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# Recurrent Geodesics in Flat Lorentz $3$-Manifolds

Published:2004-09-01
Printed: Sep 2004
• Virginie Charette
• William M. Goldman
• Catherine A. Jones
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## Abstract

Let $M$ be a complete flat Lorentz $3$-manifold $M$ with purely hyperbolic holonomy $\Gamma$. Recurrent geodesic rays are completely classified when $\Gamma$ is cyclic. This implies that for any pair of periodic geodesics $\gamma_1$, $\gamma_2$, a unique geodesic forward spirals towards $\gamma_1$ and backward spirals towards $\gamma_2$.
 Keywords: geometric structures on low-dimensional manifolds, notions of recurrence
 MSC Classifications: 57M50 - Geometric structures on low-dimensional manifolds 37B20 - Notions of recurrence

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