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 know complement.