Let $F$ be a smooth foliation of a torus. Assume that $F$ can be mapped by a homeomorphism to an irrational-straight-line foliation $L$. Does it follow that $F$ can be mapped to $L$ by a diffeomorphism?


I am interested in 2 dimensional case and higher dimensional case.



A question about regularity of foliations
As far as I understand, the answer to the question is no, you can check this in
Handbook of Dynamical Systems, Volume 1, Part 1 By Boris Hasselblatt, Anatole Katok, page 173. This question is identical to the following -- suppose we have a diffeo of S^1 toplogically conjugate to an irrational rotation, can we make this conjugation a diffeo? Here is the sitation from the book:


For smooth and analytic circle 
diffeomrophisms with extremely well approximable rotation
number, the conjugacy to a roation and hence the invariant 
measure tend to be singular. Arnold's theorem 
exposed sigularity of the conjugacies as a generic phenomenon
in typical one-parameters families of real-analityc maps


