  We introduce a method for constructing invariant probability measures of a
large class of non-singular volume-preserving flows on closed, oriented
odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from
symplectic geometry. These flows include any non-singular volume preserving
flow in dimension three, and autonomous Hamiltonian flows on closed, regular
energy levels in symplectic manifolds of any dimension. As an application, we
use our method to prove the existence of obstructions to unique ergodicity for
this class of flows, generalizing results of Taubes and Ginzburg-Niche.
