Suppose we are given an analytic divergence free vector field $(X,Y)$ on the standard torus. We can find constants $a$ and $b$ and 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 $B$ be the image under this map of the set of points $(x,y)$ at which $F_{xx}F_{yy}-(F_{xy})^2$ vanishes. For any point $(a,b)$ in the complement of the interior of $A$, the flow on the torus arising from the differential equations $dx/dt=a-F_y(x,y)$, $dy/dt=b+F_x(x,y)$ is metrically transitive 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 $a/b$ is irrational but the flow on the torus is not metrically transitive and we use our differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It is an open question whether a point $(a,b)$ in the interior of $A$ can give rise to a metrically transitive flow.
Discrete Vector Potential Representation of a Divergence-Free Vector Field in Three-Dimensional Domains: Numerical Analysis of a Model Problem
