Timeline for How quickly will billiard trajectories cluster?
Current License: CC BY-SA 2.5
4 events
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| Dec 6, 2010 at 0:55 | comment | added | Peter LeFanu Lumsdaine | @Vaughn: there is a possible middle way, between universal and probabilistic bounds — recent work by Towsner, Avigad, Tao and others (I’m a bit out of date on this) on the constructive content of ergodic theory gives a lot of strong statements along these lines. Typically, quantitative versions of the hypotheses (replacing “the trajectories are non-periodic” by some quantitative data on how far they are from being periodic) gives quantitative versions of the conclusions, such as (in this case, I expect) a hard upper bound on the clustering times. I know Henry Towsner is on MO, so hopefully… | |
| Nov 17, 2010 at 22:41 | comment | added | Vaughn Climenhaga | We can't give a universal upper bound, but we can give explicit estimates for the expected return time. Good point about not being able to say anything about every trajectory, though; the best we can do is something probabilistic. | |
| Nov 17, 2010 at 22:38 | comment | added | Joseph O'Rourke | Ah, that makes perfect sense! So I guess the question for air molecules involves random trajectories. | |
| Nov 17, 2010 at 22:34 | history | answered | Ian Morris | CC BY-SA 2.5 |