Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift

2011/11/17 by Helge Krueger, Krueger, Helge
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical functions and polynomials #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1111.4019

openalex publication_date 2011/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

I give an example of a family of orthogonal polynomials on the unit circle with Verblunsky coefficients given by the skew-shift for which the associated measures are supported on the entire unit circle and almost-every Aleksandrov measure is pure point. Furthermore, I show in the case of the two dimensional skew-shift the zeros of para-orthogonal polynomials obey the same statistics as an appropriate irrational rotation. The proof is based on an analysis of the associated CMV matrices.

Citations

• On the spectrum of skew-shift Schrödinger operators

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