Abstract: Bernoulli transformations form one of the few classes of ergodic transformations which can be classified completely up to measurable conjugacy. In a sense, Bernoulli transformations are rather boring: they are shift operators on cartesian products $(Y^\mathbb{Z}, \nu^\mathbb{Z})$ of some probability space $(Y,\nu)$. The interesting part of the story is, of course, how to recognize whether a given ergodic transformation is of this special form. There are some well-known examples: shifts of finite type, Anosov diffeomorphisms, nonhyperbolic toral automorphisms, to name only three.

Bernoulli transformations can be characterised as those maps which are most random in the sense that their `past' evolution gives least information about its `future'. Implicit in this description is the notion of a linear time evolution, which is, of course, absent when dealing with ergodic actions of countable groups other (or `bigger') than $\mathbb{Z}$. This talk will discuss a few examples of Bernoulli actions of $\mathbb{Z}^d, \ge 2$ which are not obviously recognizable as being Bernoulli. The last part of the talk (if I ever get there in time) will attempt to give a small glimpse of the intriguing world of Bernoulli actions of nonamenable countable groups.