For a while now, my bedtime reading has been a recent Dover reprint of Paul Halmos's 1956 booklet Lectures on Ergodic Theory. It's an interesting little work. At the end he lists a number of open problems in ergodic theory. Open, that is, in 1956. I imagine that some progress has since been made on at least some of these problems. So my question, directed mainly at those who have read the book and are knowledgeable about the current state of ergodic theory, is: what (major) modifications should I pencil into the margins on those pages? A somewhat related question: have any important new questions arisen in ergodic theory proper (not its applications to other mathematical areas per se) since Halmos's book?
Thanks for your time!