Regular dependence of the Peierls barriers on perturbations

Abstract

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

19 pages, 2figures

Cited by in corpus (2)

• Gevrey genericity of Arnold diffusion in a priori unstable Hamiltonian systems
• Quantitative destruction of invariant circles