Examples of transformations that are totally ergodic but not weakly mixing?

Question

The most natural examples are irrational rotations. Are there other examples that are fundamentally different from irrational rotations? By the way, if $T$ is totally ergodic, but not weakly mixing, then there must be an eigenvalue of $U_T$ (the natural unitary operator associated to $T$) that is an irrational multiple of $2\pi$; this gives at least some connection between transformations that are totally ergodic but not weakly mixing, and irrational rotations.

To be extremely specific, I would like to construct (if possible) a rank-1 transformation with bounded cutting parameter that is totally ergodic but not weak mixing. I'm looking for examples which will help me understand the ways in which a transformation can have totally ergodicity but not weak mixing, and then use that understanding in developing the construction.

Answer 1

Totally ergodic is equivalent to not having rational eigenvalues (I guess a suitable reference for this is Eli's book).

Hence basically the Kronecker factor of such a system will be "essentially" the irrational rotation you've mentioned (as any Kronecker factor is "essentially" rotation over a compact topological group).

By Furstenberg's structure theorem, as long as your system is "nice", you basically extend the Kronecker factor by a weak-mixing extension, which won't effect the "totality" (as $T^{k}$ is weak mixing as well for any $k$).