  Given a 3-manifold $M$ fibering over the circle, we investigate how the
asymptotic translation lengths of pseudo-Anosov monodromies in the arc complex
vary as we vary the fibration. We formalize this problem by defining normalized
asymptotic translation length functions $\mu_d$ for every integer $d \ge 1$ on
the rational points of a fibered face of the unit ball of the Thurston norm on
$H^1(M;\mathbb{R})$. We show that even though the functions $\mu_d$ themselves
are typically nowhere continuous, the sets of accumulation points of their
graphs on $d$-dimensional slices of the fibered face are rather nice and in a
way reminiscent of Fried's convex and continuous normalized stretch factor
function. We also show that these sets of accumulation points depend only on
the shape of the corresponding slice. We obtain a particularly concrete
description of these sets when the slice is a simplex. We also compute $\mu_1$
at infinitely many points for the mapping torus of the simplest hyperbolic
braid to show that the values of $\mu_1$ are rather arbitrary. This suggests
that giving a formula for the functions $\mu_d$ seems very difficult even in
the simplest cases.
