Is there a proof that no continuous map $f:S^2\to S^2$ can be minimal (=each orbit is dense)? Where can I find it?
I know recent works characterizing the 2-manifolds admitting minimal maps, but I'm searching for a hopefully simple proof just for the case of $S^2$.
I finally found it in the book of Dugundji and Granas: Fixed Point Theory, the following way (here $d(f)$ denotes Brouwer's degree):
Proposition 5.2: If $f,g \in C(S^{n},S^{n})$ then $d(f\circ g)=d(f)d(g)$
Theorem 5.4: If $f \in C(S^{n},S^{n})$ and $d(f) \not= (-1)^{n+1}$, then $f$ has a fixed point.
Thus if $f \in C(S^{2n},S^{2n})$, then $d(f\circ f) =d(f)^2 \not= -1=(-1)^{2n+1}$, so $f\circ f$ has a fixed point.