Renormalization for golden circles
type: Item
instance of: scholarly article
title: Renormalization for golden circles
zbMATH Open document ID: 0773.58023
DOI: 10.1007/BF02098303
author: Andreas Stirnemann
published in: Communications in Mathematical Physics
publication date: 1993-06-29T00:00:00Z
review text: This work considers one-parameter families of symmetric area preserving twist maps of the cylinder. For the standard family there seems to exist a critical value \(\kappa\) of the parameter above which a homotopically nontrivial invariant circle with the golden rotation number does not exist and below which such a circle does exist. \textit{R. S. MacKay} [Physica D 7, 283-300 (1983; Zbl 0568.70023)] has established a relation between this problem and the existence of a so-called critical fixed point of a renormalization operator for a periodic twist map. A proof of the existence of such a fixed point has not yet been constructed, though \textit{J. N. Mather} [Ergodic Theory Dyn. Syst. 4, 301-309 (1984; Zbl 0557.58019)] gives a proof that if the absolute value of \(\kappa\) is greater than 4/3 then there does not exist any nontrivial homotopically invariant circles for the dynamical system associated with the difference equation \(\Delta^ 2x_ n=(\kappa/2\pi)\sin(2\pi x_ n)\). Assuming the existence and hyperbolicity of the critical fixed point, the author studies the problem of finding a critical circle, that is, an invariant curve at precisely the critical parameter value. The author shows that any map on its stable manifold has a transitive invariant golden circle. Further, if one assumes that the linearization of the operator has exactly one expanding eigenvalue at the critical fixed point, then the stable manifold has codimension one and it follows that a generic one-parameter family will intersect it. Interpreting the critical value mentioned above as the point of intersection of the standard family with the stable manifold of the critical fixed point, the author concludes that at this parameter value, the standard family has a transitive invariant golden circle.
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 81T15
zbMATH DE Number: 222075
zbMATH Keywords: area-preserving maps
zbMATH Keywords: golden circle
zbMATH Keywords: twist maps
zbMATH Keywords: renormalization
reviewed by: Chandra Shekhar Sharma
MaRDI profile type: Publication
cites work: Q4284069
cites work: A renormalization approach to invariant circles in area-preserving maps
cites work: Non-existence of invariant circles
cites work: Critical Behavior of a KAM surface. I: Empirical results
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description: scientific article; zbMATH DE number 222075