Bezuglyi S. Dynamical systems on measurable, Borel, and Cantor spaces

Objects - hyperfinite groups of automorphisms of measurable and Borel spaces, homeomorphisms of a Cantor set. Goals - classification of transformation groups, study of topological properties of transformation groups. Methods of orbit theory and cocycle theory, group theory, general topology, measure theory, spectral operator theory. Results - the problem of outer conjugacy of countable amenable groups is solved, hyperfinite groups of automorphisms are classified with respect to weak equivalence, the structure of cocycles is found, the problem of extension and classification of ergodic actions of abelian groups is solved, the approximation theory of aperiodic transformations by means of periodic ones and odometers is constructed, the concept of Bratteli diagrams is applied for the realization of aperiodic Borel automorphisms, the closures of classes of minimal and transitive homeomorphisms is found, the structure of topological full groups is studied. Employment - the results are important for classification and investigation of properties of transformation groups in measurable, Borel, and Cantor dynamics.