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Dec 17, 2013 at 1:45 comment added Michael Zieve Just to be clear: this answer is an augmentation of item (2) from my answer, namely, that if $\sigma$ has finite order then it has fixed points outside $\mathbf{F}_3$. What is new here is the fact that the fixed point described in my answer generates all fixed points, in the sense that every fixed point is a formal power series in "my" fixed point. On the other hand, this answer doesn't address infinite-order $\sigma$'s.
Dec 16, 2013 at 19:03 history edited Lubin CC BY-SA 3.0
Expanded my response slightly.
Dec 16, 2013 at 18:55 history answered Lubin CC BY-SA 3.0