Smooth automorphisms and path - connectedness in Borel dynamics

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 [BDK1]. 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. 0 Introduction In the present paper, we continue the study of topological properties of the group Aut(X, B) of all Borel automorphisms of a standard Borel space (X, B). We consider two topologies, τ and p, on Aut(X, B) which take their origins in ergodic theory. They were defined and studied in the context of Borel and Cantor dynamics in [BDK1, BDK2, BDM, BK1, BK2]. Recall that the topology τ is defined by the base of neighborhoods a direct analogue of the well known uniform topology on the group Aut(X, B, µ) of all non-singular automorphisms of a measure space generated by the metric d(S, T) = µ(E(S, T)). It is worthwhile to mention that, in fact, Aut(X, B, µ) is formed by classes of automorphisms coinciding µ-almost everywhere. It allows one to neglect the behavior of automorphisms on sets of zero measure.