Let
be a hyperbolic fibered
–manifold. We study
properties of sequences
of fibers and monodromies for primitive integral classes in the fibered cone of
.
The main object is the asymptotic translation length
of the pseudo-Anosov
monodromy
on the curve complex. We first show that there exists a constant
depending only on the fibered cone such that for any primitive integral class
in the fibered
cone,
is bounded
from above by
.
We also obtain a moral connection between
and the normal generating
property of
in the
mapping class group on
.
We show that for all but finitely many primitive integral classes
in an arbitrary
–dimensional slice
of the fibered cone,
normally generates the mapping class group on
. In
the second half of the paper, we study if it is possible to obtain a continuous
extension of normalized asymptotic translation lengths on the curve complex as a
function on the fibered face. An analogous question for normalized entropy has been
answered affirmatively by Fried and the question for normalized asymptotic
translation length on the arc complex in the fully punctured case has been answered
negatively by Strenner. We show that such an extension in the case of the curve
complex does not exist in general by explicit computation for sequences in the fibered
cone of the magic manifold.