Quantitative destruction of invariant circles

Abstract

<p style='text-indent:20px;'>For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> of an integrable system by a trigonometric polynomial of degree <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> perturbation <inline-formula><tex-math id="M3">\begin{document}$ R_N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \|R_N\|_{C^r}&lt;\epsilon $\end{document}</tex-math></inline-formula>. We obtain a relation among <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the arithmetic property of <inline-formula><tex-math id="M8">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>, for which the area-preserving map admit no invariant circles with <inline-formula><tex-math id="M9">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>.</p>