Orthogonal polynomials on the unity circle with Fibonacci Verblunsky coefficients, I. The essential support of the measure

TL;DR

This paper investigates probability measures on the unit circle linked to orthogonal polynomials with Fibonacci-invariant Verblunsky coefficients, emphasizing the fractal nature of their essential support.

Contribution

It introduces a novel class of measures with Fibonacci symmetry and analyzes their fractal support properties.

Findings

The essential support exhibits fractal characteristics.

Fibonacci invariance influences the measure's support structure.

New insights into the spectral properties of Fibonacci-invariant measures.

Abstract

We study probability measures on the unit circle corresponding to orthogonal polynomials whose sequence of Verblunsky coefficients is invariant under the Fibonacci substitution. We focus in particular on the fractal properties of the essential support of these measures.