A topological characterization of periodic flows. (English) Zbl 1519.37008
Summary: Let \(G = \{h_t \, | \, t \in \mathbb{R}\}\) be a continuous flow of homeomorphisms of a connected \(n\)-manifold \(M\). The flow \(G\) is called periodic if: for some real \(s>0\), \(h_s=identity\). A global section for a flow \(G\) is a closed subset \(K\) of \(M\) such that every orbit under \(G\) intersects \(K\) in exactly one point. In this paper, we give a topological characterization of periodic flows with global sections for \(M = \mathbb{R}^n\). Next, we consider periodic flows defined on any connected \(n\)-manifold \(M\), and we give a similar local characterization.
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37C27 | Periodic orbits of vector fields and flows |
| 57S05 | Topological properties of groups of homeomorphisms or diffeomorphisms |
| 57S10 | Compact groups of homeomorphisms |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
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