A surface with positive curvature and positive topological entropy
type: Item
instance of: scholarly article
title: A surface with positive curvature and positive topological entropy
zbMATH Open document ID: 0809.53043
author: Gerhard Knieper
author: Howard Weiss
published in: Journal of Differential Geometry
publication date: 1994-06-19T00:00:00Z
review text: It is known, by the work of Donnay and Burns-Gerber, that there exist smooth metrics on the 2-sphere whose associated geodesic flow is Bernoulli. In the known examples, this behaviour is due to the existence of negative curvature on part of the surface. It is an open question whether there exist examples of a 2-sphere with a positively curved metric, for which the geodesic flow has complicated dynamical behaviour, for example ergodic, positive Liouville entropy, etc. In this paper, the authors give a positive answer to a question of this kind, namely, they construct real analytic Riemannian metrics on the 2-sphere with positive Gaussian curvature whose geodesic flows have positive topological entropy. Smooth metrics are constructed as small perturbations of the ellipsoid. The methods are extensions of methods due to Poincaré. The results follow from the existence of horseshoes in the dynamics of the geodesic flow. Real analytic examples follow by using approximation results.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 590829
zbMATH Keywords: geodesic flow
zbMATH Keywords: positive Liouville entropy
zbMATH Keywords: positive topological entropy
zbMATH Keywords: horseshoes
Wikidata QID: Q115179840
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214454871
OpenAlex ID: W1565074190
DOI: 10.4310/JDG/1214454871
Recommended article: Q4314291
Recommended article: Topological structure of candidates for positive curvature
Recommended article: A geometric criterion for positive topological entropy
Recommended article: Q3051931
Recommended article: Flexibility of entropies for surfaces of negative curvature
Recommended article: Entropy, minimal surfaces and negatively curved manifolds
Recommended article: Minimal Surface Entropy of Negatively Curved Manifolds
Recommended article: Positive curvature, symmetry, and topology
Recommended article: Q4332754
Recommended article: Topological uniqueness of negatively curved surfaces
description: scientific article; zbMATH DE number 590829