The pseudo-Anosov homeomorphism is a diffeomorphism away from finitely many 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 by Sakata ).

In general though there will be interior singularities of the pseudo-Anosov map of the fiber. Gerber and Katok proved 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).

The pseudo-Anosov homeomorphism 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 by Sakata ).

In general though there will be interior singularities of the pseudo-Anosov map of the fiber. Gerber and Katok proved 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).