Involutive translation surfaces and Panov planes

This dissertation is concerned with the study of Panov planes and involutive translation surfaces, motivated by questions encountered in trying to understand certain self-similar billiard trajectories in the periodic variant of the Ehrenfest wind-tree model. In particular, we outline a new approach for studying billiard trajectories in certain types of infinite billiard tables by using Panov planes. After describing how this is done in the special case of the wind-tree model, we generalize our construction to show that there are typically several Panov planes that may be associated to an involutive translation surface.;The first three chapters of the dissertation provide a brief introduction to the theory of half-translation surfaces, and are included for the convenience of the reader that may not already be familiar with this theory. The fourth chapter recalls the original example of Dmitri Panov and then generalizes this example, in particular providing criteria for the existence of a foliation of the plane with dense leaves. The fifth chapter applies Panov planes to study an infinite billiard trajectory in the Ehrenfest wind-tree model, and also explains the "self-similarity" exhibited by billiard trajectories in the eigendirection of a pseudo-Anosov map on the L-shaped surface associated to the wind-tree. The sixth chapter generalizes the relationship between Panov planes and the wind-tree model by studying involutive surfaces, particularly tori related to these surfaces by a cover-quotient relation. There is one appendix which presents two particular examples of the construction described in the sixth chapter.