Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings

Let be a probability measure with an infinite compact support on . Let us further assume that is a sequence of orthogonal polynomials for where is a sequence of nonlinear polynomials. We prove that if there is an such that 0 is a root of f (n) ' for each then the distance between any two zeros of an orthogonal polynomial for of a given degree greater than 1 has a lower bound in terms of the distance between the set of critical points and the set of zeros of some F (k) . Using this, we find sharp bounds from below and above for the infimum of distances between the consecutive zeros of orthogonal polynomials for singular continuous measures.