论文标题
熵，虚拟阿贝尔和香农轨道等效
Entropy, virtual Abelianness, and Shannon orbit equivalence
论文作者
论文摘要
我们证明，如果两个免费的P.M.P. $ \ mathbb {z} $ - 动作是香农轨道等效的，然后它们具有相同的熵。该论点也更普遍地用于免费的P.M.P.得出相同的结论。有限生成的实际上是阿贝尔群体的动作。连同Ornstein和Ornstein-Weiss的同构理论以及在非虚拟环节环境中Austin和Kerr-Li的熵不变性结果，这表明，如果他们仅在shannon等等，即使他们是conjugate conjugate，shannon imbit等等。我们还显示，在随机频谱的另一端，每个$ \ mathbb {z} $ - 里程表是Shannon Orbit等同于通用$ \ Mathbb {Z} $ - 差异计。
We prove that if two free p.m.p. $\mathbb{Z}$-actions are Shannon orbit equivalent then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein-Weiss and the entropy invariance results of Austin and Kerr-Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every $\mathbb{Z}$-odometer is Shannon orbit equivalent to the universal $\mathbb{Z}$-odometer.
