We will consider infinite translation surfaces which are abelian covers of
compact surfaces with a (singular) flat metric and focus on the dynamical
properties of their flat geodesics. A motivation come from mathematical
physics, since flat geodesics on these surfaces can be obtained by unfolding
certain mathematical billiards. A notable example of such billiards is the
Ehrenfest model, which consists of a particle bouncing off the walls of a
periodic planar array of rectangular scatterers.
The dynamics of flat geodesics on compact translation surfaces is now well
understood thanks to the beautiful connection with Teichmueller dynamics. We
will survey some recent advances on the study of infinite translation
surfaces and in particular focus on a joint work with K. Fraczek, in which
we proved that the Ehrenfest model and more in general geodesic flows on
certain abelain covers have no dense orbits. We will try to convey an
heuristic idea of how Teichmueller dynamics plays a crucial role in the
proofs.