The existence of acips for interval maps with critical points is usually related to the growth-rate of derivatives along the critical orbits. Recent results by Bruin, Shen and van Strien show that no growth is actually needed for unimodal maps, and more recently a new proof (with Rivera-Letelier) extends this result to multimodal maps. In this talk I want to present ideas of the proof(s) and context, and further open questions.