Let α be an irrational number and I an interval of R. If
α is diophantine, we show that any one-parameter group of homeomorphisms
of I whose time-1 and α maps are C∞ is in fact the flow of a
C∞ vector field. If α is Liouville on the other hand, we
construct a one-parameter group of homeomorphisms of I whose time-1 and
α maps are C∞ but which is not the flow of a C2 vector field
(though, if I has boundary, we explain that the hypotheses force it to be the
flow of a C1 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
(R,+) on I, these results follow from famous linearization
theorems for circle diffeomorphisms. The novelty of this work concerns non-free
actions.Comment: 35 pages, 8 figure