Realizing rotation numbers on annular continua

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 cas...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2017-02, Vol.285 (1-2), p.549-564
Main Author: Koropecki, Andres
Format: Article
Language:English
Subjects:
Invariants
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-016-1720-z