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)?
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 $1$.
This seems fairly widely applicable--on many manifolds it's known that every Hamiltonian diffeomorphism has infinitely many periodic points (see for instance arXiv:1006.0372 and arXiv:0912.2064), and as noted above generic Hamiltonian diffeomorphisms have all their periodic points isolated.
A time-one map of a flow is homotopic (even isotopic) to the identity, so that provides a lot of homotopy and homology obstructions. In the two dimensional case, i.e., an oriented surface with a designated area element, you can get lots of explicit examples this way.
