Medynets K. Approximation of transformations of standard Borel spaces and Cantor sets

The objects of the study are the group of Borel automorphisms of a standard Borel space and group of homeomorphisms of a Cantor set. Goals: the study of approximation theory for dynamical systems and its application to the classification of dynamical systems. Methods: methods of group theory, algebraic and set topology, measure theory and measurable functions, descriptive set theory. New theoretical results: the density of periodic ho-meomorphisms in the group of all homeomorphisms of a Cantor set, con-struction of the Bratteli-Vershik model for any aperiodic homeomorphism, it is shown that the commutator of the full group of a minimal homeomorphism is a complete invariant for orbit equivalence, topological properties of the set of smooth Borel automorphisms are studied. Employment: the obtained results are relevant for a further study of ape-riodic dynamical systems as well as operator algebras associated to them.