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# Stability for the vertical rotation interval of twist mappings

• 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 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 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 rational values. In particular we prove that for analytic area-preserving mappings these functions are locally constant at rational values.
Mathematics Subject Classification: 37E40, 37E45, 37C25, 37G15.

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