Timeline for Is it a known example of adic transformation ? (1)
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 26, 2012 at 18:44 | vote | accept | Stéphane Laurent | ||
| Aug 26, 2012 at 18:44 | comment | added | Stéphane Laurent | @RW Damned! I think I was lost because I denoted the blue path by 1101..., and then the action is $z\mapsto z-1$ instead of $z\mapsto z+1$ ! Thank you, this was really helpful ! | |
| Aug 26, 2012 at 18:23 | comment | added | R W | Re the edit of 26/08/2012. Maybe before asking any further you would care to at least read the construction of the isomorphism I am talking about? If you did, you could notice that the red and blue paths on your picture correspond to dyadic sequences 0010... and 1010..., which is precisely how it should be for the odometer. There are 3 parts in my answer: (1) the construction; (2) the claim that the tail equivalence relations coincide; (3) the claim that the lexicographic orders on the respective tail equivalence relations coincide. What precisely you don't understand? | |
| Aug 25, 2012 at 21:47 | comment | added | Stéphane Laurent | @RW I guess you know the Pascal adic and the Euler adic transformations. The Euler graph is the same as the Pascal graph but with multiple edges. Is is true that Euler adic is a suspension over the Pascal adic ? | |
| Aug 22, 2012 at 20:48 | comment | added | Stéphane Laurent | @RW That's right. I have found the definition in Einsiedler & Ward's book. Thanks ! @moderators: RW has answered the question of my 2nd post. I don't know whether I should move the 2nd post here ? Sorry for the inconvenience caused. | |
| Aug 22, 2012 at 17:35 | comment | added | Stéphane Laurent | @RW is it like "automorphism built under a function" ? (this terminology is from Nadkarni's book "Basic ergodic theory") | |
| Aug 22, 2012 at 17:11 | comment | added | Stéphane Laurent | I don't know what is a suspension. I will search but if you have a weblink to propose it would be welcomed ! | |
| Aug 22, 2012 at 17:10 | comment | added | R W | The obvious answer is then that it's a suspension over the 2-adic odometer - will this satisfy you? | |
| Aug 22, 2012 at 13:05 | comment | added | Stéphane Laurent | @RW Yes, exactly! | |
| Aug 21, 2012 at 13:50 | comment | added | R W | So, the graph itself is the same, but you need to take into account multiplicities of edges? | |
| Aug 21, 2012 at 11:54 | comment | added | Stéphane Laurent | @RW Sorry, in fact this is not the graph I am interested in. Please could you take a look at my new question: mathoverflow.net/questions/105152/… | |
| Aug 20, 2012 at 15:18 | comment | added | Stéphane Laurent | Thanks. I will try to check the details (this could take a long time since doing mathematics is not my current job) | |
| Aug 20, 2012 at 15:05 | comment | added | R W | Your graph is definitely different - however I described a coding which (with the exception of one path) establishes an isomorphism between its path space and the path space of the 2-adic odometer - I have omitted the proof that it's really an isomorphism, but you should be able to fill in the details. | |
| Aug 20, 2012 at 14:42 | comment | added | Stéphane Laurent | I meant my graph. I don't see how the odometer can be derived from a cut-and-stack approximation associated to my graph. But I am a beginner in this topic, maybe I need some time to see clearer. | |
| Aug 20, 2012 at 14:20 | comment | added | R W | You mean that the graph a described in the last comment (each level consists just of a single point, and there are two edges between any two consecutive levels) is different from the one in Petersen's lectures? Yes, it is (I don't really understand why he preferred to make things more complicated than they are) - however, the corresponding path spaces are isomorphic, so that the associated adic transformations are precisely the same. | |
| Aug 20, 2012 at 13:34 | comment | added | Stéphane Laurent | Are you sure ? This is not the usual Bratelli-Vershik graph associated to the odometer, isn't it (showed on page 17 here math.unc.edu/Faculty/petersen/lecturespdf.pdf )? | |
| Aug 20, 2012 at 11:49 | comment | added | R W | Sorry, I shouldn't have called it a shift (although it should be clear from my description that it's by no means a Bernoulli shift - one-sided or two-sided). What I mean is the "2-adic odometer transformation", i.e., $z\mapsto z+1$ on 2-adic integers, or, in other words, the adic transformation on the Bratteli diagram, for which each level consists just of a single point, and there are two edges between any two consecutive levels. | |
| Aug 20, 2012 at 11:48 | comment | added | Stéphane Laurent | ... in particular I would like to see the corresponding cut-and-stack construction. | |
| Aug 20, 2012 at 9:08 | vote | accept | Stéphane Laurent | ||
| Aug 26, 2012 at 9:16 | |||||
| Aug 20, 2012 at 9:07 | comment | added | Stéphane Laurent | After googling I find that the dyadic shift is $Tx=2x \text{ (mod $1$)}$. Please do you know a textbook or an online course on ergodic theory which gives the properties of $T$ ? | |
| Aug 20, 2012 at 5:31 | comment | added | Stéphane Laurent | Please what is the "2-adic shift" ? Is it Bernoulli shift on $\{0,1\}^\mathbb{Z}$ ? | |
| Aug 19, 2012 at 22:49 | history | answered | R W | CC BY-SA 3.0 |