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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?

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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:

  1. 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.

  2. 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.

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Thanks, Ronnie.

  1. 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)?

  1. 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.

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    $\begingroup$ Naively speaking, quantum chaos deals with geodesic flows, hence have a ''diagonal behavior'' in the terms of ergodic theory (hyperbolic) while translations over compact abelian groups are ``unipotent'' (parabolic). The analog of geodesic flows over the torus say would be the $\times 2$ map (or any endomorphism) which in this case happens to be Bernoulli. (same thing with general automorphisms over $\mathbb{T}^2$, see the Adler-Weiss paper, also Ornstein-Weiss, etc). $\endgroup$
    – Asaf
    Commented Nov 17, 2021 at 16:57
  • $\begingroup$ I understand your thinking, but here I don't think your intuition is quite right; in fact in many classes of systems (including physically defined ones like billiards), it turns out that a typical system is weakly but not strongly mixing (with respect to the natural measure). I know that the definition might seem pathological, but it ends up being the "natural" one in some sense. As one explicit example, for a typical polygon with horizontal and vertical sides, the billiard is weakly (but not strong) mixing for a typical direction. (I can give references, but this is fairly Googlable). $\endgroup$
    – Ronnie Pavlov
    Commented Nov 17, 2021 at 19:48
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    $\begingroup$ Also, to make the second example clearer (though it sounds like you might prefer the billiards examples): an interval exchange is where you split the interval [0,1] into some number of pieces (it should be more than 2 for interesting behavior), and then permute those intervals in some nontrivial way. (see picture at researchgate.net/publication/309207330/figure/fig1/…) $\endgroup$
    – Ronnie Pavlov
    Commented Nov 17, 2021 at 19:54
  • $\begingroup$ Re: Asaf, one can still ask the following: Say I take 4 different models representing the four levels (no Bernoulli); I do like billiards, so let's say we take 4 billiards and instead of solving for classical dynamics, solve the eignevalue problem of the Laplace operator (i.e., the Schrodinger equation). Is there any generic difference between the spectra say of weakly mixing billiards and K-chaotic ones. It's probably always Wigner-Dyson level statistics for any level of ergodicity, but there could be other metrics. We have conjectures, but I want to check them explicitly. Hence my question. $\endgroup$
    – Victor Galitski
    Commented Nov 17, 2021 at 20:52

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