Local Entropy and Descriptive Complexity

Blanchard introduced the concepts of Uniform Positive Entropy (UPE) and Complete Positive Entropy (CPE) as topological analogues of K-automorphism. He showed that UPE implies CPE, and that the converse is false. A flurry of recent activities study the relationship between these two notions. For example, one can assign a countable ordinal which measures how complicated a CPE system is. Recently, Barbieri and Gracia-Ramos constructed Cantor CPE system at every level of CPE. Westrick showed that natural rank associated to CPE systems is actually a Π^1_1-rank. More importantly, she showed that the collection of CPE Z_2 SFT's is a Π^1_1-complete set. In this talk, we discuss some results, where UPE and CPE coincide and others where we show that the complexity of certain classes of CPE systems is Π^1_1-complete. This is joint work with Garica-Ramos.