Rational relations as rational series

A rational relation is a rational subs et of the direct product of two free monoids: R subset of or equal to A* x B*. Consider R as a function of A* into the family of subsets of B* by posing for all u is an element of A*, R(u) = {v, is an element of B*\ (u, v) is an element of R}. Assume R(u) is a finite set for all u is an element of A*. We study how the cardinality of R(u) behaves as the length of u tends to infinity and we show that there exists an infinite hierachy of growth functions.