On regular curve homeomorphisms without periodic points. (English) Zbl 1523.37015
Summary: The purpose of this paper is to study the dynamics of regular curve homeomorphisms without periodic points. We prove mainly that they are extensions of irrational rotations of the circle via a monotone factor map collapsing all proximal pairs and we prove also the absence of Li-Yorke pairs. Furthermore, we give a full characterization of minimal sets. In particular, we get that the circle is the only regular curve supporting a minimal (or a transitive) \(\mathbb{Z}\)-action. At the end of the paper, we give some counter-examples on rational curves.
MSC:
| 37B02 | Dynamics in general topological spaces |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37B45 | Continua theory in dynamics |
| 37E10 | Dynamical systems involving maps of the circle |
| 37E45 | Rotation numbers and vectors |
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