A hierarchy of topological systems with completely positive entropy

A note on derivatives, expansions and $Π^1_1$-ranks

“…Definition 2.2 Let X be a Polish space, C ⊆ X and ϕ : C → ω 1 a rank on C. We say that ϕ is a Π 1 1 -rank if C is Π 1 1 and there are relations P, Q ⊆ X 2 , with one of them Σ 1 1 and the other Π 1 1 , such that for all y ∈ C we have that…”

“…The following are fundamental results on Π 1 1 -ranks [5]. Theorem 2.4 Every Π 1 1 set admits a Π 1 1 -rank. Theorem 2.5 Let C be a Π 1 1 set and ϕ be a…”

“…At the dawn of descriptive set theory, Lebesgue made an infamous mistake by assuming that the continuous projection of a Borel set was Borel. Suslin spotted this mistake 10 years later and began the study of analytic or Σ 1 1 sets. One of tools which was developed later to understand the complexity of Borel sets and the difference between Borel and co-analytic sets were Π 1 1 -ranks.…”

“…One of tools which was developed later to understand the complexity of Borel sets and the difference between Borel and co-analytic sets were Π 1 1 -ranks. One of the most well known Π 1 1 -ranks is the Cantor-Bendixson rank. Kechris developed a very general set up were, using the concept of derivatives (or the dual version of expansions), one can prove that the Cantor-Bendixson rank, as well as several other natural ranks, are Π 1 1 (see Theorem 2.7) [5].…”

“…Though the Γ-rank can be stated in terms of expansions it does not fit exactly in the context of the dual of Theorem 2.7. In this note, we adapt the proof given in [5] in order to show that the Γ-rank is a Π 1 1 -rank. A concrete example of the Γ-rank is the entropy rank for topological dynamical systems that was introduced by Barbieri and the second author [1] to classify dynamical systems with completely positive entropy.…”