Torsion of instability zones for conservative twist maps on the annulus

TL;DR

This paper investigates the conditions under which conservative twist maps on the annulus have positive measure sets of points with non-zero asymptotic torsion, linking instability regions and integrability properties.

Contribution

It provides sufficient conditions for positive measure of points with non-zero asymptotic torsion and offers a geometric proof characterizing integrability via conjugate points.

Findings

Bounded instability regions contain positive measure sets with non-zero asymptotic torsion

Sufficient conditions for existence of points with non-zero asymptotic torsion

Geometric proof linking $ ext{C}^0$-integrability and absence of conjugate points

Abstract

For a twist map $f$ of the annulus preserving the Lebesgue measure, we give sufficient conditions to assure the existence of a set of positive measure of points with non-zero asymptotic torsion. In particular, we deduce that every bounded instability region for $f$ contains a set of positive measure of points with non-zero asymptotic torsion. Moreover, for an exact symplectic twist map $f$, we provide a simple, geometric proof of a result by Cheng and Sun (see [CS96]) which characterizes $\mathcal{C}^0$-integrability of $f$ by the absence of conjugate points.