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.
Categories: math.DS
