Let $(M^{2n}, \omega)$ be a compact symplectic manifold. Suppose $\phi$ is a Hamiltonian diffeomorphism of $M$. In other words there exists a one-parameter family of smooth functions $H_t : M \to \mathbb R, 0 \leq t \leq 1$ such that if we define the vector fields $X_{H_t}$ via $d H_t = X_{H_t} \lrcorner \omega$, and we define a one-parameter family of diffeomorphisms $\psi_t$ via

$\frac{\partial \psi}{\partial t} = X_{H_t}\\ \psi_0 = \mbox{Id}$,

then $\phi = \psi_1$. My understanding is that it is a theorem of Banyaga that there exists a single smooth function $f$ such that $\phi$ can be expressed as the time $1$ flow of the vector field $X_f$ defined as above via $d f = X_f \lrcorner \omega$.

My question is: is this correct, and if so can you give a precise reference? Moreover, is it possible to explicitly construct the requisite function $f$ from the one-parameter family $H_t$?

generatedby autonomous Hamiltonians. There's no reason to believe the product is autonomous. Then the interesting result is that you can express any Hamiltonian diffeomorphism as the composition of finitely many autonomous Hamiltonian diffeomorphisms. $\endgroup$ – Mike Miller Nov 25 '16 at 1:11