A lower bound on the rank of mixing extensions
Let T be a measure-preserving transformation of a probability space (X,\(\mu)\). A partition \({\mathbb{P}}\) is a finite list of disjoint measurable sets whose union is all of X. For two partitions, say that \({\mathbb{P}}\epsilon\)-refines \({\mathbb{Q}}\) if for each member \(A\in {\mathbb{Q}}\) there exists a set B, a union of members of \({\mathbb{P}}\), such that \(\mu (A\Delta B)<\epsilon.\) Suppose a natural number r has the following property: Given \(\epsilon\) and any partition \({\mathbb{Q}}\), there exists r disjoint Rohlin stacks of a common height such that, letting \({\mathbb{P}}\) be the partition whose members are the levels of the stacks (and the complement of their union), \({\mathbb{P}}\epsilon\)-refines \({\mathbb{Q}}\). The rank of T is the infimum of all such numbers r. The result: If transformation R is a k-point extension of transformation T then \[ rank(R)\geq k\quad \cdot \quad rank(T) \] whenever T is mixing. Indeed, this remains true with the weaker requirement only that T have zero-rigidity. This is a second application of a technique developed in Isr. J. Math. 56, 102-122 (1986; Zbl 0626.47012)]. The fundamental lemma is proven there.
measure-preserving transformation of a probability space
scientific article; zbMATH DE number 4055966