Stabilizers and orbits of circle-valued smooth functions
  Let $M$ be a smooth compact manifold and $P$ be either $R^1$ or $S^1$. There is a natural action of the groups $Diff(M)$ and $Diff(M) \times Diff(P)$ on the space of smooth mappings $C^{\infty}(M,P)$. For $f\in C^{\infty}(M,P)$ let $S_f$, $S_{MP}$, $O_f$, and $O_{MP}$ be the stabilizers and orbits of $f$ under these actions. Recently, the author proved that under mild conditions on $f\in C^{\infty}(M,R^1)$ the corresponding stabilizers and orbits are homotopy equivalent: $S_{MR} \sim S_{M}$ and $O_{MR} \sim O_M$. These results are extended here to the actions on $C^{\infty}(M,S^1)$. It is proved that under the similar conditions (that are rather typical) we have that $S_{MS}\sim S_M$ and $O_{MS} \sim O_M \times S^1$. 
Sergey Maksymenko