CONSTRUCTION OF CURIOUS MINIMAL UNIQUELY ERGODIC HOMEOMORPHISMS ON MANIFOLDS: THE DENJOY-REES TECHNIQUE
By Franccois B'eguin, S. Crovisier, F. Roux. 2006
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Abstract In [Rees, M., A minimal positive entropy homeomorphism of the 2-torus, J. London Math. Soc. 23 (1981) 537–550], Mary Rees has 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 ⩾ 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 × 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?
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/e43a33fb511fb1994b3326320f7ea4216609f3a2
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group (2014, background): “It is known that if K ′ is another Cantor set embedded in S(2), and if ∞′ is a point of S(2)−K ′, then there exists a homeomorphism of S(2) which maps K ′ on K and∞′ on ∞ (see for example the appendix A of Béguin, Crovisier & Le Roux [BCLR07]).”