Approximation in ergodic theory, Borel, and Cantor dynamics

This survey is focused on the results related to topologies on the groups of transformations in ergodic theory, Borel, and Cantor dynamics. Various topological properties (density, connectedness, genericity) of these groups and their subgroups are studied. In this paper, we intend to present a unified approach to the study of topological properties of transformation groups in ergodic theory, Borel, and Cantor dynamics. We consider here the group Aut(X,B, μ) (and Aut0(X,B, μ)) of all non-singular (measure-preserving, resp.) automorphisms of a standard measure space, the group Aut(X,B) of all Borel automorphisms of a standard Borel space, and the group Homeo(Ω) of all homeomorphisms of a Cantor set Ω. The basic technique in the study of transformation groups acting on an underlying space is to introduce various topologies into these groups which make them topological groups and investigate topological properties of the groups and their subsets (subgroups). The study of topologies on the group of transformations of a space has a long history. The first significant results in this area are the classical results of Oxtoby and Ulam on the typical dynamical behavior of homeomorphisms which preserve a measure [O-U]. This circle of problems has attracted attention in various areas of dynamical systems, notably, in measurable and topological dynamics, where it is important for many applications to understand what kind of transformations is typical for certain dynamics. Of course, this problem assumes that a topology is defined on the group of all transformations. The topologies which are usually considered on groups Aut(X,B, μ) and Aut0(X,B, μ) were defined 60 years ago in the pioneering paper by Halmos [Hal 1]. He called them the uniform and weak topologies. Using these topologies, turned out to be very fruitful and led to many outstanding results in ergodic theory. The best known theorems concerning ergodic, mixing, and weakly mixing automorphisms of a measure space were obtained by P. Halmos and V.A. Rokhlin (see [Hal 1], [Hal 2], [Ro 1], [Ro 2]). Many further statements on approximation of automorphisms of 1991 Mathematics Subject Classification. Primary 37A40, 37B05, 03066.