Regular dependence of the Peierls barriers on perturbations

Abstract

Let $f$ be an exact area-preserving monotone twist diffeomorphism of the infinite cylinder and $P_{ω,f}(ξ)$ be the associated Peierls barrier. In this paper, we give the Hölder regularity of $P_{ω,f}(ξ)$ with respect to the parameter $f$. In fact, we prove that if the rotation symbol $ω\in (\mathbb{R}\setminus\mathbb{Q})\bigcup(\mathbb{Q}+)\bigcup(\mathbb{Q}-)$, then $P_{ω,f}(ξ)$ is $1/3$-Hölder continuous in $f$, i.e. $$|P_{ω,f'}(ξ)-P_{ω,f}(ξ)|\leq C\|f'-f\|_{C^1}^{1/3} ,~~\forall ξ\in\mathbb{R}$$ 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.