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-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. As a by-product, we derive 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.

Minkowski's question mark measure is UST--regular

“…x) with respect to the measure µ itself. Very precise numerical estimates of this dimension have been obtained with high precision arithmetics [3]; rigorous numerical lower and upper bounds [30] derived from the Jacobi matrix of µ place this value between 0.874716305108213 and 0.874716305108207. Further analytical properties of µ have been recently studied, among others, in [1,2,44].…”

“…In this context, Dresse and Van Assche [11] asked whether it is regular, in the sense of Ullmann-Stahl-Totik, as defined below. Their numerical investigation was successively refined via a more powerful technique by the present author in [30], to provide compelling numerical evidence in favor of regularity of this measure. In this paper we provide a rigorous proof of this result, which further reveals the intriguing nature of Minkowski's question mark function.…”

“…In the case of Minkowski's question mark measure, regularity and the asymptotic behavior of orthogonal polynomials have been investigated theoretically and numerically in [30], with detailed pictures illustrating the abstract properties. This investigation continues in this paper from a slightly different perspective: we do not prove regularity of µ directly from the definition, that is, orthogonal polynomials play no rôle herein, but we use a purely measure-theoretical criterion, which translates the idea that a regular measure is never too thin on its support.…”

“…The fact that Minkowski's question mark function is UST-regular is remarkable in many ways. First, it was not at all obvious how to reveal it from the numerical point of view: it required dedicated techniques [30]. From the theoretical side, regularity is required in the hypotheses of Proposition 1 and 2 of [30], that are therefore now rigorously established: these propositions describe and quantify in a precise way the local asymptotic behavior of zeros of the orthogonal polynomials p n (µ; x) and of the Christoffel functions associated with µ, and they link them to the Farey / Stern-Brocot organization of the set of rational numbers.…”

“…Further conjectures were presented in [30], on the speed of convergence in the above asymptotic behaviors and, more significantly, on the fact that Minkowski's question mark might belong to Nevai's class: numerical indication is that its Jacobi matrix elements converge to a limit value, although slowly. If confirmed, this conjecture will provide us with an example of a measure in Nevai's class which does not fulfill Rakhmanov sufficient condition [34,31] (almost everywhere positivity of the Radon Nikodyn derivative of µ with respect to Lebesgue) and might perhaps indicate a widening of such condition; it is known that Nevai's class does contain pure point [46] and singular measures [24] but these examples do not seem to indicate a general criterion on a par with Rakhmanov's.…”