Ornstein and Katznelson will investigate problems in the following areas: Groups of diffeomorphisms of the circle, application of "circle techniques" to twist maps; statistical stability; entropy, returns rates and data compression; Bernoulli systems; ergodic and topological dynamic methods in combinatorics and in particular identifying the characteristic factors of ergodic systems for given exponent polynomials; and regularity properties for maps of fractals. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.
