流形上奇异极小唯一遍历同胚的构造：Denjoy-Rees技巧
Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique
摘要：
Mary Rees构造了具有正拓扑熵的2-环面的极小同胚。这个同胚f是通过丰富无理旋转R的动力学得到的。我们改进了Rees构造，允许从任何同胚R而不是无理旋转开始，并精确控制f的可测动力学。这产生了特别是以下结果：任何维数d>1的紧致流形携带一个最小的唯一遍历同胚也携带一个最小的唯一遍历同胚与积极的拓扑熵。更一般地，给定一个（紧）流形的同胚R和一个康托尔集的同胚h，我们构造一个从拓扑观点看“像”R的同胚f，从可测观点看“像”R*h。这种构造可以看作是对以下可实现性问题的部分回答：哪些可测动力系统由流形上的同胚表示？
英文摘要：
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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