Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

TL;DR

This paper explores the behavior of geodesic flows on noncompact polygonal surfaces, demonstrating that most geodesics on typical periodic surfaces with boundary tend to recur, enriching the understanding of their geometric and dynamical properties.

Contribution

It establishes foundational results for geodesic dynamics on noncompact polygonal surfaces and proves recurrence properties for typical periodic cases.

Findings

Almost all geodesics on a topologically typical Z-periodic surface with boundary are recurrent.

Provides a framework for studying geodesic behavior on noncompact polygonal surfaces.

Highlights the significance of topology and periodicity in geodesic recurrence.

Abstract

We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on $Z$-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical $Z$-periodic surface with boundary are recurrent.