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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

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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 $I_i$).

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    $\begingroup$ The Hamiltonian diffeomorphism in (1) associated with a single function is called autonomous and, as Francois explained, has obvious fixed points. In general, the existence of fixed points is true but nontrivial to prove. $\endgroup$
    – Jarek Kędra
    Commented Jul 10, 2017 at 19:44

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