  For one-dimensional random Schr\"odinger operators, the integrated density of
states is known to be given in terms of the (averaged) rotation number of the
Pr\"ufer phase dynamics. This paper develops a controlled perturbation theory
for the rotation number around an energy, at which all the transfer matrices
commute and are hyperbolic. Such a hyperbolic critical energy appears in random
hopping models. The main result is a H\"older continuity of the rotation number
at the critical energy that, under certain conditions on the randomness,
implies the existence of a pseudo-gap. The proof uses renewal theory. The
result is illustrated by numerics.
