New universality classes associated to fractals

Abstract

The local behavior of zeros of orthogonal polynomials is determined by the local scaling behavior of Christoffel-Darboux (CD) kernels. All previously studied behaviors are described by scaling limits, and different values of the limit kernel correspond to different universality classes. In this paper, we describe new universality classes in which instead of a single limit kernel, there is a limit cycle. These are naturally suited to Cantor spectra and to fractal behaviors of the measure. We show that these new phenomena occur for two canonical models with singular measures: the middle third Cantor measure and the balanced/equilibrium measure on a real Julia set of an expanding polynomial. In particular, this is the first result on the local behavior of CD kernels for an almost periodic operator with singular spectrum. As a complementary result, we describe the asymptotics of the scaling function, and the Christoffel function, at a fixed point of a quadratic iteration. This is the first such analysis for an almost periodic model with singular spectrum. It allows us to conclude that, at the fixed point, the local scaling of zeros of the polynomial of degree $n$ is precisely of order $n^{-1/\alpha}$, where $\alpha$ is the local dimension of the measure. We also study the limit chain in this case, and prove that it can be parametrized by its asymptotics with respect to a Martin function (so-called $M$-type); this is the first result of this kind for a chain with a singular measure.