Limit-Periodic Verblunsky Coefficients for Orthogonal Polynomials on the Unit Circle

TL;DR

This paper adapts Avila's method for limit periodic potentials to orthogonal polynomials on the unit circle, analyzing measures derived from limit periodic Verblunsky coefficients represented via Cantor group orbits.

Contribution

It introduces a novel approach to study limit periodic Verblunsky coefficients by representing them as sampling functions on Cantor groups, extending techniques from Schrödinger operators.

Findings

Representation of Verblunsky coefficients as Cantor group orbit samplings

Analysis of resulting measures on the unit circle

Extension of Avila's method to orthogonal polynomials

Abstract

Avila recently introduced a new method for the study of the discrete Schr\"odinger Operator with limit periodic potential. I adapt this method to the context of orthogonal polynomials in the unit circle with limit periodic Verblunsky Coefficients. Specifically, I represent these Verblunsky Coefficients as a continuous sampling of the orbits of a Cantor group by a minimal translation. I then investigate the measures that arise on the unit circle as I vary the sampling function.