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, $τ$ 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 $Aut_0(X,\mathcal B)=Aut(X,\mathcal{B})/Ctbl(X)$ is path-connected with respect to the quotient topology $τ_0$. It is also proved that $Aut_0(X,\mathcal B)$ has the Rokhlin property in the quotient topology $p_0$, i.e., the action of $Aut_0(X,\mathcal B)$ on itself by conjugation is topologically transitive.