A geometric criterion for positive topological entropy
type: Item
instance of: scholarly article
title: A geometric criterion for positive topological entropy
zbMATH Open document ID: 0945.37003
DOI: 10.1007/BF02104512
author: Keith Burns
author: Howard Weiss
published in: Communications in Mathematical Physics
publication date: 2000-10-09T00:00:00Z
review text: We prove that a diffeomorphism possessing a homoclinic point with a topological crossing (possibly with infinite order contact) has positive topological entropy, along with an analogous statement for heteroclinic points. We apply these results to study area-preserving perturbations of area-preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections. In the heteroclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection. In the homoclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection. These results significantly simplify the application of the Poincaré-Arnold-Melnikov-Sotomayor method. The results apply even when the contraction and expansion at the fixed point is subexponential.
Mathematics Subject Classification ID: 37B40
zbMATH DE Number: 878032
zbMATH Keywords: geometric criterion
zbMATH Keywords: positive topological entropy
zbMATH Keywords: area preserving surface diffeomorphism
zbMATH Keywords: homoclinic point
zbMATH Keywords: topological crossing
zbMATH Keywords: infinite order contact
zbMATH Keywords: heteroclinic points
zbMATH Keywords: area preserving perturbations
zbMATH Keywords: homoclinic connections
zbMATH Keywords: double heteroclinic connections
zbMATH Keywords: perturbed map
zbMATH Keywords: Poincare Arnold Melnikov Sotomayor method
zbMATH Keywords: contraction
zbMATH Keywords: stable manifolds
zbMATH Keywords: unstable manifolds
MaRDI profile type: Publication
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Wikidata QID: Q125373705
full work available at URL: https://doi.org/10.1007/bf02104512
OpenAlex ID: W2023615234
description: scientific article; zbMATH DE number 878032