Return Times Distribution of Expanding Maps

We consider expanding systems with invariant measures that are uniformly expanding everywhere except on a small measure set and show that the limiting statistics of hitting times for zero measure sets are compound Poisson provided the limits for the cluster size distributions exist. This extends previous results from neighbourhoods around single points to neighbourhoods around zero measure sets. The assumptions require the correlations to decay at least polynomially and the non-uniformly expanding part of the iterates of the map also has to satisfy some decay condition. We also require some regularity conditions around the limiting zero measure target set.