t}(M^4,3)$. For n>5, one proves that the closure of any separatrix of $f\in MS(M^n,3)$ is a locally flat n/2-sphere while there is $f\in MS(M^4,3)$ such that the closure of any separatrix is a wildly embedded 2-sphere.