Minimal geodesic foliation on T² in case of vanishing topological entropy

2011/01/09 by Eva Glasmachers, Gerhard Knieper, Glasmachers, Eva +1 · 1 citation
Mathematics · Physics and Astronomy · #37C10 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Primary 37C40 #Quantum chaos and dynamical systems #Secondary 53C22 #math.DG #math.DS #msc:37C10 #msc:37C40 #msc:53C22

paper · pdf · doi:10.48550/arxiv.1101.1660

10 pages, 1 figure

openalex publication_date 2011/01/09 · arxiv created 2011/09/02 · arxiv updated 2011/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

On a Riemannian 2-torus (T²,g) we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \citeGK we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all r ∈ ℝ ∪ \∞\ the universal cover \Br² is foliated by minimal geodesics of rotation number r. For irrational r ∈ ℝ all geodesics are minimal, for rational r ∈ ℝ ∪ \∞\ all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.

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• Minimal geodesics and integrable behavior in geodesic flows

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