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Realizing rotation numbers on annular continua
| Content Provider | Semantic Scholar |
|---|---|
| Author | Koropecki, Andres |
| Copyright Year | 2015 |
| 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 $$K=A$$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. |
| Starting Page | 549 |
| Ending Page | 564 |
| Page Count | 16 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00209-016-1720-z |
| Alternate Webpage(s) | https://arxiv.org/pdf/1507.06440v2.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s00209-016-1720-z |
| Volume Number | 285 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
