Timeline for When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?
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16 events
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| Oct 3, 2023 at 11:14 | comment | added | kvicente | It is possible to define ergodicity in non-compact spaces, there are easily findable references for that. However, it is a delicate matter. Now, because of the way symplectomorphisms of the cotangent bundle work, in order for $f^*$ to be ergodic a necessary condition would be that any $K\subset M$ with $\bar{K}\neq M$ and invariant by $f$, must have empty interior. So, this forces $K$ to be a rather complicate set, probably a Cantor like set where $f$ behaves somehow like a horseshoe map. | |
| Oct 3, 2023 at 10:55 | comment | added | Ali Taghavi | @kvicente but as you interestingly indicated invariant finit measure subsets of the cotangent bundle is a necessary point which can generates new interesting question | |
| Oct 3, 2023 at 10:53 | comment | added | Ali Taghavi | @kvicente however i wonder if ergodicity can be defined in non finite measure too: a full measure set is a measurable set whose intersection with every compact set K has full measure $\mu(K)$ | |
| Oct 3, 2023 at 10:50 | comment | added | Ali Taghavi | @kvicente what about isometric condition? namely we assume that f preserves a Riemannian metric However the volum form of the cotangent bundle is independent of any metric on M | |
| Oct 3, 2023 at 10:20 | comment | added | Ali Taghavi | @kvicente very interesting and necessary point. thank you I come back very soon | |
| Oct 2, 2023 at 15:22 | comment | added | kvicente | Don't you need to assume there is compact subset of $T^*M$ invariant by $f^*$ and restrict the symplectic form to this subset in order to be able to have a finite measure space a properly define ergodicity? | |
| Sep 13, 2023 at 11:03 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| S Aug 23, 2023 at 12:02 | history | bounty ended | CommunityBot | ||
| S Aug 23, 2023 at 12:02 | history | notice removed | CommunityBot | ||
| S Aug 15, 2023 at 10:16 | history | bounty started | Ali Taghavi | ||
| S Aug 15, 2023 at 10:16 | history | notice added | Ali Taghavi | Draw attention | |
| Jul 22, 2023 at 19:38 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jul 22, 2023 at 10:48 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jul 21, 2023 at 20:32 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jul 21, 2023 at 20:20 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jul 21, 2023 at 19:52 | history | asked | Ali Taghavi | CC BY-SA 4.0 |