[Paper Review] Ergodic components and topological entropy in geodesic flows of surfaces

Jan Philipp Schröder|arXiv (Cornell University)|Jul 23, 2014

Advanced Differential Geometry Research10 references3 citations

TL;DR

This paper investigates geodesic flows on the 2-sphere and 2-torus with vanishing topological entropy under reversible Finsler metrics. Using Katok's constructions and results from Franks and Handel on Birkhoff annuli, it proves that ergodicity and dense orbits in the full unit tangent bundle are impossible on the 2-sphere if there are at least two closed geodesics and all geodesics have conjugate points—implying hyperbolicity and positive entropy. On the 2-torus, ergodicity is confined to strict subsets of flow-invariant tubes.

ABSTRACT

We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has positive flag curvatures and at least two closed geodesics. In the case of the 2-torus, we show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle of the 2-torus.

Motivation & Objective

• To understand the structure of ergodic components in geodesic flows on the 2-sphere and 2-torus under reversible Finsler metrics with zero topological entropy.
• To determine whether dense orbits or full ergodicity can occur in the unit tangent bundle when the flow has zero entropy.
• To investigate the dynamical implications of conjugate points and multiple closed geodesics on the 2-sphere.
• To analyze the constraints on ergodicity in the 2-torus, particularly within invariant tubes in the unit tangent bundle.
• To establish sharp dynamical restrictions using tools from surface dynamics and monotone twist maps.

Proposed method

• Applies Katok's construction of Finsler metrics on the 2-sphere with two closed geodesics and vanishing topological entropy.
• Uses the Birkhoff annulus map to analyze the dynamics of geodesic flows on the 2-sphere with at least two closed geodesics.
• Applies results from Franks and Handel on periodic points and invariant sets in monotone twist maps to constrain the structure of the flow.
• Employs the Poincaré section technique and first-return maps to relate topological entropy of the flow to that of the return map.
• Analyzes the unit tangent bundle of the 2-torus as a union of invariant tori and studies ergodic components within tubes between them.
• Uses the fact that zero entropy in the flow implies zero entropy in return maps, leveraging uniform boundedness of return times.

Experimental results

Research questions

• RQ1 Can the geodesic flow on the 2-sphere with reversible Finsler metric and zero topological entropy be ergodic in the entire unit tangent bundle if there are at least two closed geodesics and all geodesics have conjugate points?
• RQ2 What dynamical consequences arise when a geodesic flow on the 2-sphere has dense orbits and vanishing topological entropy under positive flag curvature?
• RQ3 To what extent can ergodic components fill the unit tangent bundle of the 2-sphere in reversible Finsler metrics with zero entropy?
• RQ4 How is ergodicity constrained in the unit tangent bundle of the 2-torus under zero entropy Finsler metrics?
• RQ5 What role do flow-invariant tori and tubes in the unit tangent bundle play in limiting the size of ergodic components on the 2-torus?

Key findings

• On the 2-sphere, if a reversible Finsler metric has at least two closed geodesics and all geodesics have conjugate points, then the geodesic flow cannot be ergodic in the entire unit tangent bundle if it has zero topological entropy.
• Under the same conditions, the existence of a dense orbit in the unit tangent bundle of the 2-sphere implies positive topological entropy, contradicting the zero entropy assumption.
• The geodesic flow on the 2-sphere cannot have dense orbits in the full unit tangent bundle if the metric has positive flag curvature and at least two closed geodesics.
• On the 2-torus, ergodicity for the geodesic flow with zero topological entropy is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle.
• The construction of Katok provides examples of reversible Finsler metrics on the 2-sphere with zero entropy and two ergodic components that are symmetric under the antipodal map and can be made arbitrarily large in measure.
• The results imply that hyperbolicity—evidenced by positive topological entropy—must emerge when ergodicity or dense orbits occur in the presence of conjugate points and multiple closed geodesics on the 2-sphere.