In order to obtain a good statistical theory for a system with a hole in it, the heuristic is that the (exponential) speed of mixing must dominate the (exponential) rate at which mass leaks from the system: so the hole must be appropriately `small'. I'll present joint work with Mark Demers where we analysed this idea for a simple class of systems (Manneville-Pomeau maps with certain `geometric' equilibrium states), giving a complete picture of how the competition between mixing and escape lead to different statistical behaviour. We show a transition from the usual picture of good statistical properties, through a (non-trivial) zone where mixing and escape match exactly, with a terminal transition to subexponential mixing.