Timeline for Can the Reeb foliation of $S^3$ be realized as stable manifold foliation of a smooth hyperbolic discrete dynamic on $S^3$?
Current License: CC BY-SA 4.0
7 events
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| Aug 30 at 13:31 | comment | added | Ali Taghavi | but the torus admit both continuous and discrete(Arnold cat map) hyperbolic dynamics | |
| Jul 20, 2023 at 10:19 | comment | added | Martin M. W. | The unique torus leaf is mapped to itself. Picture what happens if it's tangent to a center and a stable direction: its area would have to shrink, which is impossible. On the other hand, if all you want is a non-identity map whose dynamics respect the Reeb foliation, there's a whole family of rotations (which of course have no stable or unstable directions). | |
| Jul 20, 2023 at 8:03 | comment | added | Ali Taghavi | BTW I mean does sphere admit a Partially hyperbolic diffeomorphism and is there one for which the reeb foliation is invariant(Maybe a union of stable and center manifolds)? | |
| Jul 20, 2023 at 7:56 | comment | added | Ali Taghavi | Thank you very much for your answer. I just relaized that manifolds with spher rational homology do not admit anasov diffeomorphism. So $S^3$ does not admit any Anasov diffeomorphism. What partial hyperbolic structure? Just another question, regardless of the above obstruction, why does your answer implies that we have no the required hyperbolic dynamic? Because the nearby leaves approaches the torus?Do you mean that an Anasov diffeomorphism on a compact manifold does not admit an invariant compact proper submanifold? If yes , why? | |
| Jul 20, 2023 at 0:50 | history | edited | Martin M. W. | CC BY-SA 4.0 |
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| Jul 19, 2023 at 23:28 | history | edited | Martin M. W. | CC BY-SA 4.0 |
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| Jul 19, 2023 at 23:23 | history | answered | Martin M. W. | CC BY-SA 4.0 |