EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

Given a finite field F-p = (0, ..., p - 1) of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation xi -> psi(xi) associated with a rational function psi is an element of F-p (X). We use bounds of exponential sums to show that if N >= p(1/2+epsilon) some fixed E 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 psi(x)= (aX + b)/(cX + d) is an element of F-p(X), with ad not equal bc and c not equal 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.