Strictly invariant sets of the rotation transformation on a discrete space.

Question

Fix an integer $n>0$. Consider the space $X=\{a_0,a_1,...,a_{n-1}\}$ with transformation $T:X\to X$ defined by $T(a_i)=a_{i+1(\text{ mod n})}$.

What are the strictly invariant sets of this space? (i.e the $A\subset X$ with $T^{-1}A=A$.)

I think the only strictly invariant sets are $X$ and $\emptyset$, but I am having some trouble proving this claim.

Answer 1

Let $A$ be a non-empty invariant set. If $a_j\in A$, then $a_j\in T^{-1}(A)\cap T(A)$, hence $a_{j+1\mod n}$ and $a_{j-1\mod n}$ also are in $A$. This proves that $A=X$.