Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings

We first study a family of invariant transformations for the integer moment problem. The fixed point of these transformations generates a positive measure with support on a Cantor set depending on a parameter q. We analyze the structure and properties of the set of orthogonal polynomials with respect to this measure. Among these polynomials, we find the iterates of the canonical quadratic mapping: F(x)=(x–q) ², q2. It appears that the measure is invariant with respect to this mapping. Algebraic relations among these polynomials are shown to be analytically continuable below q=2, where bifurcation doubling among stable cycles occurs. As the simplest possible consequence we analyze the neighborhood of q=2 (transition region) for q<2.