A prime system with many self-joinings (Q2026757)

The authors study some indecomposable structures of measurable dynamics, the prime transformations. These are transformations with no (non-trivial) measurable factors. An important concept is the set of self-joinings of the system, that defines a Poulsen simplex (a simplex such that the extreme points are dense). The authors show the existence of a prime system \((Y,\mathcal{B},\nu, T)\), which has rank \(1\), is rigid, and has an ergodic self-joining \(\eta\) different from the product measure, such that \((Y\times Y,\mathcal{B}\times\mathcal{B},\eta,T\times T)\) is not a distal extension of \((Y,\mathcal{B},\nu, T)\), and the set of self-joinings is a Poulsen simplex. The system provides an elegant construction of an infinite product of finite symbols. A lot of ingenious analysis is provided.