  We study circle actions with the Rokhlin property, in relation to their
restrictions to finite subgroups. We construct examples showing the following:
the restriction of a circle action with the Rokhlin property (even on a real
rank zero $C^*$-algebra), need not have the Rokhlin property; and even if every
restriction of a given circle action has the Rokhlin property, the circle
action itself need not have it. As a positive result, we show that the
restriction of a circle action with the Rokhlin property to the subgroup
$\mathbb{Z}_n$ has the Rokhlin property if the underlying algebra absorbs
$M_{n^\infty}$. The condition on the algebra is also necessary in most cases of
interest.
  Despite the fact that there are no circle actions with the Rokhlin property
on UHF-algebras, we construct many such actions on certain UHF-absorbing simple
AT-algebras. Additionally, we show that circle actions with the Rokhlin
property on $\mathcal{O}_2$-absorbing $C^*$-algebras are generic, in a suitable
sense.
