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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)-action. At the end of the paper, we give some counter-examples on rational curves.

On regular curve homeomorphisms without periodic points