Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$

Question

Let $Y$ be a hyperbolic manifold that fibers over $S^1$, with fibration $\pi:Y \to S^1$ with fiber $\Sigma$. Thurston states that the monodromy $\phi:\Sigma \to \Sigma$ of this projection is then pseudo-Anosov. Moreover, we can write $Y = N/\mathbb{Z}$ where $N \to \mathbb{R}$ is a hyperbolic manifold fivering over $\mathbb{R}$ with fiber $\Sigma$ and $\mathbb{Z}$ acts on $N$ by isometries. Let $\Phi$ be the generator of the $\mathbb{Z}$-action.

Question. Is there a way of constructing a canonical smooth respresentative of the monodromy $\phi$ of the fiber bundle using the deck transformation $\Phi$?

For instance, you could imagine that there is a projection $N \to \Sigma$ (perhaps with geodesic fibers) where $\Phi$ permutes the fibers of the projection, and where $\Phi$ descends to a diffeomorphism of $\Sigma$ isotopic to $\phi$.

I'm particularly interested in the case of a fibered hyperbolic knot complement.

Answer 1

Here is an answer of sorts; it is not completely canonical though. First of all, you have to pick a conformal or hyperbolic structure on the fiber $\Sigma$. This can be made almost canonical, since there is a unique $\phi$-invariant (real) Teichmuller geodesic. In any case, you get $\Sigma={\mathbb H}^2/\Gamma$, where $\Gamma$ is $\pi_1(\Sigma)$ acting on the hyperbolic plane (isometrically, freely, properly). The map $\phi$ induces a canonical $\Gamma$-equivariant homeomorphism $f: S^1\to S^1$ (of the boundary circle of the hyperbolic plane). Now, use the Douady-Earle extension to extend $f$ to a $\Gamma$-equivariant diffeomorphism $F$ of the hyperbolic plane. Lastly, project $F$ to a diffeomorphism $\phi$ of the surface $\Sigma$.

Answer 2

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).