Timeline for Uncountable preimage of every point
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Nov 29, 2010 at 20:32 | comment | added | Nikita Kalinin | Do you know something about Brownian preimages? Does Brownial be a example for my question? | |
| Nov 29, 2010 at 3:22 | comment | added | Pablo Shmerkin | For some random fractals, on the other hand, it is possible to show that the dimension of all linear (and more general) fibers is the dimension of the fractal minus $1$. This is closely related to some of my research in progress. However, for the question as hand, it is much easier to do an ad-hoc construction, as done by Sergei Ivanov (and generalized in my answer). | |
| Nov 29, 2010 at 3:17 | comment | added | Pablo Shmerkin | It is an extremely difficult problem to find the dimension of all (say, linear) fibers of a non-trivial deterministic fractal. Several open questions by Furstenberg (motivated by deep dynamical problems) can be reinterpreted as asking for the dimension of the intersections of certain fractals with all lines in a given direction (or in all directions, depending on the problem). In particular, I guess it is very hard to do this for the Koch snowflake. (continued) | |
| Nov 28, 2010 at 14:24 | comment | added | gowers | I have the same problem with the Koch snowflake: I think abstract results about projecting sets with given Hausdorff dimension will give almost sure statements. But it feels plausible that it should be true. | |
| Nov 28, 2010 at 13:56 | comment | added | Nikita Kalinin | Brownian motion almost surely has uncountable sero set. The similar is true for any other point. But I am not assured that we can commutate these words. I don't now whether is Brownian motion almost surely have uncountable preimage of every point. | |
| Nov 28, 2010 at 13:46 | comment | added | S. Carnahan♦ | I don't know much about probability, but it sounds like Brownian motion gives you dimension 1 almost surely, but makes it hard to say things about pointwise lower bounds. (Also, congratulations for reaching 10k.) | |
| Nov 28, 2010 at 10:49 | history | answered | gowers | CC BY-SA 2.5 |