Abstract: 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 ? 7? ?(? ) associated with a rational function ? ? Fp(X). We use bounds of exponential sums to show that if N ? p1/2+? 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 ?(X) = (a X + b)/(cX + d) ? Fp(X), with ad 6= bc and c 6= 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

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