Timeline for Are almost all measure-preserving flows on compact manifolds ergodic?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2021 at 20:34 | vote | accept | Panopticon | ||
| Nov 30, 2021 at 18:30 | answer | added | D. J. Obata | timeline score: 6 | |
| Nov 19, 2021 at 21:26 | comment | added | user103342 | I think Steve Alpern's work gives a positive answer to your question. Take a look at the book "Typical Dynamics of Volume Preserving Homeomorphisms" by Alpern and Prasad. | |
| Nov 16, 2021 at 21:31 | comment | added | Moishe Kohan | Taking your comment as a definition, I think, the answer to your question depends on the details such as degree of differentiability and the precise topology you use. For instance, if you consider Hamiltonian flows on $S^2$, then KAM theory tells you that the density you are expecting fails, at least if you control enough derivatives, see here. On the opposite extreme, as a geometer, I like Lohkamp's theorem that Riemannian metrics with ergodic geodesic flows are dense in $C^0$-topology. | |
| Nov 16, 2021 at 21:05 | comment | added | Christian Remling | For a slightly silly, but concrete example, consider a finite phase space, of cardinality $n$, with counting measure. Only a small fraction $1/n$ of the measure preserving transformations (in other words, all bijections) give you an ergodic transformation (the permutations consisting of a single cycle). | |
| Nov 16, 2021 at 20:23 | comment | added | Panopticon | @MoisheKohan I suppose I mean "almost all" in the sense of dense with respect to an appropriate topology, in the same sense as one would say "almost all functions are Morse functions." | |
| Nov 16, 2021 at 19:34 | comment | added | Moishe Kohan | What do you mean by "almost all" in this case? The space of flows is infinite-dimensional, hence, there is no canonical measure on it. | |
| Nov 16, 2021 at 18:50 | comment | added | Christian Remling | I'm not sure if this sounds plausible. If you do it the other way around (fix the dynamics, vary the measure), then the ergodic measures are exactly the extreme points of the invariant measures, so would be rare if you have more than one. | |
| S Nov 16, 2021 at 18:10 | review | First questions | |||
| Nov 16, 2021 at 18:22 | |||||
| S Nov 16, 2021 at 18:10 | history | asked | Panopticon | CC BY-SA 4.0 |