Let $k$ be a field of characteristic zero and let $p=p(x,y),q=q(x,y) \in k[x,y]$. Let $F: k[x,y] \to k[x,y]$ be defined by $F: (x,y) \mapsto (p,q)$ and let $f: k^2 \to k^2$ be its corresponding map, namely the map defined by $f: (a,b) \mapsto (p(a,b),q(a,b))$, $a,b \in k$.
What is the connection between (non-constant) fixed points of $F$ and fixed points of $f$?
BTW, if, in addition, the Jacobian of $p$ and $q$, $p_xq_y-p_yq_x$, is a non-zero constant, then there is a nice result that if $F$ has a fixed point, then $F$ is an automorphism, see Kraft's paper and Shpilrain and yu's paper.
Any comments are welcome!