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

##### 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
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##### References:
 [1] S. V. Tikhonov. A complete metric in the set of mixing transformations. Sb. Math. 198 (2007), No. 4, 575–596. · Zbl 1140.37005 · doi:10.1070/SM2007v198n04ABEH003850 [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
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