Smooth automorphisms and path-connectedness in Borel dynamics

Abstract

Let $Aut(X,\mathcal{B})$ be the group of all Borel automorphisms of a standard Borel space $(X,\mathcal{B})$. We study topological properties of $Aut(X,\mathcal{B})$ with respect to the uniform and weak topologies, $\tau$ and $p$, defined in [Bezuglyi S., Dooley A.H., Kwiatkowski J., Topologies on the group of Borel automorphisms of a standard Borel space, Preprint, 2003]. It is proved that the class of smooth automorphisms is dense in $(Aut(X,\mathcal{B}),p)$. Let $Ctbl(X)$ denote the group of Borel automorphisms with countable support. It is shown that the topological group $Au{t}_0(X,\mathcal{B})=Aut(X,\mathcal{B})/Ctbl(X)$ is path-connected with respect to the quotient topology ${\tau }_0$. It is also proved that $Au{t}_0(X,\mathcal{B})$ has the Rokhlin property in the quotient topology $p_0$, i.e., the action of $Au{t}_0(X,\mathcal{B})$ on itself by conjugation is topologically transitive.

arXiv comment17 pages. Indag. Mathem., to appear