Minkowski's Question Mark Measure

TL;DR

This paper investigates Minkowski's question mark measure using potential theory and orthogonal polynomials, providing numerical evidence for its regularity and Nevai class membership, and analyzing zeros and Christoffel functions.

Contribution

It offers new numerical and theoretical insights into the measure's regularity, Nevai class, and the behavior of its orthogonal polynomials using Iterated Function Systems.

Findings

Numerical evidence supports the measure's regularity.

The measure likely belongs to a Nevai class.

Asymptotic formulas for zeros and Christoffel functions are derived.

Abstract

Minkowski's question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense of Ullman--Stahl--Totik and moreover it belongs to a Nevai class: we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Mobius maps.