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Realizing rotation numbers on annular continua

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Abstract

An annular continuum is a compact connected set K which separates a closed annulus A into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where \(K=A\), showing that if K is an invariant annular continuum of a homeomorphism of A isotopic to the identity, then the rotation set in K is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in K (and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum K is minimal with the property of being annular (what we call a circloid), then every rational number between the extrema of the rotation set in K is realized by a periodic orbit in K. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in K. This improves a previous result of Barge and Gillette.

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Notes

  1. The author is grateful to T. Jäger for pointing this out.

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Acknowledgments

I would like to thank A. Passeggi and T. Jäger for the discussions that motivated this paper and for their suggestions, and M. Handel for his availability to answer my questions and for the helpful comments. I also thank the anonymous referee for pointing out a mistake in the statement of Theorem A and for other corrections.

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  1. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, Niteroi, RJ, 24020-140, Brazil

    Andres Koropecki

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  1. Andres Koropecki
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Correspondence to Andres Koropecki.

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The author was supported by research grants from CNPq-Brasil and FAPERJ-Brasil.

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Koropecki, A. Realizing rotation numbers on annular continua. Math. Z. 285, 549–564 (2017). https://doi.org/10.1007/s00209-016-1720-z

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