Sets of simply-invariance

Let $X$ be a space of smooth functions on the unit circle $\mathbb T$. Suppose that the operator of multiplication by $z$ is invertible on $X$. A closed set $E$, $E\subset\mathbb T$, is (by definition) the set of simply-invariance for the space $X$ if there exists a function $f$, $f\in X$, such that $f|_E\equiv0$ and $z^{-1}\not\in\operatorname{span}\{z^nf:n\ge0\}$, It is proved that the class of sets of simply-invariance for the spaces $C^n$, $W_p^n$ ($p<\infty$), $\lambda_\omega^n$, coincides with the class of sets of zero Lebesgue measure, for the space $C^\infty$, with the class of Carleson sets, for the space $\Lambda_\omega^n$ with the class of all nowhere dense closed sets. Some related problems are also considered.