Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic) I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the hierarchy:
$$
\mbox{Bernoulli} \subset \mbox{K-ergodic} \subset \mbox{strongly mixing} \subset \mbox{weakly mixing}\subset    \mbox{merely ergodic}
$$
and know plenty of examples of K-ergodic (e.g., Bunimovich stadium) and strongly mixing (e.g., irrational trianglular billiard) systems, I have not actually seen any reasonable examples of the weakly mixing and merely ergodic levels. By "reasonable," I mean something like a billiard or a Hamiltonian dynamical system (for my purposes, I need something, which would be straightforward enough to quantize).
My question is: is somebody familiar  with concrete examples of classical dynamical systems representing weakly mixing and merely ergodic levels?
 A: I'm not sure if these examples are generalizable for your purposes (I do symbolic dynamics, and the examples I like the most probably have nothing to do with quantum mechanics...), but:

*

*Every aperiodic translation action on a compact abelian group (e.g. irrational circle rotation) is ergodic with respect to Haar measure, but not weakly mixing.


*A typical interval exchange transformation (i.e. a piecewise defined slope $1$ self-map of $[0,1]$) with more than two intervals is weakly mixing but not strongly mixing with respect to Lebesgue measure.
A: Thanks, Ronnie.

*

*So, basically the first example takes a distribution on say a circle and just moves it around at an irrational angle every step, so that the distribution doesn't change at all, but "covers" densely the entire circle?
Is there anything qualitatively different if instead we translate it by a Liouville number or it's still merely ergodic?

Either way, a translation on an Abelian group may be generalizable to quantum dynamics. Ideally though, I'd like to see a billiard like that, but I suspect that it may not actually exist. Is it known if such examples exist, is it an interesting question from the mathematical point of view of studies of dynamical systems (an obstruction to having certain ergodic behaviors for billiards)?


*I don't understand the second example. My general intuition about weak mixing, you take a distribution and it does spread (mix) but "every once in a while" (measure zero set in terms of time dynamics) collapses back to  its original form or just anything with a finite measure < 1 (the area of the entire phase space). So, if you integrate over time, you don't notice such collapses. It seems a pretty pathological situation, and suspect that there may be no example of it  for billiards (or "reasonable" Hamiltonian dynamics vaguely defined) either.

My unprofessional "conjecture" is that for billiards (and a certain class of Hamiltonian dynamical systems, to be defined), the ergodic hierarchy collapses to Kolmogorov $\subset$ strongly mixing  and that's all (+ integrable, which is not ergodic at all: e.g., just a ball bouncing off the walls of a square). I would be very interested to see a counterexample.
