Consider the automorphism $\sigma$ on ${\Bbb F}_3[[T]]$ such that $T \mapsto c_1T + f(T)$ with $c_1 = 1$ or $-1$, and $f(T) \not=0$ and the non-zero leading term $c_mT^m$ of $f(T)$ satisfies $m \geq 2$.

Question: Is there any fixed element $t \in {\Bbb F}_3[[T]]$ other than those in the constant field ${\Bbb F}_3$? Namely does such $t$ exist as $\sigma(t) = t$ but $t \notin {\Bbb F}_3$?

Pierre