链接扭曲映射的遍历性
Ergodic properties of linked-twist maps
摘要：
我们研究一类同胚曲面统称为链接扭曲映射。我们引入了一个抽象的定义，使我们能够给一个精确的表征的财产观察到的其他作者，即这种地图属于两个类称为合作和反扭曲。我们挑出三个具体的链接扭曲地图，每个上的两个环面，在平面上和两个球，并证明了一个定理，其遍历性质的不变勒贝格测度。对于环面上的映射，我们证明了存在一个不变的零测度康托尔集，其上的动力学与符号序列空间上的全移位拓扑共轭。这些特征通常被称为拓扑马蹄铁。对于平面上的映射，我们证明了存在一个满测度集，其上的动力学在测度理论上同构于符号序列空间上的一个满移位。这通常被称为伯努利性质，并在一定条件下验证了沃伊特科夫斯基的一个猜想。我们在球面上引入映射，并证明它也具有伯努利性质。最后，我们得出一些结论，从我们的经验，关于如何可能扩展的结果，我们有特定的链接扭曲地图，我们已经定义的抽象链接扭曲地图。
英文摘要：
We study a class of homeomorphisms of surfaces collectively known as linked-twist maps. We introduce an abstract definition which enables us to give a precise characterisation of a property observed by other authors, namely that such maps fall into one of two classes termed co- and counter-twisting. We single out three specific linked-twist maps, one each on the two-torus, in the plane and on the two-sphere and for each prove a theorem concerning its ergodic properties with respect to the invariant Lebesgue measure. For the map on the torus we prove that there is an invariant, zero-measure Cantor set on which the dynamics are topologically conjugate to a full shift on the space of symbol sequences. Such features are commonly known as topological horseshoes. For the map in the plane we prove that there is a set of full measure on which the dynamics are measure-theoretically isomorphic to a full shift on the space of symbol sequences. This is commonly known as the Bernoulli property and verifies, under certain conditions, a conjecture of Wojtkowski's. We introduce the map on the sphere and prove that it too has the Bernoulli property. We conclude with some conjectures, drawn from our experience, concerning how one might extend the results we have for specific linked-twist maps to the abstract linked-twist maps we have defined.
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