[Paper Review] Minimal geodesic foliation on T^2 in case of vanishing topological entropy

Eva Glasmachers, Gerhard Knieper|arXiv (Cornell University)|Jan 9, 2011

Mathematical Dynamics and Fractals15 references3 citations

TL;DR

This paper establishes that on a Riemannian 2-torus with zero topological entropy, the universal cover ℝ² is foliated by globally minimizing geodesics for every rotation number r ∈ ℝ ∪ {∞}. For irrational r, the foliation is unique and all geodesics with that rotation number are minimal and non-self-intersecting under lattice translations; for rational r, uniqueness holds only if the metric is flat. The result extends Aubry-Mather theory to geodesic flows via curve-shortening techniques.

ABSTRACT

On a Riemannian 2-torus $(T^2,g)$ 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 \cite{GK} 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 $r \in \mathbb{R} \cup \{\infty\}$ the universal cover $\Br^2$ is foliated by minimal geodesics of rotation number $r$. For irrational $r \in \mathbb{R}$ all geodesics are minimal, for rational $r \in \mathbb{R} \cup \{\infty\}$ 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.

Motivation & Objective

• To characterize the structure of geodesic flows on a 2-torus when topological entropy vanishes.
• To determine whether the absence of orbit complexity implies a nearly integrable structure in terms of minimal geodesic foliations.
• To establish the existence and uniqueness of minimal geodesic foliations for every rotation number r ∈ ℝ ∪ {∞} under zero entropy.
• To clarify the relationship between the uniqueness of such foliations and the flatness of the metric.
• To extend results from monotone twist maps (Aubry-Mather theory) to geodesic flows on T² using geometric flow techniques.

Proposed method

• The authors use the curve-shortening flow to globally control geodesic segments and analyze their intersection patterns.
• They define asymptotic direction and rotation number for geodesics, proving existence via prior results and extending to foliation structure.
• By analyzing the boundary of sets S_x^r of geodesics with rotation number r, they show that boundary geodesics are minimal and asymptotic to neighboring minimal axes.
• They apply Theorem 3.4 and Corollary 3.3 to prove that interior points of strips contain no minimal rays, ensuring foliation structure.
• They use topological conjugacy invariance and properties of minimal geodesics to show that for irrational r, the foliation is unique and geodesics do not intersect their translates.
• The proof distinguishes between irrational and rational rotation numbers, showing that rational foliations are unique iff the metric is flat.

Experimental results

Research questions

• RQ1 Does the existence of a minimal geodesic foliation for every rotation number r ∈ ℝ ∪ {∞} on T² imply zero topological entropy for the geodesic flow?
• RQ2 For a Riemannian metric on T² with zero topological entropy, are there only finitely many rational rotation numbers with non-unique minimal geodesic foliations?
• RQ3 Is the geodesic flow on such a torus integrable in the Liouville-Arnold sense?
• RQ4 Does the uniqueness of minimal geodesic foliations for all rational r imply that the metric is flat?
• RQ5 Are there metrics on T² with integrable geodesic flows other than Liouville metrics?

Key findings

• For every r ∈ ℝ ∪ {∞}, the universal cover ℝ² of the 2-torus is foliated by globally minimizing geodesics of rotation number r.
• When r is irrational, the foliation is unique and all geodesics with that rotation number are minimal and do not intersect their nontrivial translates under ℤ².
• For rational r, the foliation is unique if and only if the metric is flat, implying that non-flat metrics admit multiple distinct foliations for the same rational rotation number.
• In strips between neighboring minimal axes with rational rotation number r, minimal geodesics are asymptotic to the boundary axes and generate two distinct foliations.
• The existence of a minimal geodesic foliation for every rotation number r implies that the geodesic flow is not chaotic, consistent with zero topological entropy.
• The only known metrics on T² with zero topological entropy are Liouville metrics of the form ds² = (f(x) + g(y))(dx² + dy²), and among these, non-unique foliations occur only for directions (1,0) and (0,1) when f and g are non-constant.