Every action of a non-amenable group is the factor of a small action
Every action of a non-amenable group is the factor of a small action
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不服从群体的每一个行动都是小行动的因素
DOI：
10.3934/jmd.2014.8.251
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Brandon Seward
中科院分区：
文献类型：
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作者：
Brandon Seward
众所周知，如果 $G$ 是可数服从群，且 $G \curvearrowright (Y, \nu)$ 因式分解到 $G \curvearrowright (X, \mu)$ 上，则第一个动作的熵必须大于或等于第二个动作的熵。特别是，如果 $G \curvearrowright (X, \mu)$ 具有无限熵，则动作 $G \curvearrowright (Y, \nu)$ 不允许任何有限生成分区。另一方面，我们证明，如果$G$是一个可数的不可数群，则存在一个有限整数$n$，其具有以下属性：对于每个概率测度保留动作$G \curvearrowright (X, \mu)$，在$n^G$上存在一个$G$不变概率测度$\nu$，使得$G \curvearrowright (n^G, \nu)$因式分解到$G \curvearrowright (X, \mu)$上。对于许多不服从的群体，$n$ 可以选择为 $4$ 或更小。对于紧凑空间和连续因子图上的连续动作，我们也获得了类似的结果。
It is well known that if $G$ is a countable amenable group and $G \curvearrowright (Y, \nu)$ factors onto $G \curvearrowright (X, \mu)$, then the entropy of the first action must be greater than or equal to the entropy of the second action. In particular, if $G \curvearrowright (X, \mu)$ has infinite entropy, then the action $G \curvearrowright (Y, \nu)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable non-amenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G \curvearrowright (X, \mu)$ there is a $G$-invariant probability measure $\nu$ on $n^G$ such that $G \curvearrowright (n^G, \nu)$ factors onto $G \curvearrowright (X, \mu)$. For many non-amenable groups, $n$ can be chosen to be $4$ or smaller. We also obtain a similar result with respect to continuous actions on compact spaces and continuous factor maps.