Generic Dynamics

The roots of studying generic properties in dynamical systems can be derived from the article by Oxtoby and Ulam from 1941 in which they showed that for a finite-dimensional compact manifold with a non-atomic measure which is positive on open sets, the set of ergodic measure-preserving homeomorphisms is generic in the strong topology. Subsequently, Halmos in 1944 introduced approximation techniques to a purely metric situation: the study of interval maps which are invertible almost everywhere and preserve the Lebesgue measure. He showed that the generic invertible map is weakly mixing, i.e., has continuous spectrum. Then, Rohlin in 1948 showed that the set of (strongly) mixing measure preserving invertible maps is of the first category. Two decades later, Katok and Stepin in 1967 introduced the notation of a speed of approximations. One of the notable applications of their method is the genericity of ergodicity and weak mixing for certain classes of interval exchange transformations. One of the most outstanding result using approximation theory is the Kerckhoff, Masur, Smillie result on the existence of polygons for which the billiard flow is ergodic, as well as its quantitative version by Vorobets. Many more details on the history of approximation theory can be found in surveys. In a recent series of articles the members of the project have studies the dynamical properties of generic continuous maps of the interval and circle which preserve the Lebesgue measure. Our investigations have lead to many new and interesting questions which we propose to study during this research in residence program.