Stability for the vertical rotation interval of twist mappings

Abstract

In this paper we consider twist mappings of the torus, $\overline{T}:T^2\rightarrow T^2,$ and their vertical rotation intervals $\rho _V(T)=[\rho _V^{-},\rho _V^{+}],$ which are closed intervals suchthat 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 case $\omega $ is a rational number, $\overline{Q}_\omega $ is a periodic orbit(this study began in [1] and [2]). Here we analyze how $\rho _V^{-}$ and $\rho _V^{+}$ behave as we perturb $\overline{T}$ when they assume rationalvalues.In particular we prove that for analytic area-preserving mappings thesefunctions are locally constant at rational values.