Monotonicity of entropy and positively oriented transversality for   families of interval maps
  In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps. 
Genadi Levin, Weixiao Shen and Sebastian van Strien
math.DS