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Qualitative Theory of Dynamical Systems

, Volume 4, Issue 2, pp 125–137 | Cite as

On properties of the vertical rotation interval for twist mappings II

  • Salvador Addas-Zanata
Article
  • 30 Downloads

Abstract

In this paper we consider twist mappings of the tours,\(\bar T:T^2 \to T^2 \), 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\(\bar T - invariant\) set\(\bar Q_\omega \) with\(\rho v(\bar x) = \omega \) for any\(\bar x \in \bar Q_\omega \), where\(\rho v(\bar x)\) is the vertical rotation number of\(\bar x\). In case ω is a rational number,\(\bar Q_\omega \) is a periodic orbit (this study began in [1] and [2]). Here we analyze how\(\rho _v^ - \) and\(\rho _v^ + \) behave as we perturb\(\bar T\) when they assume rational values. In particular we prove that, for an interesting class of mappings, these functions are locally constant at rational values.

Key Words

twist mappings vertical rotation set topological methods, saddlenodes analyticity bifurcations 

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Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSao PauloBrazil

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