Twist interval for twist maps

TL;DR

This paper proves that non-wandering twist maps on the annulus have a non-degenerate twist interval and that disjoint invariant curves must have different rotation numbers, extending understanding of their dynamical structure.

Contribution

It establishes conditions under which the twist interval is non-degenerate and invariant curves have distinct rotation numbers for general twist maps.

Findings

Non-wandering twist maps have non-degenerate twist intervals.

Disjoint invariant curves of a twist map have different rotation numbers.

Results extend classical area-preserving twist map properties to more general cases.

Abstract

The twist interval of a twist map on the annulus $A=\mathbb{T}\times [0,1]$ has nonempty interior if $f$ preserves the area, but could be degenerate for general twist maps. In this note, we show that if a twist map $f$ is non-wandering, then the twist interval of $f$ is non-degenerate. Moreover, if there are two disjoint invariant curves of $f$, then their rotation numbers must be different (no matter if they are rational or irrational).