The study of return time statistics of a dynamical system (X,f,μ)
involves fixing a typical point xєX and looking at the
distribution of return times to progressively smaller and smaller sets
around x. In hyperbolic situations it has been known for some years
that these distributions will be exponential. In the non-uniformly
hyperbolic situation, much less is known. In this talk I will present
a result, joint with Henk Bruin, showing that for a very large class
of multimodal interval maps we have exponential return time
statistics. This holds for a variety of measures on these systems
including the absolutely continuous invariant measures.