I was reminded of this topic by some of the answers to this question, where it was noted that "typical" measure-preserving transformations are weak-mixing but not strong-mixing for several senses of "typical". As a result, it occurred to me that I do not know of any very natural, explicit examples of transformations which are weakly but not strongly mixing. So,
What are some good examples of measure-preserving transformations which are weak-mixing but not strong-mixing?
To clarify "good": I'm particularly interested in examples where it can be proved in a concise and self-contained manner that weak mixing occurs and strong mixing does not, in examples which arise constructively, and in examples which arise directly from a continuous transformation of a compact metric space (as opposed to abstract measure-theoretic constructions).
Thanks in advance!
Best Answer
The Chacon transformation has a nice and fairly explicit description as a uniquely ergodic subshift: set $B_0=0$ and set $B_{k+1}=B_kB_k1B_k$.
The subshift is the set of all infinite words, all of whose finite subwords occur as a someword of some $B_k$.
From this it is easy to see why the lengths $n=|B_k|$ fail to be good times for strong mixing. As mentioned previously Parry shows the weak mixingness.