Pseudo-Anosov eigenfoliations on Panov planes

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 [33]. 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.   Possible strategies to generalize our main dynamical result to larger sets of directions are discussed. Particularly we include recent results of Frączek and Ulcigrai [17, 18] and Delecroix [6] for the wind-tree model. Implicitly Panov planes appear in Frączek and Schmoll [15], where the authors consider Eaton Lens distributions.