Timeline for Universal structure of fractal spaces
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 3, 2019 at 17:59 | comment | added | Wojowu | For $\alpha=1$, you have fat Cantor sets which have Hausdorff-dimension $1$ and the interval doesn't map to it nontrivially, so $[0,1]$ won't work. I think by considering a suitable family of fat Cantor sets we might be able to prove such a set doesn't exist. | |
| May 3, 2019 at 17:55 | comment | added | Pietro Majer | There are instead structure theorems available for spaces of given Hausdorff dimension. You may check Federer's book. | |
| May 3, 2019 at 17:53 | history | edited | QCD_IS_GOOD | CC BY-SA 4.0 |
added 30 characters in body
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| May 3, 2019 at 17:52 | comment | added | QCD_IS_GOOD | @Wojowu sorry I didn't mean an object with unique morphisms to every object - I got confused with terminology. I just meant an object with a morphism to every object | |
| May 3, 2019 at 17:51 | comment | added | Pietro Majer | Of course there is a huge variety of possible topological local types (or bi-lipschitz if you prefer), even if you assume local connectedness | |
| May 3, 2019 at 17:50 | comment | added | Wojowu | For $\alpha=1$ there is no initial object - if $X$ where one, then there should be a unique map $X\to [0,1]$. But since $X$ is nonempty, we can compose this with a map $[0,1]\to [0,1/2]\subseteq [0,1]$ and get a different such map. Same idea should work for any $\alpha>0$, just find a fractal of a suitable dimension. | |
| May 3, 2019 at 17:39 | history | asked | QCD_IS_GOOD | CC BY-SA 4.0 |