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@SamNead thank you for the reference by Ghys. I think Daniel Asimov meant two smooth foliation which are not smooth equivalent but are topological equivalent
@DanielAsimov May be there is a counterexample in the Novikov book or Toender book? Thank you for your chalenging question. (Existence of two foliation topological equivalent but not smooth equivalent)
@DanielAsimov I think I was mistaken, Sorry. Because an smooth equivalent need not preserve the derivative of Poincare return map. My mistake initiated from the following confusion: Two smooth equivalent singularity have conjugate(similar) linear part. So do you agree my argument and hence your argumenyt do not work?
Two Kronecker foliations of tori with slops $\alpha, \beta$ are not topological equiivalent if $\alpha , \beta$ do not lie on the same orbits of the action of $SL(2,\mathbb{R}$ via action $\pmatrix{a&b\\c&d}.z=\frac{az+b}{cz+d}$. Now Consider the product foliations by $F_1\times S^1$ and $F_2\times S^1$. In this way, can we obtain two topological equivalent foliation of $\mathbb{T}^3$ which are not smoothly equivalent?
