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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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