Continuous orbit equivalence rigidity
Continuous orbit equivalence rigidity
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DOI：
10.1017/etds.2016.98
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发表时间：
2015-03
影响因子：
0.9
通讯作者：
Xin Li
中科院分区：
文献类型：
--
作者：
Xin Li
我们迈出了更好地理解连续轨道等价的第一步，即具有连续共环的拓扑轨道等价。首先，我们用C^{\ast}$ -交叉积保持Cartan子代数的同构刻画了连续轨道等价。这是Singer和Feldman-Moore在可测量环境下的经典结果的拓扑模拟。其次，我们转向连续轨道等价刚性，即对于某些类型的拓扑动力系统，连续轨道等价是否意味着共轭的问题。我们通过构造连续轨道等价但不共轭的拓扑动力系统（自由阿贝尔群和非阿贝尔自由群的作用）来证明这种情况并非总是如此。此外，我们证明了正的刚性结果。例如，对于可解对偶群，一般拓扑伯努利作用和有限字母上全位移的某些子位移是刚性的。
We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterize continuous orbit equivalence in terms of isomorphisms of $C^{\ast }$ -crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Second, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this is not always the case by constructing topological dynamical systems (actions of free abelian groups and also non-abelian free groups) that are continuously orbit equivalent but not conjugate. Furthermore, we prove positive rigidity results. For instance, for solvable duality groups, general topological Bernoulli actions and certain subshifts of full shifts over finite alphabets are rigid.