Minkowski's Question Mark Measure
For researchers in orthogonal polynomials and potential theory, this work provides conjectures and numerical evidence for the regularity of a singular continuous measure, but the results are largely incremental and not yet proven.
This paper studies Minkowski's question mark measure from the perspective of logarithmic potential theory and orthogonal polynomials, providing numerical evidence that it is regular and belongs to a Nevai class, and deriving asymptotic formulas for zeros and Christoffel functions.
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.