an:06138184
Zbl 1260.37004
Tikhonov, S. V.
Complete metric on mixing actions of general groups
EN
J. Dyn. Control Syst. 19, No. 1, 17-31 (2013).
00314689
2013
j
37A25 28D05 37A05 37A15
complete separable metric space
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.
Geoffrey R. Goodson (Towson)
Zbl 1140.37005