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

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.

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Notes

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    In the whole text “transverse” will mean “positively transverse”.

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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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Correspondence to Jonathan Conejeros.

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The first author was supported by CNPq-Brasil. The second author was partially supported by Fapesp, CNPq-Brasil and CAPES.

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Conejeros, J., Tal, F.A. Existence of non-contractible periodic orbits for homeomorphisms of the open annulus. Math. Z. 294, 1413–1439 (2020). https://doi.org/10.1007/s00209-019-02309-6

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