[Paper Review] Chaos and integrability in SL(2,R)-geometry

Alexey V. Bolsinov, А. П. Веселов+1 more|arXiv (Cornell University)|Jun 19, 2019

Mathematical Dynamics and Fractals42 references4 citations

TL;DR

This paper investigates the geodesic flow on 3-manifolds with $SL(2,\mathbb{R})$-geometry, particularly quotients $\mathcal{M}^3_\Gamma = \Gamma \backslash PSL(2,\mathbb{R})$ where $\Gamma$ is a Fuchsian group. It demonstrates that the phase space $T^*\mathcal{M}^3_\Gamma$ contains distinct open regions of integrable and chaotic dynamics, with zero and positive topological entropy, respectively. The key result is that periodic geodesics on the modular 3-fold (with $\Gamma = PSL(2,\mathbb{Z})$) correspond to trefoil cable knots in the integrable limit and to Lorenz-like knots in the chaotic regime.

ABSTRACT

The integrability of the geodesic flow on the three-folds $\mathcal M^3$ admitting $SL(2,\mathbb R)$-geometry in Thurston's sense is investigated. The main examples are the quotients $\mathcal M^3_Γ=Γ\backslash PSL(2,\mathbb R)$, where $Γ\subset PSL(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the corresponding phase space $T^*M_Γ^3$ 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 $\mathcal M^3_Γ$ 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 $\mathcal M^3_Γ$ 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 $\mathcal M^3_Γ$ they are replaced by the simple class of cable knots of trefoil.

Motivation & Objective

• To investigate the integrability and chaotic behavior of geodesic flows on 3-manifolds with $SL(2,\mathbb{R})$-geometry, particularly in the context of Thurston's geometrization program.
• To analyze the phase space structure of the geodesic flow on quotients $\mathcal{M}^3_\Gamma = \Gamma \backslash PSL(2,\mathbb{R})$ for cofinite Fuchsian groups $\Gamma$.
• To clarify the topological and dynamical distinction between integrable and chaotic regions in the phase space, especially in relation to knot invariants.
• To explore the correspondence between periodic geodesics and knot types, particularly in the case of the modular 3-fold with $\Gamma = PSL(2,\mathbb{Z})$.
• To examine the implications of topological obstructions to integrability, especially in light of Taimanov’s and Dinaburg’s theorems on fundamental groups and entropy.

Proposed method

• Utilizes left-invariant, naturally reductive metrics on $PSL(2,\mathbb{R})$, parameterized by $\alpha > 0 > \beta$, with the Sasaki metric as a special case ($\alpha = -\beta = 2$).
• Applies the Cartan decomposition of the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ into symmetric and skew-symmetric parts to define the metric structure.
• Analyzes the geodesic flow via the angular velocity $\Omega = g^{-1}\dot{g} \in \mathfrak{g}$, reducing the dynamics to a system on the cotangent bundle $T^*\mathcal{M}^3_\Gamma$.
• Employs the unit tangent bundle structure $\mathcal{M}^3 = S\mathcal{M}^2_\Gamma$ to relate geodesic dynamics to the geometry of the base surface $\mathcal{M}^2_\Gamma = \Gamma \backslash \mathbb{H}^2$.
• Uses topological invariants such as the fundamental group and first homology to assess obstructions to integrability, applying results from Taimanov and Butler.
• Connects periodic geodesics to knot theory by lifting them to the universal cover and identifying their isotopy classes, particularly in the modular case.

Experimental results

Research questions

• RQ1 What is the structure of the phase space $T^*\mathcal{M}^3_\Gamma$ for $\Gamma$ a cofinite Fuchsian group, and how do integrable and chaotic regions coexist?
• RQ2 How do periodic geodesics on the modular 3-fold $\mathcal{M}^3_\Gamma$ with $\Gamma = PSL(2,\mathbb{Z})$ relate to knot types in 3-space?
• RQ3 What topological constraints prevent the existence of analytically integrable geodesic flows on $SL(2,\mathbb{R})$-manifolds, and how do they relate to the fundamental group?
• RQ4 What is the dynamical transition between integrable and chaotic behavior in the geodesic flow on $\mathcal{M}^3_\Gamma$, and how is it reflected in the topology of periodic orbits?
• RQ5 How do the knot types of periodic geodesics evolve from cable knots of the trefoil in the integrable limit to Lorenz-like knots in the chaotic regime?

Key findings

• The phase space $T^*\mathcal{M}^3_\Gamma$ for $\Gamma$ a cofinite Fuchsian group contains two open regions: one with integrable dynamics and zero topological entropy, and another with chaotic dynamics and positive topological entropy.
• For the modular 3-fold with $\Gamma = PSL(2,\mathbb{Z})$, periodic geodesics in the integrable limit correspond to cable knots of the trefoil knot.
• In the chaotic regime, the same periodic geodesics lift to knots that are isotopic to those arising in the Lorenz system, as established by Ghys and Birman-Williams.
• The fundamental group of the modular 3-fold violates Taimanov’s condition for analytic integrability ($\dim H_1 > \dim M^3$), confirming the absence of analytically integrable geodesic flows.
• The existence of positive topological entropy in the chaotic region is consistent with Dinaburg’s theorem, as the fundamental group has exponential growth.
• The results demonstrate a smooth Liouville integrable system with positive topological entropy, extending earlier examples from the $Sol$-geometry case, and highlighting the role of knot theory in classifying periodic orbits.