A condition for minimal interval exchange maps to be uniquely ergodic. (English) Zbl 0602.28009
A certain property (Property P) is considered which an interval exchange map may satisfy. (The definition of Property P is motivated by the observation that minimal interval exchange maps having Property P must be uniquely ergodic.) It is proved that “typical” interval exchange maps of rank 2 must satisfy Property P and hence cannot be minimal but not uniquely ergodic. The above results are applied for ascertaining the unique ergodicity of some finite group extensions of irrational rotations. Then “billiards” dynamical systems on the “rational” polygons are considered. It is observed that these billiards are closely related to certain finite group extensions of irrational rotations and deduced that the billiards on the “rational” polygons are uniquely ergodic in all irrational directions.
In the next parts a new proof of the “Keane conjecture” is given and the techniques developed in the previous paragraphs are employed to investigations of “rational billiards” dynamical systems.
In the next parts a new proof of the “Keane conjecture” is given and the techniques developed in the previous paragraphs are employed to investigations of “rational billiards” dynamical systems.
Reviewer: Jaromir Šiška (Praha)
MSC:
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 37E05 | Dynamical systems involving maps of the interval |
| 28D15 | General groups of measure-preserving transformations |
| 28D10 | One-parameter continuous families of measure-preserving transformations |
Keywords:
minimal interval exchange maps; unique ergodicity; finite group extensions of irrational rotations; Keane conjecture; rational billiards dynamical systemsReferences:
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