Timeline for Transitive geodesics on closed surfaces of genus greater than one
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 6, 2013 at 20:09 | vote | accept | alvarezpaiva | ||
| Aug 8, 2013 at 13:59 | comment | added | Vladimir S Matveev | You better ask Victor about it -- he explained me the idea of the proof but I did not check the whole proof/construction. As far I remember the metric is smooth and quadratically convex. | |
| Aug 8, 2013 at 8:15 | comment | added | alvarezpaiva | @VladimirSMatveev: but is this example smooth and quadratically convex? There are some really odd (and interesting) things that can happen (Zoll tori, for example) if you depart a bit from the orthodox definition of a Finsler metric. | |
| Aug 7, 2013 at 21:28 | comment | added | Vladimir S Matveev | to @Andrey Gogolev As far as I know the example itself was not discussed in the literature but the idea (which you and Misha Katz also explained in your answers) is definitely not new and the additional information came from study of Liouville metrics on the torus which is not a big deal and can be obtained by hands | |
| Aug 7, 2013 at 21:24 | comment | added | Vladimir S Matveev | to @alvarezpaiva Everybody wants to see this example of Bangert which is actually not an explicit example but a proof of the existence. The example is not real anylytic of course and is obtained as a limit of a sequence of finsler metrics | |
| Aug 7, 2013 at 20:21 | comment | added | Andrey Gogolev | This is a much better example because it is robust. In fact, if you put negative curvature on the necks you can probably say a something about the "chaotic part". This is a difficult subject in dynamics: coexistence of KAM and hyperbolic dynamics. Your example is geometric and I wonder if it was discussed in the literature. I know several references (on coexistence), but none talks about coexistence for the geodesic flow. | |
| Aug 7, 2013 at 19:44 | comment | added | alvarezpaiva | I think Gabriel Paternain proved that a real analytic Finsler metric on a surface of genus g > 1 cannot be integrable. That makes me quite curious about this example of Bangert !! | |
| Aug 7, 2013 at 18:12 | comment | added | Vladimir S Matveev | Of course. Sorry. As always I skipped the introduction and went to the question immediately. By the way, Victor Bangert claimed that he could proved the existence of a finsler metric on a closed surface of genus >1 whose geodesic flow is integrable -- he did not publish the proof as far as I know though | |
| Aug 7, 2013 at 18:00 | comment | added | alvarezpaiva | @Matveev: ?? that's the question posted. | |
| Aug 7, 2013 at 17:30 | comment | added | Vladimir S Matveev | What is Morse-Hedlund theorem and why one can not make it better? | |
| Aug 7, 2013 at 17:16 | comment | added | alvarezpaiva | Thanks Vladimir. It does seems then that the Morse-Hedlund theorem is as good as it gets. By the way that theorem has now been extended to Finsler metrics by Gomes and Ruggiero (for reversible metrics it is just the same techniques, but the extension is not trivial for non-reversible metrics). | |
| Aug 7, 2013 at 16:33 | history | answered | Vladimir S Matveev | CC BY-SA 3.0 |