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| Content Provider | Springer Nature Link |
|---|---|
| Author | Boyland, Philip Hall, Toby Carvalho, André |
| Copyright Year | 2015 |
| Abstract | We construct a family $$\{\varPhi _t\}_{t\in [0,1]}$$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $$\rho (\varPhi _t)$$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of [0, 1], consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rotation set in the Hausdorff topology has infinitely many extreme points, accumulating on a single totally irrational extreme point at which there is a unique supporting line; and that, although $$\rho (\varPhi _t)$$ varies continuously with t, the set of extreme points of $$\rho (\varPhi _t)$$ does not. The family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set, or by any directional ergodic measure. |
| Ending Page | 937 |
| Page Count | 43 |
| Starting Page | 895 |
| File Format | |
| ISSN | 00209910 |
| e-ISSN | 14321297 |
| Journal | Inventiones mathematicae |
| Issue Number | 3 |
| Volume Number | 204 |
| Language | English |
| Publisher | Springer Berlin Heidelberg |
| Publisher Date | 2015-10-14 |
| Publisher Place | Berlin, Heidelberg |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Homeomorphisms and diffeomorphisms of planes and surfaces Rotation numbers and vectors Mathematics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
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