Let be $f \in Diff(M)$. When is finite the subgroup $span\{f\}< Diff(M)$? What are implications in the structure of $f$ and $M$?



Let be $f \in Diff(M)$. What we can say about the subgroup $span{f}< Diff(M)$? What are implications in the structure of $f$ and $M$?
If $f$ generates a finite group, say of order $p$, then the action of $f$ can be linearized in a neighborhood of any fixed point, turning $f$ into an orthogonal matrix with eigenvalues that are all $p$-th roots of unity. So the dynamics are ``boring''. The action of $f$ on real homology must also be by diagonalizable linear maps whose eigenvalues are $p$-th roots of unity, so again boring. If $f$ generates an infinite group, the dynamics near a fixed point can be very complicated. 


