Closures of asymptotic curves in a two-dimensional symplectic map
type: Item
instance of: scholarly article
title: Closures of asymptotic curves in a two-dimensional symplectic map
zbMATH Open document ID: 0943.37011
DOI: 10.1023/A:1022812415121
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: Let \(f\) be a \(C^1\)-smooth area-preserving diffeomorphism of a 2D manifold \(M\) and \(z^*\in M\) a hyperbolic fixed point of the map \(f\). Such point \(z^*\in M\) generates four asymptotic curves. Following the author we call these curves separatrix branches and denote the stable branches by \(\Gamma_{1,2}^s\) and the unstable ones by \(\Gamma_{1,2}^u\). The author deals with the following questions: Suppose that (a) these branches lie in a compact invariant domain, (b) these branches intersect at some point \(w^*\), \(w^*\neq z^*\). Do the closures of these curves coincide? The author gives a positive answer to this question.
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 1431864
zbMATH Keywords: area-preserving map
zbMATH Keywords: hyperbolic fixed point
zbMATH Keywords: invariant domain
author: D. Treschev
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: a:1022812415121
OpenAlex ID: W189036413
description: scientific article; zbMATH DE number 1431864