Минковски сұрақ белгісі функциясының ерекше қасиеттері

Function with unusual fractal properties

In mathematics, Minkowski's question mark function, denoted size=120%, is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. It also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.

The question mark function provides a one to one mapping from the non dyadic rationals to the quadratic irrationals, thus allowing an explicit proof of countability of the latter. These can, in fact, be understood to correspond to the periodic orbits for the dyadic transformation. This can be explicitly demonstrated in just a few steps.

The question mark function is a strictly increasing and continuous, but not absolutely continuous function. The derivative is defined almost everywhere, and can take on only two values, 0 (its value almost everywhere, including at all rational numbers) and There are several constructions for a measure that, when integrated, yields the question mark function. One such construction is obtained by measuring the density of the Farey numbers on the real number line. The question mark measure is the prototypical example of what are sometimes referred to as multi fractal measures. The question mark function maps rational numbers to dyadic rational numbers, meaning those whose base two representation terminates, as may be proven by induction from the recursive construction outlined above. It maps quadratic irrationals to non dyadic rational numbers. In both cases it provides an order isomorphism between these sets, making concrete Cantor's isomorphism theorem according to which every two unbounded countable dense linear orders are order isomorphic. It is an odd function, and satisfies the functional equation ; consequently x ↦ ? (x) − x is an odd periodic function with period one. If ? (x) is irrational, then x is either algebraic of degree greater than two, or transcendental. The question mark function has fixed points at 0, and 1, and at least two more, symmetric about the midpoint. One is approximately 0.42037. It was conjectured by Moshchevitin that they were the only 5 fixed points. In 1943, Raphaël Salem raised the question of whether the Fourier–Stieltjes coefficients of the question mark function vanish at infinity. In other words, he wanted to know whether or not

This was answered affirmatively by Jordan and Sahlsten, as a special case of a result on Gibbs measures. The graph of Minkowski question mark function is a special case of fractal curves known as de Rham curves.

Restricting the Minkowski question mark function to ? :[0,1] → [0,1], it can be used as the cumulative distribution function of a singular distribution on the unit interval. This distribution is symmetric about its midpoint, with raw moments of about m1 = 0.5, m2 = 0.290926, m3 = 0.186389 and m4 = 0.126992, and so a mean and median of 0.5, a standard deviation of about 0.2023, a skewness of 0, and an excess kurtosis about −1.147.