Persistence of invariant graphs for twist maps under analytic perturbations

Abstract

We construct high-frequency perturbations of an integrable exact symplectic twist map which preserve an analytic invariant graph with a prescribed rotation number. Their oscillation frequencies may be any positive integers tending to infinity. For every fixed Gevrey parameter $L>0$, the perturbations tend to zero in a Gevrey-$\alpha$ norm and diverge in a Gevrey-$(\alpha -\epsilon )$ norm. The frequency class strictly contains the $\alpha$-Brjuno--Rüssmann class, and their difference has cardinality \(2^{\aleph_0}\).