In a joint work with Daniel Litt we borrow ideas from complex dynamics to show that for any normal variety X/k there is an integer N, such that any l-adic arithmetic local system on X which is trivial modulo l^N is trivial. In this talk I will explain the meaning of all of the words in the previous sentence, how this result relates to concrete questions in arithmetic, and what all of this has to do with Siegel disks inside noncommutative balls.