Geodesics on Margulis spacetimes

TL;DR

This paper explores the relationship between geodesics on Margulis spacetimes and hyperbolic surfaces, revealing a correspondence for spacelike geodesics and the non-recurrence of timelike geodesics.

Contribution

It establishes an orbit equivalence between recurrent spacelike geodesics on Margulis spacetimes and those on hyperbolic surfaces, generalizing known correspondences.

Findings

Recurrent spacelike geodesics on M correspond to those on S

No timelike geodesic recurs in either time direction

Generalization of geodesic correspondence for Margulis spacetimes

Abstract

Let M be a Margulis spacetime whose associated complete hyperbolic surface S has compact convex core. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on M and recurrent geodesics on S. In contrast, no timelike geodesic recurs in either forward or backwards time.