Timeline for Symmetries of the standard probability space
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23, 2013 at 0:20 | comment | added | Henrique de Oliveira | @TomLaGatta Sorry, my understanding of Nguyen's paper is probably worse than yours. I had asked myself a similar question and I found that paper, but I have yet to understand it. Unfortunately I had to put this to the side for now; if you find the answer let me know. | |
| Oct 19, 2013 at 4:21 | answer | added | Jason Rute | timeline score: 3 | |
| Oct 19, 2013 at 2:58 | comment | added | Nate Eldredge | @TomLaGatta: I'm confused; the identity map is not ergodic. | |
| Oct 19, 2013 at 2:33 | comment | added | Jason Rute | I realized I was mistaken on the ergodic maps being a closed subset. I think they are actually dense. (Let me check for sure, then I will write up an answer.) | |
| Oct 19, 2013 at 2:30 | comment | added | Tom LaGatta | @HenriquedeOliveira: Nguyen's paper looks amazing, thank you. It states that $\Gamma$ is an absolute retract, hence homeomorphic to a separable Hilbert space. Can you describe in an answer what the absolute-retract property is, and why this implies that $\Gamma$ is homeomorphic to a separable Hilbert space? | |
| Oct 19, 2013 at 2:19 | comment | added | Tom LaGatta | @JasonRute: that's a great list of structures. Could you please expand your comment into an answer? In particular, I do not know anything about the Ky Fan metric; Wikipedia doesn't even have an article on it! If you can describe it, I can start an article. en.wikipedia.org/wiki/Ky_Fan_metric | |
| Oct 19, 2013 at 2:15 | comment | added | Tom LaGatta | @MikaeldelaSalle: The subgroup of ergodic automorphisms consists of those automorphisms which are ergodic. Sorry if this wasn't clear; I edited the post to emphasize it. | |
| Oct 19, 2013 at 2:13 | history | edited | Tom LaGatta | CC BY-SA 3.0 |
added 19 characters in body
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| Oct 18, 2013 at 23:58 | comment | added | Jason Rute | Actually, the topology I mentioned can also be given by any of the $L^p$ norms ($1\le p < \infty$) (since the maps are bounded and real-valued). | |
| Oct 18, 2013 at 23:14 | comment | added | Jason Rute | I don't know what you are looking for. $\Gamma$ has a natural metric structure on it under the Ky-Fan metric which is the metric of convergence in probability. The space $\Gamma$ is a closed sunspace (under this metric) of the space of measurable maps from the standard probability space to [0,1]. The ergodic maps are a closed subspace of $\Gamma$. | |
| Oct 18, 2013 at 21:18 | comment | added | Mikael de la Salle | The ergodic transformations do not form a group... | |
| Oct 18, 2013 at 21:17 | comment | added | Henrique de Oliveira | This paper seems relevant. ams.org/journals/proc/1990-110-02/S0002-9939-1990-1009997-6/… | |
| Oct 18, 2013 at 21:15 | comment | added | Henrique de Oliveira | @YvesCornulier My point was that I don't know the answer to the question even with more structure. So a first exercise could be to characterize with the extra metric structure, and then try to make sense of that when we go to a more general space. | |
| Oct 18, 2013 at 20:00 | comment | added | YCor | If $G$ is a semisimple connected Lie group, then it admits a lattice $\Lambda$ and the action of $G$ on $G/\Lambda$ is faithful, this probably embeds $G$ into your group $\Gamma$ (since $G/\Lambda$ is a standard probability space). | |
| Oct 18, 2013 at 19:55 | comment | added | YCor | @Henrique: certainly you don't want to use the metric structure on $[0,1]$, since you don't want something sensitive on the topology. | |
| Oct 18, 2013 at 19:41 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Oct 18, 2013 at 18:56 | comment | added | Henrique de Oliveira | If we're allowed to use the metric structure of $[0,1]$, we can use the supnorm to define a metric on $\Gamma$. I'm not sure if this makes sense when we only consider the Borel $\sigma$-algebra though; just an idea. | |
| Oct 18, 2013 at 18:33 | comment | added | YCor | Usually in $Aut(I,B,\lambda)$, one identifies things that coincide outside a null subset. Besides, for your question with Lie groups, in "embeds as a subgroup" do you have some topology in mind on the automorphism group? You probably want the function $g\mapsto g(A)\cap B$ to be continuous for all measurable $A,B$. | |
| Oct 18, 2013 at 18:20 | history | asked | Tom LaGatta | CC BY-SA 3.0 |