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arXiv · 1612.07422

Regular dependence of the Peierls barriers on perturbations

Qinbo Chen·Chong-Qing Cheng
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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 H\"{o}lder 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 $1/3$-H\"{o}lder continuous in $f$, i.e. $$|P_{\omega,f'}(\xi)-P_{\omega,f}(\xi)|\leq C\|f'-f\|_{C^1}^{1/3} ,~~\forall \xi\in\mathbb{R}$$ 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.

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Qinbo Chen, Chong-Qing Cheng. 2016-12-22. Regular dependence of the Peierls barriers on perturbations. https://doi.org/10.1016/j.jde.2016.12.018

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