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October  2006, 14(4): 631-642. doi: 10.3934/dcds.2006.14.631

Stability for the vertical rotation interval of twist mappings

Salvador Addas-Zanata 1,

1. 

Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil

Received  November 2004 Revised  October 2005 Published  January 2006

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.
Keywords: Twist mappings, analyticity, vertical rotation set, topological methods, bifurcations..
Mathematics Subject Classification: 37E40, 37E45, 37C25, 37G1.
Citation: Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631
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