Combinatorial Constructions in Ergodic Theory and Dynamics
The book is an updated version of the unpublished notes “Constructions in ergodic theory”, written by the author in collaboration with E.A.Robinson Jr., in 1982-83. The first part entitled “Approximations and genericness in ergodic theory”, deals with the approximations of measure-preserving transformations by periodic processes, which are permutations of partitions of the space. Several types of such periodic processes are considered and the speed of approximation is evaluated. The main theorem says that the set of measure-preserving transformations of a Lebesgue space, which admit a periodic approximation of a given type and speed, is residual. The second part entitled “Cocycles, cohomology and combinatorial constructions”, deals with algebraic constructions in ergodic theory. The concepts of rigidity, stability and effectiveness are investigated. Main applications include Diophantine translations of the torus, Anosov diffeomorphisms, horocycle flows and interval exchange transformations.