Geodesics in Margulis spacetimes (Q2908150)

A complete flat affine \(3\)-manifold \(M\) with free nonabelian fundamental group is called a Margulis spacetime. It admits a unique parallel Lorentz metric. Associated to \(M\) is a noncompact complete hyperbolic surface \(\Sigma\), which serves as a moduli space for equivalence classes of parallel timelike geodesics on \(M\). This article contributes to the study of the relation between the dynamics of the geodesic flows on \(M\) and \(\Sigma\).NEWLINENEWLINEThe authors restrict themselves to the case that \(\Sigma\) has a compact convex core. The homotopy-equivalence of \(\Sigma\) and \(M\) yields that free homotopy classes of essential closed curves in \(M\) correspond to free homotopy classes of essential closed curves in \(\Sigma\). While it is well known that every such homotopy class in \(\Sigma\) contains a unique closed geodesic, and that this property translates to \(M\), V. Charette showed in 2000 that geodesics in \(M\) exist which spiral around closed geodesics both in forward and backward time, and that they correspond to geodesics in \(\Sigma\) with the same property.NEWLINENEWLINEIn this article, the authors establish this kind of correspondence for recurrent geodesics. More precisely, they show that the recurrent part of the geodesic flow on \(\Sigma\) is topologically orbit-equivalent to the recurrent spacelike part of the geodesic flow on \(M\), the set of recurrent spacelike geodesics on \(M\) is the closure of the set of periodic geodesics, and timelike geodesics are not recurrent.