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A Crossing Lemma for Annular Regions and Invariant Sets with an Application to Planar Dynamical Systems

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  • Anna Pascoletti
  • Fabio Zanolin

Abstract

We present a topological result, named crossing lemma , dealing with the existence of a continuum which crosses a topological space between a pair of “opposite” sides. This topological lemma allows us to obtain some fixed point results. In the works of Pascoletti et al., 2008, and Pascoletti and Zanolin, 2010, we have widely exposed the crossing lemma for planar regions homeomorphic to a square, and we have also presented some possible applications to the theory of topological horseshoes and to the study of chaotic-like dynamics for planar maps. In this work, we move from the framework of the generalized rectangles to two other settings (annular regions and invariant sets), trying to obtain similar results. An application to a model of fluid mixing is given.

Suggested Citation

  • Anna Pascoletti & Fabio Zanolin, 2013. "A Crossing Lemma for Annular Regions and Invariant Sets with an Application to Planar Dynamical Systems," Journal of Mathematics, Hindawi, vol. 2013, pages 1-12, July.
    • Handle: RePEc:hin:jjmath:267393
    DOI: 10.1155/2013/267393
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    1. Anna Pascoletti & Fabio Zanolin, 2011. "A Topological Approach to Bend-Twist Maps with Applications," International Journal of Differential Equations, Hindawi, vol. 2011, pages 1-20, October.
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