Smooth automorphisms and path-connectedness in Borel dynamics

Abstract

Let 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 Aut 0(X, B ) = Aut(X, B)/Ctbl(X) is path-connected with respect to the quotient topology τ 0. It is also proved that Aut 0(X, B) has the Rokhlin property in the quotient topology p 0, i.e., the action of Aut 0(X,B) on itself by conjugation is topologically transitive.