Entropy and Independence

Abstract

As Chapters 2 and 8 illustrate, the basic dichotomy between weak mixing and compactness for p.m.p. actions can be expressed not only using the probabilistic notion of independence and its representation-theoretic translation as orthogonality in the case that two sets are at play, but also through the combinatorial notion of independence, which connects it in a precise technical way to a corresponding theory of tameness and weak mixing in topological dynamics. From each of these viewpoints the concept of weak mixing can be formulated as asymptotic independence along a sequence of group elements, and so the dichotomy with compactness can be seen as one of infiniteness versus finiteness. In the combinatorial set-up, weak mixing reduces precisely to the question of whether or not independence occurs across an infinite subset of the group as it acts on tuples of sets.