Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:
$\sigma_k: X \mapsto X + S + X^k$
$\sigma_k: S \mapsto S + S^3$.
Conjecture: There exists a principal ideal $\langle a\rangle$ other than $\langle S\rangle$ such that $\langle a\rangle$ is stable by $\sigma$. Namely $\sigma(a) = ab$ with some element $b$ of $\mathbb{F}_3[[X,S]]$.
$\mathbb{F}_3[[X,S]]$ is a $2$-variable power series ring over the finite field $\mathbb{F}_3$, but I guess $\langle a\rangle$ could be found somehow.
I tried to apply Brouwer's fixed point formula somehow, but already $\langle S\rangle$ occupies one fixed point in Brouwer's theorem.
Please just help me with any of your ideas.