I have previously encountered a strong working definition of complexity in ergodic theory (divide a wandering set into subsets based on their behavior, graph th

I have previously encountered a strong working definition of complexity in ergodic theory (divide a wandering set into subsets based on their behavior, graph their respective measures over time and do some analysis) and I'm surprised to see no use of its tools referred to here. If anybody has followed along with this project, do you know if they've since investigated the applicability of the notions of topological entropy and Hopf decomposition to their question?