Groups generated by iterations of polynomials over finite fields

Given a finite field of Fq elements, we consider a trajectory of the map u→ f(u) associated with a polynomial f ∈ Fq[X]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ≥ Cq^1/2 distinct consecutive elements of such a trajectory are contained in a small subgroup G of F∗q, improving the trivial lower bound #G ≥ N. Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.