A criterion for the non-existence of invariant circles
type: Item
instance of: scholarly article
title: A criterion for the non-existence of invariant circles
zbMATH Open document ID: 0603.58028
DOI: 10.1007/BF02831625
author: John N. Mather
published in: Publications Mathématiques
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104011
review text: The purpose of this paper is to give a criterion for the existence of invariant circles for a certain class of area-preserving and monotone twist diffeomorphisms of the annulus. The author provides a numerical invariant, for each irrational number \(\omega\), which is always nonnegative, and which vanishes if and only if there is an invariant circle of frequency \(\omega\). The main idea of this paper comes from the version of Birkhoff's arguments for the billiard ball problem. However, the author has to introduce new concepts and to prove a series of propositions for the principal result of this paper, which will be omitted here because of their long statements. The interested reader is referred to this well- written paper.
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 3973899
zbMATH Keywords: minimax principle
zbMATH Keywords: area-preserving mappings
zbMATH Keywords: invariant circles
zbMATH Keywords: monotone twist diffeomorphisms
zbMATH Keywords: Birkhoff's arguments
reviewed by: Ding Tongren
MaRDI profile type: Publication
cites work: Q4497419
cites work: The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states
cites work: Q5785431
cites work: Q3249421
cites work: Q5768571
cites work: Q3670242
cites work: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus
cites work: Concavity of the Lagrangian for quasi-periodic orbits
cites work: Glancing billiards
cites work: Non-existence of invariant circles
cites work: Topologies on Spaces of Subsets
cites work: Definite paths and upper bounds on regular regions of velocity phase space
cites work: A variational principle for invariant tori of fixed frequency
cites work: Q4482959
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description: scientific article; zbMATH DE number 3973899