Timeline for Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)
Current License: CC BY-SA 4.0
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| Nov 17, 2021 at 20:57 | history | edited | Victor Galitski | CC BY-SA 4.0 |
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| Nov 17, 2021 at 20:52 | comment | added | Victor Galitski | 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. | |
| Nov 17, 2021 at 19:54 | comment | added | Ronnie Pavlov | 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/…) | |
| Nov 17, 2021 at 19:48 | comment | added | Ronnie Pavlov | 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). | |
| Nov 17, 2021 at 16:57 | comment | added | Asaf | 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). | |
| Nov 17, 2021 at 15:46 | history | edited | Victor Galitski | CC BY-SA 4.0 |
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| Nov 17, 2021 at 15:40 | history | edited | Victor Galitski | CC BY-SA 4.0 |
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| Nov 17, 2021 at 15:23 | history | answered | Victor Galitski | CC BY-SA 4.0 |
