  For a product of i.i.d. random maps or a memoryless stochastic flow on a
compact space $X$, we find conditions under which the presence of locally
asymptotically stable trajectories (e.g. as given by negative Lyapunov
exponents) implies almost-sure mutual convergence of any given pair of
trajectories ("synchronisation"). Namely, we find that synchronisation occurs
and is stable if and only if the system exhibits the following properties: (i)
there is a smallest deterministic invariant set $K \subset X$, (ii) any two
points in $K$ are capable of being moved closer together, and (iii) $K$ admits
asymptotically stable trajectories. Our first condition (for which unique
ergodicity of the one-point transition probabilities is sufficient) replaces
the intricate vector field conditions assumed in Baxendale's similar result of
1991, where (working on a compact manifold) sufficient conditions are given for
synchronisation to occur in a SDE with negative Lyapunov exponents.
