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A characterization of the standard Reeb flow. (English) Zbl 1296.37034
This article is devoted to a characterization of the conjugation class of the Reeb flow among all possible flows topologically equivalent to it. The Reeb flow is example of a planar flow without equilibria, not conjugated to a translation. A flow, topologically conjugated to it can be made by: take two translations in opposite directions, along two disjoint and parallel half-planes; on the middle strip, any flow topologically conjugated to that given by the equations $$x '= x$$, $$y' = -y$$ considered in $$[0, \infty) ^ 2 - \{(0,0)\}$$. Two flows are topologically equivalent if there is a homeomorphism taking orbits in orbits, keeping the orientation but not necessarily the temporal evolution. The author shows that a flow topologically equivalent to the Reeb flow, is also topologically conjugated to it, if and only if $$\phi^t$$ and $$\phi^{\lambda t}$$ are topologically conjugated for any positive $$\lambda$$. The author also shows an example in which this this condition is fulfilled for a single value of $$\lambda$$ and the flow is not topologically conjugated to the Reeb flow. The proofs uses a characterization of the topological conjugation class of flows that are topologically equivalent to Reeb flow, given by F. Le Roux [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 1, 45–50 (1999; Zbl 0922.58069)].

##### MSC:
 37E35 Flows on surfaces 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
##### Keywords:
Reeb Flow; topological conjugacy; topological equivalence
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