I don't know. Is there any use for such a function? Do you know an $F$ such that you get a polynomial of degree at least 2 with a real fixed point?
– Gerry MyersonNov 7 '10 at 4:10

Ya one gets the Chebychev polynomials $T_n$ of degree $n$ with $F(x)=\cos(n^x)$. For example $\cos(2^{1+log_2(\arccos(x))})=\cos(2 \arccos(x))=2 x^2-1 = T_2(x)$.
– bo198214Nov 7 '10 at 4:43

And I forgot to mention: All Chebychev polynomials have fixpoint 1.
– bo198214Nov 7 '10 at 12:07