Seminar 3.28.19 Chaika

Title: A prime transformation with many and big self-joinings

SpeakerJon Chaika (University of Utah)

Abstract: Let (X,μ,T) be a measure preserving system. A factor is a system (Y,ν,S) so that there exists F with SF=FT and so that F pushes μ forward to ν. A measurable dynamical system is prime if it has no non-trivial factors. A classical way to prove a system is prime is to show it has few self-joinings, that is, few T×invariant measures on X×X that project to μ. We show that there exists a prime transformation that has many self-joinings which are also large. In particular, its ergodic self-joinings are dense in its self-joinings and it has a self-joining that is not a distal extension of itself. As a consequence we construct the first known rank 1 transformation that is not quasi-distal and show that being quasi-distal is a meager property in the set of measure preserving transformations, which answers a question of Danilenko. This talk will not assume previous familiarity with joinings or prime transformations. This is joint work with Bryna Kra.