Expansion of orbits of some dynamical systems over finite fields

Given a finite field Fp={0,p-1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation 〈→φ(〈) associated with a rational function φ ε Fp(X). We use bounds of exponential sums to show that if N≥p^1/2+ for some fixed then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions φ (X)=(aX+b)(cX+d)εF_p(X), with ad≠ and c≠0 we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.