ANALYTICITY AND METRIC TRANSITIVITY ON THE TORUS

Suppose we are given an analytic divergence free vector field (X, Y) o n the standard torus. We can find constants a and b sind a function F( x, y) of period one in both x and y such that (X, Y) = (a - F-y, b + F -x). For a given F, let P be the map sending (x, y) into (F-y(x, y), - F-x(x, y)). Let A be the image of the torus under this map and let B b e the image under this map of the set of points (x, y) at which FxxFyy - (F-xy)(2) vanishes. For any point (a, b) in the complement of the i nterior of A, the flow on the torus arising from the differential equa tions dx/dt = a - F-y(x, y), dy/dt = b + F-x(x, y) is metrically trans itive if and only if a/b is irrational. For any point in A but not in B the flow is not metrically transitive. Moreover, if alb is irrationa l but the flow on the torus is not metrically transitive and we use ou r differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It i s an open question whether a point (a, b) in the interior of A can giv e rise to a metrically transitive flow.