Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: The Denjoy-Rees technique. (English. French summary) Zbl 1132.37003
Summary: M. Rees [J. Lond. Math. Soc. 23, 537–550 (1981; Zbl 0451.58022)] constructed a minimal homeomorphism of the \(n\)-torus with positive topological entropy. This homeomorphism \(f\) is obtained by enriching the dynamics of an irrational rotation \(R\). We improve Rees construction, allowing to start with any homeomorphism \(R\) instead of an irrational rotation and to control precisely the measurable dynamics of \(f\). This yields in particular the following result: Any compact manifold of dimension \(d\geq 2\) which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy.
More generally, given some homeomorphism \(R\) of a compact manifold and some homeomorphism \(h_C\) of a Cantor set, we construct a homeomorphism \(f\) which “looks like” \(R\) from the topological viewpoint and “looks like” \(R\times h_C\) from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds?
More generally, given some homeomorphism \(R\) of a compact manifold and some homeomorphism \(h_C\) of a Cantor set, we construct a homeomorphism \(f\) which “looks like” \(R\) from the topological viewpoint and “looks like” \(R\times h_C\) from the measurable viewpoint. This construction can be seen as a partial answer to the following realisability question: which measurable dynamical systems are represented by homeomorphisms on manifolds?
MSC:
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
Keywords:
representation by homeomorphisms on manifolds; control the measurable dynamics; measurable dynamical systemsCitations:
Zbl 0451.58022References:
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