  We study the coarse geometry of the moduli space of dilation tori with two
singularities and the dynamical properties of the action of the Teichmuller
flow on this moduli space. This leads to a proof that the vertical foliation of
a dilation torus is almost always Morse-Smale. As a corollary, we get that the
generic piecewise affine circle homeomorphism with two break points -with
respect to the Lebesgue measure- is Morse-Smale.
