Orthogonal polynomials on Cantor sets of zero Lebesgue measure

TL;DR

This survey explores orthogonal polynomials on Cantor sets of zero Lebesgue measure, emphasizing equilibrium measures and their role in understanding measures similar to Szeg\

Contribution

It highlights the importance of equilibrium measures in developing a general theory for orthogonal polynomials on totally disconnected sets.

Findings

Emphasizes the role of equilibrium measures in polynomial theory.

Identifies open problems, some approachable via numerical methods.

Connects orthogonal polynomials with spectral theory on Cantor sets.

Abstract

In this survey article, we review some results and conjectures related to orthogonal polynomials on Cantor sets. The main purpose of this paper is to emphasize the role of equilibrium measures in order to have a general theory of sufficiently good measures (measures that behave similarly to measures which are in the Szeg\H{o} class and the isospectral torus in the finite gap case) supported on totally disconnected subsets of $\mathbb{R}$. We present some open problems a number of which can be studied numerically.