Consider a compact manifold *M*. For a vector field *X* on *M*, let $\phi_X$ denote the diffeomorphism of *M* given by the time 1 flow of *X*.

If *X* and *Y* are two vector fields, is $\phi_X \circ \phi_Y$ necessarily of the form $\phi_Z$ for some vector field *Z*?

Since $X\mapsto \phi_X$ can be thought of as the exponential map from the Lie algebra of vector fields to the group of diffeomorphisms, an obvious candidate is that *Z* should be given by the Baker-Campbell-Hausdorff formula $B(X, Y) = X+Y+\frac{1}{2}[X,Y]+\cdots$. But does this hold in this infinite-dimensional setting? If so, in which sense does the series converge to *Z*?

Also, I'm interested in the case where *M* is a symplectic manifold and we consider only symplectic vector fields (ie. vector fields for which the contraction with the symplectic form is a closed 1-form). Locally, *X* and *Y* are the Hamiltonian vector fields associated to smooth functions *f* and *g*, so I assume that asking whether *B(X, Y)* makes sense/is symplectic corresponds to asking whether *B(f, g)* makes sense/defines a smooth function (where, of course, we use the Poisson bracket in the expansion of *B(f, g)*). The right-hand side of *B(f,g)* consists of lots of iterated directional derivatives of *f* and *g* in the *X _{f}* and

*X*directions; it is not clear to me that the coefficients in the BCH formula make the series converge (uniformly, say) for any choice of

_{g}*f*and

*g*.