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@PietroMajer according to definition PS condition you mentioned above I think $exp$ satisfies this condition. On the other hand I think "exp" is not a counter example to the question in my last comment. Because in my last comment i search for an example of two non empty non diffeomorphic regular level set with values a and b such that [a b] consist of regular values. You said that in PS condition it is impossible but what about non PS condition
@PietroMajer Am I mistaken on diffeomorphicity of every two regular level set as above without PS condition? May be in Hirsch book we had the compactness of M which automatically implies the PS condition?
So I wonder if every two regular level set as above have similar dynamics? If there would be a counter example then it would be interesting to prove it with extra assumption Palais Smale condition.
@PietroMajer Thank you very much for your comment and the interesting point of Palais Smale condition. No I did not assume the Palais Smale condition. But I remember a theorem in Hirsch book differential topology that every two non empty level set are diffeomorphic provided there is no critical value between a and b. He use normalized gradient vector field to prove this theorem.
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)?
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?
