Abstract
We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Moreover, in case (1) bounded disks are non-periodic and in case (2) all disks are non-periodic. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.
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Notes
Here, we identify \(\mathcal A \) with \(\mathbb{A }\) to define unboundedness.
This is true in any \(\sigma \)-compact connected Hausdorff space.
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Acknowledgments
We thank the referee for thoughtful comments and suggestions on the manuscript. We are indebted to Andres Koropecki and Patrice Le Calvez for helpful comments and remarks. Our results were first presented at the Visegrad Conference of Dynamical Systems 2011 in Banska Bystrica, and we would like to thank the organisers Roman Hric and Lubomir Snoha for creating this opportunity. T. Jäger and A. Passeggi acknowledge support by an Emmy-Noether-grant Ja 1721/2-1 of the German Research Council.
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Jäger, T., Kwakkel, F. & Passeggi, A. A classification of minimal sets of torus homeomorphisms. Math. Z. 274, 405–426 (2013). https://doi.org/10.1007/s00209-012-1076-y
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DOI: https://doi.org/10.1007/s00209-012-1076-y