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Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique

François Béguin·Sylvain Crovisier·Frédéric Le Roux
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Abstract

Mary Rees has constructed a minimal homeomorphism of the 2-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>1 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 of a Cantor set, we construct a homeomorphism f which "looks like" R from the topological viewpoint and "looks like" R*h 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 ?

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François Béguin, Sylvain Crovisier, Frédéric Le Roux. 2006-05-16. Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique. https://arxiv.org/abs/math/0605438

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