Symplectic invariants of elliptic fixed points
type: Item
instance of: scholarly article
title: Symplectic invariants of elliptic fixed points
zbMATH Open document ID: 0985.37054
author: Karl Friedrich Siburg
published in: Commentarii Mathematici Helvetici
publication date: 2002-05-27T00:00:00Z
review text: This paper is devoted to the dynamics of an area-preserving diffeomorphism in a neighborhood of an elliptic fixed point. The author constructs a new ``local'' symplectic invariant which includes the Birkhoff invariants, but in addition, reflects some part of the dynamical behaviour near the fixed point. To this end the author introduces a variational principle analogous to that in Aubry-Mather theory for monotone twist maps; more precisely he numerizes the Lagrangian action over all orbits with a given rotation number. This yields the minimal action, a function canonically associated to the given symplectic map and invariant under symplectic coordinate changes. The author proves that this function is real valued, and hence a nontrivial invariant. He applies his result to the Reeb flow.
Mathematics Subject Classification ID: 37J10
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 37J50
zbMATH DE Number: 1562866
zbMATH Keywords: symplectic invariants
zbMATH Keywords: elliptic fixed points
zbMATH Keywords: minimal action
zbMATH Keywords: Reeb flow
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: s000140050145
OpenAlex ID: W2017381768
DOI: 10.1007/S000140050145
description: scientific article; zbMATH DE number 1562866