# Special Hamiltonian diffeomorphisms

Is there any obstruction that prevents a Hamiltonian diffeomorphism on some symplectic manifold to be realized as the time-one map of the Hamiltonian flow of an autonomous Hamiltonian?

In the same spirit, is there any obstruction that prevents a Hamiltonian diffeomorphism on $T^*M$ (with the canonical symplectic structure) to be realized as the time-one map of the Hamiltonian flow of a Tonelli Hamiltonian (i.e. a Hamiltonian which is fiberwise convex and superlinear)?

• I'm confused, what definition of Hamiltonian diffeomorphism are you using? I've seen them defined to be the time-one map of a Hamiltonian flow in Banyaga's on-line lecture notes. – Ryan Budney Oct 4 '10 at 23:12
• @Ryan: could you please give a link for the notes? I can't find anything like that on the Banyaga's home page. Thanks in advance! – mathphysicist Oct 4 '10 at 23:38
• Oh, it's probably because they're not on his web-page, sorry if I was confusing: math.psu.edu/wade/dakar.pdf – Ryan Budney Oct 4 '10 at 23:41
• @Ryan: Note the word "autonomous" in the question. The standard definition of a Hamiltonian diffeomorphism involves a Hamiltonian function that may depend on time--he's asking if/how you can tell that a given Hamiltonian diffeo can be generated by a time-independent Hamiltonian. – Mike Usher Oct 5 '10 at 0:37
• Banyaga's work has some relevance here. He showed that for closed symplectic manifolds, the Hamiltonian group is simple. Yet the subgroup generated by the autonomous Hamiltonians is normal. Hence every Hamiltonian diffeo is a composite of autonomous Hamiltonian diffeos. – Tim Perutz Oct 5 '10 at 16:11

Can't you get such an obstruction by looking at periodic points? It's not difficult to show that generic Hamiltonian diffeomorphisms have only isolated fixed points, and indeed only isolated periodic points. On the other hand, if $\phi$ is a Hamiltonian diffeomorphism generated by an autonomous Hamiltonian $H$ (so $\phi^{\circ k}$ is generated by $kH$), then fixed points of $\phi^{\circ k}$ come in two types: either critical points of $H$ (which are also fixed points of $\phi$), or else points lying on nontrivial $k$-periodic orbits of the Hamiltonian vector field, and these latter fixed points are non-isolated (since every other point on the orbit is fixed as well).
Hence, given $\phi$, you can tell that $\phi$ isn't generated by an autonomous Hamiltonian if you find an isolated periodic point of $\phi$ whose minimal period is larger than $1$.