Are there examples of diffeomorphisms (or homeomorphisms) $f:S^2\rightarrow S^2$ on the 2-sphere which have an odd number of critical points and thus a finite, odd-size set of non-wandering points?
I have seen the classical examples using a linear map $f(x)=Ax/||Ax||$ given by a $3\times 3$ matrix $A$, but there each eigenspace produces a pair of critical points.
Is there a way to "compactify" a homeomorphism $g:{\mathbb R}^2\rightarrow {\mathbb R}^2$ having a saddle point to get such examples?
What other examples are there? Is there a restriction on the cardinality of critical points and/or non-wandering points on the 2-sphere (the 2-torus or other orientable surfaces)?
Edit: A point $x\in X$ is called a non-wandering point, if for every open neighborhood $U$ of $x$ and every $N\in{\mathbb N}$ there exists $n>N$ such that $f^n(U)\cap U\neq \emptyset$. Periodic points are non-wandering.