Entropy and mixing for amenable group actions

Daniel J. Rudolph and Benjamin Weiss

Review from Zentralblatt MATH:

This paper extends the well-known result that completely positive entropy implies mixing of all orders for actions of ${\Bbb Z}^d$ to actions of countable amenable groups -- by transferring these notions through an orbit equivalence in an unexpected way. Let $G$ be a countable amenable group with an action $T=\{T_{g}\}_{g\in G}$ of $G$ on a standard probability space $(X,\cal{B},\mu)$. The action $T$ has {\sl completely positive entropy} if, for any partition $P$ of $X$, $H(X)>0\Rightarrow h(T,P)>0$ (here the entropy of $T$ is defined using Følner sequences in $G$). The action $T$ is {\sl mixing of all orders} if for any $k\ge 1$ and sets $A_0,\dots,A_k\in\cal{B}$, $$ \mu\left( T_{g_0^{-1}}(A_0)\cap T_{g_1^{-1}}(A_1)\cap\dots\cap T_{g_k^{-1}}(A_k) \right)\longrightarrow\mu(A_0)\mu(A_1)\dots\mu(A_k) $$ as $g_ig_j^{-1}\to\infty$ in $G$ for $i\neq j$. For groups in which there is a convenient notion of past, actions with completely positive entropy are known to be mixing of all orders. For amenable group actions, both properties make sense, but in general there is no natural past to use in developing the entropy theory. Despite this, a good entropy theory has been developed for amenable group actions by {\it D. Ornstein} and {\it B. Weiss} [J. Anal. Math. 48, 1-141 (1987; Zbl 0491.28018)] this action is orbit equivalent to an action of $\Bbb Z$. The orbit equivalence lifts to give a $\Bbb Z$-action $S$ orbit equivalent to $T\times T''$, with the orbit equivalence measurable in the second coordinate. The relative Pinsker algebra over the second coordinate -- the smallest algebra containing all partitions $P$ with $h(T\times T'',P|{\cal B}'')= 0$ -- is known to be the span of ${\cal B}''$ and the Pinsker algebra of $T$, and so is just ${\cal B}''$ in the case of completely positive entropy. Thus $S$ has relatively completely positive entropy over ${\cal B}''$.

The existing theory of relatively completely positive entropy maps is extended and complemented by new ideas in relative mixing and relative orbit equivalence theory to give the main result. Call a finite set $S\subset G$ is {\sl$K$-spread} for a finite set $K\subset G$ if $g\neq h$ in $S$ implies $gh^{-1}\notin K$. Then the main result (Theorem 2.3) states that for any finite partition $P$ and $\epsilon>0$, there is a finite set $K\subset G$ with the property that $$ \left|{1\over{|S|}}h\left( \bigvee_{g\in S}T_{g^{-1}}(P)\right) -H(P) \right|<\epsilon $$ for any $K$-spread finite set $S$.

As the authors mention, this seems to be the first significant application of orbit equivalence methods to classical aspects of entropy theory. The possibility that such methods may be of use in resolving the open problems concerning the spectral theory of amenable group actions with completely positive entropy, and the nature of Følner sequences along which pointwise ergodic theorems may hold, is raised and discussed briefly.

Reviewed by Thomas Ward

Keywords: entropy; amenable group action; mixing

Classification (MSC2000): 37A20 37A25

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