Exponentially Small Residues Near Analytic Invariant Circles

  • R. S. MacKay
Part of the NATO ASI Series book series (NSSB, volume 284)


For analytic area-preserving twist maps, we sketch a proof that the “residues” of “good” sequences of periodic orbits with rotation number converging to that of an invariant circle with analytic conjugacy to rotation converge to zero exponentially, with decay rate at least the “analyticity width” of the conjugacy. This confirms numerical observations of Greene and Percival, and provides an important part of a mathematical foundation for Greene’s residue criterion, relating existence of an invariant circle of given rotation number to the behaviour of the residues of periodic orbits with rotation number converging to the given one.


Periodic Orbit Rotation Number Invariant Torus Exponential Dichotomy Continue Fraction Expansion 
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Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • R. S. MacKay
    • 1
  1. 1.Nonlinear Systems Laboratory, Mathematics InstituteUniversity of WarwickCoventryUK

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