Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique
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 ?
Keywords
Cite
@article{arxiv.math/0605438,
title = {Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique},
author = {François Béguin and Sylvain Crovisier and Frédéric Le Roux},
journal= {arXiv preprint arXiv:math/0605438},
year = {2016}
}