Regular dependence of the Peierls barriers on perturbations

Abstract

Let f be an exact area-preserving monotone twist diffeomorphism of the infinite cylinder and Pω,f(ξ) be the associated Peierls barrier. In this paper, we give the Hölder regularity of Pω,f(ξ) with respect to the parameter f. In fact, we prove that if the rotation symbol ω∈(R∖Q)⋃(Q+)⋃(Q−), then Pω,f(ξ) is 1/3-Hölder continuous in f, i.e.|Pω,f′(ξ)−Pω,f(ξ)|≤C‖f′−f‖C11/3, ∀ξ∈R where C is a constant. Similar results also hold for the Lagrangians with one and a half degrees of freedom. As application, we give an open and dense result about the breakup of invariant circles.