Entropy and mixing for amenable group actions.
Entropy and mixing for amenable group actions.
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DOI：
10.2307/121130
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发表时间：
2000-05
影响因子：
4.9
通讯作者：
D. Rudolph;B. Weiss
中科院分区：
文献类型：
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作者：
D. Rudolph;B. Weiss
对于可数服从群i，把i的作用看作标准概率空间的保测度变换，记作fT ∞g∞ 2 i作用于（X;F; n）。我们称f T ∞g∞ 2 i有完全正熵（或简称cpe），如果对X的任意有限非平凡划分P，熵h（T;P）不为零.我们的目标是证明什么是众所周知的行动，甚至d，即行动的完全正熵有很强的混合性质。设Si是i的有限子集的列表。我们称Si为扩散，如果任何特定的∞6 id属于至多有限多个集合Si i 1 i。定理0.1.对于fT∞g∞ 2 i是i的完全正熵的作用，P是任意有限划分，对于任意扩展的有限集合序列Siµ i，我们有
For i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space, written asfT∞g∞2i acting on (X;F;„). We sayfT∞g∞2i has completely positive entropy (or simply cpe for short) if for any flnite and nontrivial partition P of X the entropy h(T;P) is not zero. Our goal is to demonstrate what is well known for actions of and even d , that actions of completely positive entropy have very strong mixing properties. Let Si be a list of flnite subsets of i. We say the Si spread if any particular ∞6 id belongs to at most flnitely many of the sets SiS i1 i . Theorem 0.1. For fT∞g∞2i an action of i of completely positive entropy and P any flnite partition, for any sequence of flnite sets Siµ i which spread we have