The structure of orbits in dynamical systems. (English) Zbl 0664.54026

Aarts, J. M.

Fundam. Math. 129, No. 1, 39-58 (1988).

The orbits which are not locally are Poisson-stable and aperiodic. The topological structure of such orbits is studied in this paper. The following are the main results.
Theorem. Let \(\pi\) : \(X\times {\mathbb{R}}\to X\) be a flow. Then each orbit of the flow \(\pi\) is homeomorphic to one of the following (1) a singleton, (2) the unit circle \(S^ 1\), (3) the reals \({\mathbb{R}}\), or (4) an orientable P-manifold. Here a P-manifold is defined to be the space which is locally homeomorphic to \({\mathbb{Q}}\times {\mathbb{R}}\) and arcwise connected. Theorem. A space X is homeomorphic to the orbit of an aperiodic and Poisson-stable motion in some flow if and only if X is an orientable P- manifold. Theorem. Let \(\pi\) : \(X\times {\mathbb{R}}\to X\) be a flow. The motion \(\pi_ X\) is aperiodic and Poisson-stable if and only if \(\pi_ X\) is topologically equivalent to the suspension of a discrete system (Q,h), where h is a universally transitive homeomorphism.

Reviewer: S.Kono

MSC:

54H20

Topological dynamics (MSC2010)

Keywords:

orbits; aperiodic and Poisson-stable motion; orientable P-manifold; suspension of a discrete system; universally transitive homeomorphism