Minkowski’s question mark measure

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–Saff–Stahl–Totik and moreover that 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. As a by-product, we compute upper and lower bounds to the Hausdorff dimension of Minkowski’s measure. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Möbius maps.