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

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)
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