Smooth automorphisms and path-connectedness in Borel dynamics

AbstractLet Aut(X, B) be the group of all Borel automorphisms of a standard Borel space (X, B). We study topological properties of Aut(X, 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, B), p). Let Ctbl(X) denote the group of Borel automorphisms with countable support. It is shown that the topological group Aut0(X, B) = Aut(X, B)/Ctbl(X) is path-connected with respect to the quotient topology τ0. It is also proved that Aut0(X, B) has the Rokhlin property in the quotient topology p0, i.e., the action of Aut0(X,B) on itself by conjugation is topologically transitive