Smooth times of a flow in dimension 1

Abstract

Let $\alpha$ be an irrational number and $I$ an interval of $\mathbb{R}$. If $\alpha$ is Diophantine, we show that any one-parameter group of homeomorphisms of $I$ whose time-$1$ and $\alpha$ maps are $C^{\infty }$ is in fact the flow of a $C^{\infty }$ vector field. If $\alpha$ is Liouville on the other hand, we construct a one-parameter group of homeomorphisms of $I$ whose time-$1$ and $\alpha$ maps are $C^{\infty }$ but which is not the flow of a $C^2$ vector field (though, if $I$ has boundary, we explain that the hypotheses force it to be the flow of a $C^1$ vector field). We extend both results to families of irrational numbers, the critical arithmetic condition in this case being simultaneous "diophantinity". For one-parameter groups defining a free action of $(\mathbb{R},+)$ on $I$, these results follow from famous linearization theorems for circle diffeomorphisms. The novelty of this work concerns non-free actions.

arXiv comment37 pages, 8 figures. This version takes into account requests from referees. Most changes concern the introduction and aim at clarifying the novelty of this work. To appear in Annales Scientifiques de l'ENS