Abstract

One of the main discoveries in the theory of dynamical systems in the twentieth century is that deterministic systems can behave randomly. We survey what is known, and what is not known, about the interplay between the properties which are used to measure how random a smooth dynamical system is. On the qualitative side these are ergodicity, mixing, the $K$ property and the Bernoulli property; on the quantitative side they are decay of correlations, large deviations and the central limit theorem. We describe two rigidity results. The first is the Bernoulli property for homogeneous systems of positive entropy, which settles a conjecture of Dani. The second is the theorem that exponential mixing implies the Bernoulli property for smooth measure preserving diffeomorphisms. In both cases we explain the underlying geometric mechanism, which is based on the Ornstein--Weiss machinery and on equidistribution of unstable leaves. We then turn to flexibility. Using generalized $(T,T^{-1})$ transformations we construct smooth systems which satisfy the classical central limit theorem and have zero entropy, or are $K$ but not Bernoulli, or are mixing at an arbitrarily fast polynomial rate and still not Bernoulli. A number of open problems is collected along the way.