Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Pseudo-Anosov eigenfoliations on Panov planes

Pages: 89 - 108, January 2014      doi:10.3934/era.2014.21.89

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Chris Johnson - Clemson University, E-1b Martin Hall, Clemson, SC 29634, United States (email)
Martin Schmoll - Clemson University, O-229 Martin Hall, Clemson, SC 29634, United States (email)

Abstract: 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.

Mathematics Subject Classification:  14H15, 14H52, 30F30, 30F60, 37A60, 37C35, 37N20, 58D15, 58D27.

Received: March 2012;      Revised: March 2014;      Available Online: May 2014.