Timeline for Dynamics of a random stretch map
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 27 at 7:34 | comment | added | Nate River | ... in fact probably even probability theory. But if I knew what the deep link was, I would presumably already be writing a paper on it! | |
| Jul 27 at 7:33 | comment | added | Nate River | Ah, thank you. I don't know anything about their physical significance, but in my opinion these are more than a serious way of studying the world of mathematics. Since I don't know the surrounding areas very well, I cannot say anything specific, but I am sure there are deep links to number theory, ergodic theory, harmonic analysis and fractals/geometric measure theory. | |
| Jul 27 at 6:09 | comment | added | Anthony Quas | More recently people generalized this to sequences of $\beta$. I’d say that this is more of a toy model than a serious way to understand the world. | |
| Jul 27 at 6:07 | comment | added | Anthony Quas | Renyi started studying expansions of real numbers in non-integer bases. It turns out you can algorithmically write an $x\in[0,1]$ as $\sum d_n\beta^{-n}$ for any $\beta>1$, where the “digits” are allowed to take values in 0,…,$\lceil\beta\rceil-1$. This expansion may be non-unique, but there is a unique “greedy expansion”. That is what Renyi studied. It turns out it is closely related to the dynamics of $x\mapsto \beta x\bmod 1$. Renyi showed these dynamics could be studied using Ergodic theory. | |
| Jul 26 at 11:39 | vote | accept | Nate River | ||
| Jul 26 at 11:39 | comment | added | Nate River | Interesting, I am not sure if I have understood your proof yet, but it checks out. Nice! Intuitively, what do random $\beta$-shifts represent? | |
| Jul 25 at 5:33 | history | answered | Anthony Quas | CC BY-SA 4.0 |