Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique |
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Authors: | Franç ois Bé guin |
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Affiliation: | Université Paris-Sud, Laboratoire de mathématiques, Bâtiment 425, 91405 Orsay Cedex, France; Université Paris 13, CNRS - Laboratoire d'Analyse, Géométrie et Applications, Avenue J.-B. Clément, 93430 Villetaneuse, France; Université Paris-Sud, Laboratoire de mathématiques, Bâtiment 425, 91405 Orsay Cedex, France |
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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 dimensiond?2which 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 hC of a Cantor set, we construct a homeomorphism f which “looks like” R from the topological viewpoint and “looks like” R×hC 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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