Central sequences in flows on 2-manifolds of finite genus

Abstract:

Let $\phi$ be a continuous flow on the metric space $X$ and let ${X^1},{X^2}, \ldots$ denote the “central” sequence of closed $\phi$-invariant subsets of $X$ obtained by iterating the process of taking nonwandering points of $\phi$. A. Schwartz and E. Thomas have proved that, if $X$ is an orientable $2$-manifold of finite genus, then this sequence can have not more than two distinct elements. We extend this result to include the nonorientable case; then this sequence can have at most three distinct elements. Analogous results are derived for the sequences obtained by iterating the processes of taking $\alpha$ and $\omega$ limit sets, or closures of $\alpha$ and $\omega$ limit sets.