The [pseudo-Anosov homeomorphism][1] is a diffeomorphism away from finitely many singular points, so in general will not be a smooth diffeomorphism. However, for certain fibered knots there are not singularities away from the punctures, and hence the map is a smooth diffeomorphism and unique up to smooth conjugacy. This works for the figure 8 knot and more generally certain 2-bridge knots (considered in [this paper][2] by 
Sakata ). 

In general though there will be interior singularities of the pseudo-Anosov map of the fiber. Gerber and Katok [proved][3] that a pseudo-Anosov map is topologically conjugate to a smooth diffeomorphism. So this gives a kind of canonical smooth diffeomorphism realization, but only up to topological conjugacy (I do not know if diffeomorphism realizations are smoothly conjugate; there are also [analytic realizations][4]). 


  [1]: https://en.wikipedia.org/wiki/Pseudo-Anosov_map
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=3482493
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=672479
  [4]: https://mathscinet.ams.org/mathscinet-getitem?mr=783000