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This is a continuation of A question about regularity of foliations>this question answered by Dmitri.

Let $F$ and $F'$ be smooth ($C^\infty$) foliations of a manifold $M$. Assume that there is a homeomorphism $h$ that takes $F$ to $F'$.

  1. Assume $F$ has a compact leaf, can one conclude there is a diffeomorphism that takes $F$ to $F'$?
  2. Assume that $h$ is $C^1$, can one conclude there is a ($C^\infty$) diffeomorphism that takes $F$ to $F'$?
  3. Assume that $M$ is a torus and $F'=L$ is a Diophantine-irrational-line foliation of the torus. Then, in case of $M=T^2$ following the outline of Sam Nead one can show that there's a diffeomorphism that takes $F$ to $L$, does this hold for higher dimensional tori?
  4. Dmitri gives us certain obstruction to existence of a diffeomorphism. Are there other interesting obstructions?
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1 Answer

After some pondering, here is a negative answer to 1.

Consider the foliation from Figure 2 of these notes http://www.crm.cat/Conferences/0910/Acfoli/Hurder.pdf

The return map on the parallel is $C^1$ only if the derivative of the return map at the unique fixed point is 1. However if we draw a similar picture with two compact leaves (circles) then we can get a smooth foliation such that the return map has two fixed points with derivatives a>1 and b<1. And it is possible to perturb this foliations so that a and b are perturbed to different numbers a' and b'. This perturbation can be carefully designed so that the new foliation is continuously conjugate to the old one.

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