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*.



Does the Baker-Campbell-Hausdorff formula hold for vector fields on a (compact) manifold?
To answer your first question, the composition of two time-1 flows won't necessarily be another time-1 flow.


One way to see this is to note that when a time-1 flow $\phi\_X$ has a periodic point $P$ (period > 1), then $P$ can't be hyperbolic since it lies on a closed orbit of the flow for $X$. (The eigenvector of $D\phi\_X$ tangent to this orbit has corresponding eigenvalue 1.)


Now, take a flow on $S^2$ whose time-1 map rotates the sphere, switching the north and south poles. Take a second flow for which both poles are hyperbolic attracting fixed points. Composing the two time-1 maps gives you a new diffeomorphism with hyperbolic points of period 2.


