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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.

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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 $k$).

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