On the Minkowski Measure

TL;DR

This paper investigates the derivative of the Minkowski Question Mark function, providing a measure-theoretic construction, an exact infinite product representation, and linking it to transfer operators and invariant measures.

Contribution

It introduces a measure-theoretic framework for the Minkowski measure, expresses it as an infinite product of Möbius transformations, and connects it to transfer operators and invariant measures.

Findings

The derivative of the Minkowski Question Mark function can be represented as a Lebesgue-Stieltjes measure.

The measure is expressed as an infinite product of piecewise Möbius transformations.

The Minkowski measure is shown to be the Haar measure of a transfer operator.

Abstract

The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is 'singular'. One can show by classical techniques that its derivative must vanish on all rationals. Since the Question Mark itself is continuous, one concludes that the derivative must be non-zero on the irrationals, and is thus a discontinuous-everywhere function. This derivative is the subject of this essay. Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to manipulate in various familiar ways. Next, an exact result is given, expressing the measure as an infinite product of piece-wise continuous functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d). This construction is then shown to be the Haar measure of a certain transfer operator. A general proof is given that any transfer operator can be understood to be nothing more nor less than a push-forward on a Banach space; such push-forwards induce an invariant measure, the Haar measure, of which the Minkowski measure can serve as a prototypical example. Some minor notes pertaining to it's relation to the Gauss-Kuzmin-Wirsing operator are made.