We study dynamical properties of direction foliations on the complex plane pulled back from
direction foliations on a half-translation torus $T$, i.e., a torus equipped with a strict
and integrable quadratic differential.
If the torus $T$ admits a pseudo-Anosov map we give a homological criterion for the appearance of dense leaves and leaves with bounded deviation on the universal covering of $T$, called Panov plane.
Our result generalizes Dmitri Panov's explicit construction of dense leaves for certain
arithmetic half-translation tori . Certain Panov planes are related to
the polygonal table of the periodic wind-tree model. In fact, we show that the dynamics
on periodic wind-tree billiards can be converted to the dynamics on a pair of singular planes.
Mathematics Subject Classification: 14H15, 14H52, 30F30, 30F60, 37A60, 37C35, 37N20, 58D15, 58D27.
Received: March 2012; Revised: March 2014; Available Online: May 2014.