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 $O_{ MS} \sim O_M \times S^1$.
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