We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on $\mathbb{Z}$-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical $\mathbb{Z}$-periodic surface with a boundary are recurrent.
Keywords:
(periodic) polygonal surface, geodesic, skew product, cross-section, displacement function, recurrence, transience, ergodicity
Citation:
Gutkin E., Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces, Regular and Chaotic Dynamics,
2010, Volume 15, Numbers 4-5,
pp. 482-503