Again a very simple question. I currently hold two contradictory ideas in my head


1) A hamiltonian diffeomorphism of a torus necessarily has fixed points


2) most hamiltonian actions on a torus in an integrable system result in quasi-periodic (and, in particular, not periodic) orbits


But fixed points of the diffeomorphism generated by the Hamiltonian should result in periodic orbits. I have misunderstood one situation or the other, would anyone clarify? Thanks 



Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus
1) is correct: by saying “hamiltonian diffeomorphism of” you imply that the torus has a symplectic structure $\omega$ and the diffeo is (something like) time 1 flow of a hamiltonian vector field $X$: $\omega(X,\cdot)=-dH$. As the torus is compact, $H$ has extrema, so $X$ has zeros, which the flow fixes.


2) is not: the flow of an integrable system is quasi-periodic on tori, but those are not symplectic, they are the lagrangian level sets (assumed compact) of “action” coordinates $I\_i$ on a larger ambient symplectic manifold. Only on the latter is the flow *hamiltonian* (with $H$ a function of the $I\_i$).


