Exponentially Small Residues Near Analytic Invariant Circles
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.
KeywordsPeriodic Orbit Rotation Number Invariant Torus Exponential Dichotomy Continue Fraction Expansion
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