Pure Maths Colloquium: Nicolai Haydn

We consider maps with invariant measures and look at the time it takes for random points to enter a given (positive measure) set. For return times a theorem of Kac states that on average the return times are the reciprocal of the size of the return set. The actual return times however have a distribution which typically allows for arbitrarily long returns. We want to consider a sequence of shrinking sets that contract to a single point or, possibly, to some more complicated zero measure set. With good control over the shrinking target sets one can prove meaningful limiting results which tend to be either a straight Poisson distribution or a compound Poisson distribution.