Timeline for A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 8, 2018 at 19:42 | comment | added | Ali Taghavi | @thedude You are well come and thank you for your attention to my question. | |
| Jul 8, 2018 at 19:37 | answer | added | coudy | timeline score: 5 | |
| Jul 8, 2018 at 19:19 | comment | added | thedude | @AliTaghavi I thought the geodesic flow was defined on the manifold, but now I see it is defined on the tangent bundle. So I learned something from your question. Thank you. | |
| Jul 8, 2018 at 16:33 | vote | accept | Ali Taghavi | ||
| Jul 8, 2018 at 15:20 | comment | added | R W | @thedude No - please check carefully the definition of ergodicity and the definition of the geodesic flow. | |
| Jul 8, 2018 at 15:11 | comment | added | thedude | @AliTaghavi A torus geodesic with irrational slope will come arbitrarily close to any point. Is this not your definition of ergodic? | |
| Jul 8, 2018 at 15:03 | comment | added | Lee Mosher | I believe that if $M$ is a closed hyperbolic surface, and if you then alter the metric to be negatively curved except at a single point of zero curvature, the resulting geodesic flow will still be ergodic. I do not have a proof at hand. | |
| Jul 8, 2018 at 14:52 | answer | added | R W | timeline score: 13 | |
| Jul 8, 2018 at 14:45 | comment | added | R W | @Ali Taghavi - Oups - sorry - of course the geodesic flow on the torus is not ergodic - since the slope is preserved | |
| Jul 8, 2018 at 14:29 | comment | added | Ali Taghavi | @RW What is the argument for ergodicity? May I ask you to read my previous comment? | |
| Jul 8, 2018 at 14:28 | comment | added | Ali Taghavi | @thedude but am I mistaken to think that it is not ergodic because the unit tangent bundle possess two disjoint copy of the torus $T^2$ which separate the unit tangent bundle and is invariant under the geodesic flow? | |
| Jul 8, 2018 at 12:57 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
edited title
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| Jul 8, 2018 at 12:29 | history | asked | Ali Taghavi | CC BY-SA 4.0 |