Abstract
In this paper we investigate the location of zeros of Hankel determinants with iterated polynomial entries. Such determinants arise within the context of numerical algorithms for acceleration of functional iteration. If p is a monic polynomial of degree ≥2 and Hn(p, z) = det [p(1+1)(z)]0<i,j<n, we show that as n→∞ ‘most’ of the zeros of Hn(p, z) tend to the filled-in Julia set for p. Moreover, we exhibit classes of polynomials p for which the normalised zero counting measures for the Hn(p, z) converge in the weak-star sense to the invariant balanced measure (equilibrium distribution) of the Julia set p. We also study the class of analytic functions f for which the qth iterate f−(q) is the identity; such functions are called generalized roots of unity (GRU). We characterise all rational functions that are GRU as specific Mobius maps and we determine all second order GRUs that are real-analytic in a neighbourhood of the origin.