Let f be a polynomial defined over the integers. Then f induces a map from Z/pZ to itself for all p. To a somewhat surprising extent, the maps one obtains in this way often resemble random maps in terms of their behavior for large p. We will make this resemblance precise and discuss generalizations to more general self-maps of varieties defined over number fields. The main techniques here are Galois representations on trees and the Chebotarev density theorem for function fields.