Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure?


I've seen examples of higher-dimensional foliations not admitting transverse invariant measures, but I'd imagine the same question is much easier to address one way of the other in the one-dimensional case.



Transverse invariant measures to vector fields
I think that the best reference for this question is still the (relatively) old paper by Plante *Foliations with measure preserving holonomy* Ann. of Math. (2) 102 (1975), no. 2, 327–361, although it is a bit of an overkill for one-dimensional foliations. For instance, by Theorem 4.1 holonomy invariant measures exist for any foliation with a subexponential leaf. There are many other ways, of course.


