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

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 homeomorphisin 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 x 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? (c) 2007 Published by Elsevier Masson SAS.