Abstract. We consider families of holomorphic maps defined on subsets of the complex plane,
and develop a technique to treat unfolding of (generalized) critical relations aiming in particular to obtain monotonicity of entropy for families of interval maps. Main result can be stated (informally) as follows: under a curtain condition (called "the lifting property") either the critical relations unfold transversally or they persist along a non-trivial complex manifold whose dimension is equal to the geometric multiplicity of 1 as an eigenvalue of the associated transfer operator. More recently, we show that this technique can also be used to deal with cases where the (infinite) critical orbit converges to a hyperbolic attracting or a parabolic cycle. As an application, we show that periodic points on the real line do not disappear after born for many families of real maps, from the real quadratic x^2+c (where this fundamental fact was known though our approach gives a local and simpler proof) to new families of the form g(x)+c and cf(x).
Joint work with Weixiao Shen and Sebastian van Strien.