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In this paper we will develop a general approach which shows that generalized"critical relations" of families of locally defined holomorphic maps on thecomplex plane unfold transversally. The main idea is to define a transferoperator, which is a local analogue of the Thurston pullback operator, usingholomorphic motions. Assuming a so-called lifting property is satisfied, weobtain information about the spectrum of this transfer operator and thus abouttransversality. An important new feature of our method is that it is notglobal: the maps we consider are only required to be defined and holomorphic ona neighbourhood of some finite set.  We will illustrate this method by obtaining transversality for a wide classof one-parameter families of interval and circle maps, for example for mapswith flat critical points, but also for maps with complex analytic extensionssuch as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1},for real maps we obtain {\em positive} transversality (where $>0$ holds insteadof 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.

Positive Transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps