A note on a modified version of the Poincaré-Birkhoff theorem
type: Item
instance of: scholarly article
title: A note on a modified version of the Poincaré-Birkhoff theorem
zbMATH Open document ID: 1052.37033
author: Yanyan Li
published in: Journal of Differential Equations
publication date: 2004-08-20T00:00:00Z
review text: Consider the universal covering of the plane \({\mathcal C}=\{(\theta,r):\theta\in{\mathbb R}\), \(r\geq 0\}\). The authors prove the existence of an area-preserving homeomorphism \(\widetilde\psi:{\mathcal C}\to{\mathcal C}\) devoid of fixed points in the interior of \({\mathcal C}\) and such that \(\widetilde\psi(\theta,r)=(\theta+\Theta(\theta,r),{\mathcal R}(\theta,r))\), where \({\mathcal R}\) and \(\Theta\) are \(2\pi\)-periodic in \(\theta\), \({\mathcal R}(\theta,0)=0\) \(\forall\theta\in{\mathbb R}\), and satisfy (1) \(\exists \widetilde\theta\) s.t.\ \(\limsup_{r\to+\infty}\Theta(\widetilde\theta,r)<0\), (2) \(\Theta(\theta,0)>0\), \(\forall\theta\in {\mathbb R}\). This is particularly significant as it shows that the ``twist'' conditions at zero and at infinity in the following asymptotic version of the Poincaré-Birkhoff theorem cannot be interchanged (Asymptotic P-B theorem): Let \(\psi:{\mathcal C}\to{\mathcal C}\) be an area-preserving homeomorphism such that \(\psi(\theta,r)=(\theta+\Theta(\theta,r),{\mathcal R}(\theta,r))\), where \({\mathcal R}\) and \(\Theta\) are \(2\pi\)-periodic in \(\theta\), \({\mathcal R}(\theta,0)=0\) \(\forall\theta\in{\mathbb R}\). Assume also that the following conditions hold: (i\(_\infty\)) \(\forall\theta\in {\mathbb R}\) \(\liminf_{r\to+\infty}\Theta(\theta,r)>0\) (or \(\limsup_{r\to+\infty}\Theta(\theta,r)<0\)), (ii) \(\exists\overline\theta\in{\mathbb R}\) s.t.\ \(\Theta(\overline\theta,0)<0\) (respectively \(\Theta(\overline\theta,0)>0\)). Then, \(\psi\) admits at least one fixed point \((\theta^*,r^*)\) in the interior of \({\mathcal C}\) with \(0\leq\theta^*<2\pi\). If such a fixed point is unique then its fixed point index is nonzero. Considering the restriction of \(\widetilde\psi\) to the stripe \({\mathbb R}\times [0,R]\) for sufficiently large \(R\), one can conclude that the ``twist'' conditions on the curves \(\Gamma_1\) and \(\Gamma_2\) cannot be interchanged also in the following version of the Poincaré-…
reviewed by: Marco Spadini
Mathematics Subject Classification ID: 37E40
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 2093538
zbMATH Keywords: Poincaré-Birkhoff theorem
zbMATH Keywords: asymptotic Poincaré-Birkhoff theorem
zbMATH Keywords: twist conditions
MaRDI profile type: Publication
full work available at URL: j.jde.2004.03.022
OpenAlex ID: W2107711928
cites work: A Generalization of the Poincare-Birkhoff Theorem
cites work: Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems
cites work: A note on the Poincaré—Birkhoff fixed point theorem and periodic solutions of planar systems
DOI: 10.1016/J.JDE.2004.03.022
description: scientific article; zbMATH DE number 2093538