Can the Reeb foliation of $S^3$ be realized as foliation associated to stable(or unstable) manifolds of a hyperbolic discrete dynamic on $S^3$?If yes what is a precise formulation for that Hyperbolic diffeomorphism?
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asked Jul 19, 2023 at 22:53
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No, because it has a unique leaf that is a torus. That can't be a stable or unstable manifold; one reason is that it can't get either bigger or smaller under the action of the hyperbolic dynamical system.
answered Jul 19, 2023 at 23:23
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$\begingroup$ 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? $\endgroup$ Commented Jul 20, 2023 at 7:56
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$\begingroup$ 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)? $\endgroup$ Commented Jul 20, 2023 at 8:03
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1$\begingroup$ 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). $\endgroup$ Commented Jul 20, 2023 at 10:19
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$\begingroup$ but the torus admit both continuous and discrete(Arnold cat map) hyperbolic dynamics $\endgroup$ Commented Aug 30 at 13:31