TITLE: Does the Baker-Campbell-Hausdorff formula hold for vector fields on a (compact) manifold?
QUESTION [13 upvotes]: 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 Xf and Xg directions; it is not clear to me that the coefficients in the BCH formula make the series converge (uniformly, say) for any choice of f and g.

REPLY [2 votes]: The BCH formula holds in a finite-dimensional Lie algebra for $X,Y$ in some open neighborhood of $0$.  Therefore, the coefficients of BCH are bounded by $a^n$ for some $0&lt;a&lt;\infty$.  So suppose that $X,Y$ are vector fields for which degree-$n$ bracket monomials (things like $[X,[X,Y]]$, with $n$ many $X$s and $Y$s total) don't grow too fast in your topology — not faster than $b^n$ for some $0 &lt; b &lt; \infty$.  Then for $s &lt; b/a$, ${\rm BCH}(sX,sY)$ converges, in your topology, if your topology is complete.
Now, I'm not much of one for infinite-dimensional topological vector spaces, so I can't say which topologies are great to use.