# Smooth times of a flow in dimension 1

## TL;DR

This paper investigates when one-parameter groups of homeomorphisms in one dimension are smooth flows, revealing that Diophantine conditions ensure smoothness, while Liouville conditions can produce non-smooth examples, extending to families of irrationals.

## Contribution

It characterizes the smoothness of flows generated by irrational rotations, especially distinguishing Diophantine and Liouville cases, and extends results to families of irrationals with simultaneous Diophantinity.

## Key findings

- Diophantine irrationals guarantee the flow is smooth.
- Liouville irrationals can produce non-smooth flows.
- Results extend to families of irrationals with simultaneous Diophantinity.

## 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.