Smooth Realization and Conjugation By Approximation

Going back to the foundation work of von Neuman there is a question of whether there are smooth models of the models of classical ergodic theory. When both measure and map are required to be smooth there is only one known obstruction but essentially no general results. Within the class of zero entropy transformation we have a method called conjugation by approximation that can be used to realize many interesting properties. I will describe the method and some of the classical and modern consequences of this.