Completely unstable dynamical systems

Abstract:

We associate with the ${C^r} (r \ge 1)$ dynamical system $\phi$ on an $m$-manifold $M$, the orbit space $M/\phi$, defined to be the set of orbits of $\phi$ with the quotient topology. If $\phi$ is completely unstable, $M/\phi$ turns out to be a ${C^r} (m - 1)$-nonseparated manifold. It is known that for a completely unstable flow $\phi$ on a contractible manifold $M, M/\phi$ is Hausdorff if and only if $\phi$ is parallelizable. In general, we place an order on the non-Hausdorff points of $M/\phi$ (essentially) by setting $\bar p < \bar q$ if and only if ${\pi ^{ - 1}}(\bar q) \subseteq {J^ + }({\pi ^{ - 1}}(\bar p))$. Our result is that $(M, \phi )$ is topologically equivalent to $(M’, \phi ’)$ if and only if $M/\phi$ is order isomorphic to $M’/\phi ’$.