Timeline for Is a random circle rotation weak mixing almost surely?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Oct 28 at 4:12 | comment | added | Nate River | In any case, if random dynamical systems are of interest at all, I can’t imagine that their ergodic properties would not be of interest. | |
| Oct 28 at 4:08 | comment | added | Nate River | I think at least ergodicity of random transformations is interesting in the context of “card shuffling”, or in general trying to uniformly mix a space with random operations… | |
| Oct 28 at 4:06 | comment | added | Nate River | @Anthony Quas Only one way to find out 😁 | |
| Oct 28 at 4:02 | comment | added | Anthony Quas | I would say the definitions are good definitions if they lead to interesting behaviour. For a regular system, ergodicity is interesting, because, for example, the conclusion of Birkhoff's theorem is particularly nice. The regular definition of weak-mixing is nice because it relates to eigenvalues, and can be leveraged to prove theorems such as Furstenberg/Szemerédi's theorem about arithmetic progressions in dense subsets of the integers. It may be that the random versions are similarly consequential. Or it may not... | |
| Oct 27 at 17:59 | comment | added | Nate River | I am not too familiar with the formal theory of random dynamical systems, but asking for ergodic/mixing properties of the random system feels rather natural no? @AnthonyQuas | |
| Oct 27 at 14:09 | comment | added | Anthony Quas | I guess I’m wondering whether the notion of “weak mixing for a random composition of maps” has been defined elsewhere, and whether it has proved to be useful. | |
| Oct 27 at 4:40 | comment | added | Nate River | So for every choice of $\omega$, you get a fixed sequence of transformation and we ask if this is weak mixing almost surely wrt $\omega$. Will Sawin's answer shows in fact it's almost surely not weak mixing! @AnthonyQuas | |
| Oct 27 at 4:38 | comment | added | Anthony Quas | I would have asked what you meant by weak-mixing. It's normally defined for a fixed transformation, when it is equivalent to the non-existence of non-constant eigenfunctions. Did you mean the condition that Will Sawin showed does not hold in his answer? (this seems the most reasonable guess). Maybe it would make more sense to define the random rotation as a skew product $T\colon [0,1)^\mathbb Z\times [0,1)\to [0,1)^\mathbb Z\times [0,1)$ that is $T(\omega,t)=(\sigma(\omega),t+\omega_0\bmod1)$. | |
| Oct 26 at 16:23 | history | became hot network question | |||
| Oct 26 at 12:03 | vote | accept | Nate River | ||
| Oct 26 at 11:24 | vote | accept | Nate River | ||
| Oct 26 at 11:47 | |||||
| Oct 26 at 11:22 | vote | accept | Nate River | ||
| Oct 26 at 11:23 | |||||
| Oct 26 at 11:13 | answer | added | Will Sawin | timeline score: 7 | |
| Oct 26 at 9:10 | comment | added | Nate River | If you take $[0, 1]^{\mathbb N}$ as a model for the probability space, then the natural measure is the countable product of the uniform measure, and almost surely means with respect to this. | |
| Oct 26 at 9:07 | comment | added | Nate River | Note that this is a different $[0, 1]$ than the underlying space of the dynamics. | |
| Oct 26 at 9:06 | comment | added | Nate River | I mean almost surely with respect to the probability space underlying the independent random variables $Z_i$. @AlessandroDellaCorte | |
| Oct 26 at 8:48 | comment | added | Alessandro Della Corte | What do you mean by "almost surely" on $[0,1]^{\mathbb{N}}$? | |
| Oct 26 at 8:37 | history | edited | Nate River | CC BY-SA 4.0 |
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| Oct 26 at 8:22 | history | asked | Nate River | CC BY-SA 4.0 |