Invariant probability measures from pseudoholomorphic curves i

We introduce a method for constructing invariant probability mea-sures of a large class of non-singular volume-preserving flows on closed, ori-ented 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 dimen-sion. As an application, we use our method to prove the existence of obstruc-tions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg-Niche.