Settled polynomials over finite fields (Q2884407)

If \(K\) be a field and \(f,g\in K[X]\), then \(g\) is \(f\)-stable, if the composition \(g\circ f^n\) (\(f^n\) denoting the \(n\)-th iterate of \(f\)) is irreducible over \(K\). Moreover let \(s_n\) be the sum of degrees of \(f\)-stable polynomials dividing \(f^n\) (according to their multiplicity as factors of \(f^n)\). The polynomial \(f\) is called \textit{settled} if the ratio \(s_n/\deg f^n\) tends to \(1\). It is conjectured that if \(K\) is a finite field of odd characteristic and \(f=aX^2+bX+c\), (\(a\neq0, f\neq X^2\)), then \(f\) is settled. The authors show that such \(f\) is stable if and only if the adjusted critical orbit of \(f\), i.e. the set \(\{-f(\gamma),f^2(\gamma),f^3(\gamma),\dots\}\) with \(\gamma=-b/(2a)\), contains no squares. They show also that if \(f\) is quadratic with all iterates separable, then factorizations of the sequence of iterates of \(f\) can be described by an irreducible absorbing Markov process. A conjecture which makes this description precise is presented (Conjecture 3.6) and computational evidence of it is given.