Genericity in topological dynamics
Genericity in topological dynamics
复制标题
拓扑动力学的通用性
DOI：
10.1017/s0143385707000521
复制
发表时间：
2006
影响因子：
0.9
通讯作者：
M. Hochman
中科院分区：
文献类型：
--
作者：
M. Hochman
摘要我们研究了Cantor集的同胚空间和适当大的移位空间的子移位空间中动力学性质的通用性。格拉斯纳-金类型的对应将这些相当不同的背景联系在一起：其中一个的共性等同于另一个的共性。通过在移位空间模型中应用符号技巧，我们得到了关于Cantor集的传递和完全传递同胚的动力学性质的通用性的新结果。我们证明了万能里程表的同构类在传递系统空间中是一般的。另一方面，完全传递系统的空间表现出更加多样的动力学。特别地，我们证明了在这个空间中，每一个没有周期点的Cantor系统的同构类是稠密的，并且具有如下性质：极小性、零熵、固定完全传递系统的不交性、弱混合、强混合和极小自并.后两种情况与保值措施类别的情况形成鲜明对比。我们还证明了保测范畴中动力性质的通用性与支持具有相同性质的不变测度的系统的通用性之间的对应关系。
Abstract We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner–King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing and minimal self joinings. The latter two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.