On irreducible divisors of iterated polynomials

D. Gomez-Perez, A. Ostafe, A. P. Nicolas and D. Sadornil have recently shown that for almost all polynomials f is an element of F-q[X] over the finite field of q elements, where q is an odd prime power, their iterates eventually become reducible polynomials over IF,. Here we combine their method with some new ideas to derive finer results about the arithmetic structure of iterates of f. In particular, we prove that the nth iterate of f has a square-free divisor of degree of order at least n(1+0) (1) as n -> infinity (uniformly in q).