Olof Sisask: On the orbits of points under polynomials in finite fields

Given a point x in the finite field F_p, and a polynomial f over that field, one can naturally look at the orbit x, f(x), f(f(x)), .... Eventually this must become periodic, and the orbits can be of varying sizes. But how concentrated can such an orbit be in space (thinking of F_p as {0,1,2,...,p-1}) -- can all the points lie bunched up somewhere in a small interval, say as one follows the orbit for some number M of steps? In this expository talk I shall discuss this problem and results by several authors.

Note: the room is in the old library.