Characterization of geodesic flows on \(T^{2}\) with and without positive topological entropy
type: Item
instance of: scholarly article
title: Characterization of geodesic flows on \(T^{2}\) with and without positive topological entropy
zbMATH Open document ID: 1211.37023
author: Eva Glasmachers
author: Gerhard Knieper
published in: Geometric and Functional Analysis. GAFA
publication date: 2010-12-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/0707.0666
review text: The authors consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus for an arbitrary Riemannian metric. They formulate necessary and sufficient conditions that the topological entropy be zero, and show that in the case of zero entropy the flow has properties like those of integrable systems. Specifically, they prove the existence of a continuous constant of motion that measures the direction of geodesics lifted from \(T^2\) to \(\mathbb{R}^2\).
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 37C40
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 5822834
zbMATH Keywords: geodesic flow
zbMATH Keywords: topological entropy
zbMATH Keywords: torus
MaRDI profile type: Publication
OpenAlex ID: W2068700477
arXiv ID: 0707.0666
cites work: Monotone recurrence relations, their Birkhoff orbits and topological entropy
cites work: Parabolic equations for curves on surfaces. II: Intersections, blow-up and generalized solutions
cites work: The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states
cites work: Q3814339
cites work: Geodesic flows on the two-dimensional torus and phase transitions ``commensurability-noncommensurability''
cites work: Some geometrical conditions for the existence of chaotic geodesics on a torus
cites work: Entropy for Group Endomorphisms and Homogeneous Spaces
cites work: Entropy-Expansive Maps
cites work: Consequences of contractible geodesics on surfaces
cites work: Q3598184
cites work: Shortening embedded curves
cites work: Closed Surfaces Without Conjugate Points
cites work: Q3763215
cites work: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
cites work: Q4840149
cites work: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus
cites work: Q3931654
cites work: Q3049561
DOI: 10.1007/S00039-010-0087-2
description: scientific article