Quantitative destruction of invariant circles

TL;DR

This paper investigates how certain small perturbations can destroy invariant circles in area-preserving twist maps, providing quantitative conditions based on perturbation degree, smoothness, and arithmetic properties of the frequency.

Contribution

It establishes explicit relations among perturbation degree, smoothness, and frequency arithmetic properties for the destruction of invariant circles.

Findings

Invariant circles can be destroyed under specific quantitative conditions.

Relations among perturbation degree, smoothness, and frequency are derived.

Conditions depend on the arithmetic property of the frequency .

Abstract

For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency $\omega$ of an integrable system by a trigonometric polynomial of degree $N$ perturbation $R_N$ with $\|R_N\|_{C^r}<\epsilon$. We obtain a relation among $N$, $r$, $\epsilon$ and the arithmetic property of $\omega$, for which the area-preserving map admit no invariant circles with $\omega$.