Complete metric on mixing actions of general groups. (English) Zbl 1260.37004

Tikhonov, S. V.

J. Dyn. Control Syst. 19, No. 1, 17-31 (2013).

Let \((X, \Sigma, \mu)\) be a Lebesgue probability space and let \({\mathcal A}\) be its group of invertible measure-preserving transformations with the weak topology. In this paper, continuous homomorphisms of a topological group \({\mathcal G}\) into \({\mathcal A}\), are studied (\({\mathcal G}\)-actions).
The measure preserving \({\mathcal G}\)-action \(\{ T^g \}_{g\in {\mathcal G}}\) is “mixing” if for any \(A, B \in \Sigma\) , \[ \mu(T^gA \cap B) \to \mu(A)\mu(B) \;\;\; \text{as}\;\;\;g\to \infty . \] Continuing the author’s study of \({\mathbb Z}\)-actions in [Sb. Math. 198, No. 4, 575–596 (2007; Zbl 1140.37005)], the general case of countably infinite groups is considered. It is the purpose of this paper to give a metric on the set \({\mathcal M}_{\mathcal G}\) of the set of mixing actions of \({\mathcal G}\) so that \({\mathcal M}_{\mathcal G}\) is a complete and separable metric space.

Reviewer: Geoffrey R. Goodson (Towson)

MSC:

37A25	Ergodicity, mixing, rates of mixing
28D05	Measure-preserving transformations
37A05	Dynamical aspects of measure-preserving transformations
37A15	General groups of measure-preserving transformations and dynamical systems

Keywords:

complete separable metric space

References:

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[2]	D. S. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1–141. · Zbl 0637.28015 · doi:10.1007/BF02790325
[3]	S. V. Tikhonov. Mixing transformations with homogeneous spectrum. Sb. Math. 202 (2011), No. 8, 1231–1252. · Zbl 1247.37008 · doi:10.1070/SM2011v202n08ABEH004185