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Existence of non-contractible periodic orbits for homeomorphisms of the open annulus

  • Jonathan ConejerosEmail author
  • Fábio Armando Tal
Article
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Abstract

In this article we consider homeomorphisms of the open annulus \(\mathbb {A}=\mathbb {R}/\mathbb {Z}\times \mathbb {R}\) which are isotopic to the identity and preserve a Borel probability measure of full support, focusing on the existence of non-contractible periodic orbits. Assume f is such homeomorphism such that the connected components of the set of fixed points of f are all compact. Further assume that there exists \(\check{f}\) a lift of f to the universal covering of \(\mathbb {A}\) such that the set of fixed points of \(\check{f}\) is non-empty and that this set projects into an open topological disk of \(\mathbb {A}\). We prove that, in this setting, one of the following two conditions must be satisfied: (1) f has non-contractible periodic points of arbitrarily large periodic, or (2) for every compact set K of \(\mathbb {A}\) there exists a constant M (depending on the compact set) such that, if \(\check{z}\) and \(\check{f}^n(\check{z})\) project on K, then their projections on the first coordinate have distance less or equal to M.

Notes

Acknowledgements

We are very grateful for Patrice Le Calvez, whose several suggestions helped to improve the exposition of the paper and to greatly simplify some proofs.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Matématica e EstatísticaUniversidade de São PauloSão PauloBrazil

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