  In this paper we will develop a general approach which shows that generalized
"critical relations" of families of locally defined holomorphic maps on the
complex plane unfold transversally. The main idea is to define a transfer
operator, which is a local analogue of the Thurston pullback operator, using
holomorphic motions. Assuming a so-called lifting property is satisfied, we
obtain information about the spectrum of this transfer operator and thus about
transversality. An important new feature of our method is that it is not
global: the maps we consider are only required to be defined and holomorphic on
a neighbourhood of some finite set.
  We will illustrate this method by obtaining transversality for a wide class
of one-parameter families of interval and circle maps, for example for maps
with flat critical points, but also for maps with complex analytic extensions
such as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1},
for real maps we obtain {\em positive} transversality (where $>0$ holds instead
of just $\ne 0$), and thus monotonicity of entropy for these families, and also
(as an easy application) for the real quadratic family.
  This method additionally gives results for unimodal families of the form
$x\mapsto |x|^\ell+c$ for $\ell>1$ not necessarily an even integer and $c$
real.
