Renormalization for golden circles
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.
scientific article; zbMATH DE number 222075
scientific article