  We describe a method for constructing Teichm\"uller geodesics where the
vertical measured foliation $\nu$ is minimal but is not uniquely ergodic and
where we have a good understanding of the behavior of the Teichm\"uller
geodesic. The construction depends on various parameters, and we show that one
can adjust the parameters to ensure that the set of accumulation points of such
a geodesic in the Thurston boundary is exactly the set of all possible measured
foliations in the homotopy class of $\nu$. With further adjustment of the
parameters, one can even take $\nu$ to be an ergodic measure on a non-uniquely
ergodic foliation.
