A trichotomy for hitting times and escape rates for a class of unimodal maps

We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz-Thurston type. We prove that for any point $z$ in the interval there is a local escape rate and hitting time statistics which is one of three types. While it is key that we cover all points $z$, the particular interest here is when $z$ is periodic and in the postcritical orbit which yields the third part of the trichotomy. We also prove generalised asymptotic escape rates of the form first shown by Bruin, Demers and Todd.