On properties of the vertical rotation interval for twist mappings

Abstract

In this paper we consider twist mappings of the torus, $\overline{T}:{\rm T^2\rightarrow T^2}$, and their vertical rotation intervals $\rho _V(T)=[\rho _V^{-},\rho _V^{+}]$, which are closed intervals such that for any $\omega \in\, ]\rho _V^{-},\rho _V^{+}[$ there exists a compact $\overline{T}$-invariant set $\overline{Q}_\omega $ with $\rho _V(\overline{x})=\omega$ for any $\overline{x}\in \overline{Q}_\omega $, where $\rho _V(\overline{x})$ is the vertical rotation number of $\overline{x}$. In the case when $\omega $ is a rational number, $\overline{Q}_\omega $ is a periodic orbit. Here we analyze how $\rho _V^{-}$ and $\rho _V^{+}$ behave as we perturb $\overline{T}$ and which dynamical properties for $\overline{T}$ can be obtained from their values.