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A zero-one law for dynamical properties. (English) Zbl 0909.28014
Nerurkar, M. G. (ed.) et al., Topological dynamics and applications. A volume in honor of Robert Ellis. Proceedings of a conference in honor of the retirement of Robert Ellis, Minneapolis, MN, USA, April 5–6, 1995. Providence, RI: American Mathematical Society. Contemp. Math. 215, 231-242 (1998).
Let $$\Gamma$$ be a countable group and $$Y$$ a probability space. An action of $$\Gamma$$ is a collection $$TT= \{T^\gamma:\gamma\in \Gamma\}$$ of measure-preserving transformations so that $$T^{\alpha+\beta}= T^\alpha\circ T^\beta$$. Let $$\mathbb{A}$$ denote the set of $$\Gamma$$-actions, $$\Phi$$ the group of bi-measure-preserving transformations $$\varphi$$ and $$\varphi(T):= \varphi T\varphi^{-1}$$. We assume that $$\Gamma$$ has the “weak Rohlin property”: $$\Phi(TT)$$ is dense in $$\mathbb{A}$$ for some action $$TT$$. The zero-one law asserts that for each “dynamical property”, the set of $$\Gamma$$-actions with that property is either residual or meager. The class of groups with the weak Rohlin property includes all countable discrete amenable groups.
For $$\Gamma$$ an arbitrary countable group, and $$\mathbb{A}$$ the set of $$\Gamma$$-actions on the unit circle, it is shown that a dynamical property is Baire/meager/residual in $$\mathbb{A}$$ if and only if it is Baire/meager/residual in the set of shift invariant measures on $$Y^\Gamma$$.
For the entire collection see [Zbl 0882.00043].

##### MSC:
 28D15 General groups of measure-preserving transformations 60F20 Zero-one laws
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