Stable and unstable manifolds asymptotic to the outermost KAM curve
The authors' summary: ``A one-parameter family of area-preserving piecewise linear maps is considered. Behavior of the stable and unstable manifolds starting from a hyperbolic fixed point is investigated in relation to the outermost KAM (Kolmogorov-Arnold-Moser) curve of a neighboring elliptic fixed point. It is proved that, at least for certain specific values of the bifurcation parameter, each of the stable and unstable manifolds has a sequence of points that tends to the outermost KAM curve of the stable fixed point. As a corollary, homoclinic points also accumulate to the outermost KAM curve. Though simple, these seem to be the first rigorous examples of such an asymptotic behavior of stable and unstable manifolds. It is found that stable and unstable manifolds form a polygon for a sequence of parameter values even if the dynamical system given by the map is not integrable''. Reviewer's comment: This paper seems to be an important contribution to the theory of dynamical systems induced by linear maps.
scientific article; zbMATH DE number 62830