2019/06/19 by Alexey V. Bolsinov, Bolsinov, A. V., А. П. Веселов +3
Mathematics · Physics and Astronomy · #37D40 #37J35 #57M50 #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1906.07958
openalex publication_date 2019/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The integrability of the geodesic flow on the three-folds \mathcal M3 admitting SL(2,\mathbb R)-geometry in Thurston's sense is investigated. The main examples are the quotients \mathcal M3Γ=Γ\backslash PSL(2,\mathbb R), where Γ⊂ PSL(2,\mathbb R) is a cofinite Fuchsian group. We show that the corresponding phase space T^*MΓ3 contains two open regions with integrable and chaotic behaviour with zero and positive topological entropy respectively. As a concrete example we consider the case of modular 3-fold with the modular group Γ=PSL(2,\mathbb Z), when \mathcal M3Γ is known to be homeomorphic to the complement of a trefoil knot \mathcal K in 3-sphere. Ghys proved a remarkable fact that the lifts of the periodic geodesics to the modular surface to \mathcal M3Γ produce the same isotopy class of knots, which appeared in the chaotic version of the celebrated Lorenz system and were extensively studied by Birman and Williams. We show that in the integrable limit of the geodesic system on \mathcal M3Γ they are replaced by the simple class of cable knots of trefoil.