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

Let μ be a probability measure with an infinite compact support on R. Let us further assume that Fn: = fn∘ ⋯ ∘ f1 is a sequence of orthogonal polynomials for μ where (fn)n=1 ∞ is a sequence of nonlinear polynomials. We prove that if there is an s0∈ N such that 0 is a root of fn′ for each n> s0 then the distance between any two zeros of an orthogonal polynomial for μ 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 Fk. 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. © 2016, Akadémiai Kiadó, Budapest, Hungary.