Related papers: Minkowski's Question Mark Measure

Hermann Minkowski introduced a function in 1904 which maps quadratic irrational numbers to rational numbers and this function is now known as Minkowski's question mark function since Minkowski used the notation $?(x)$. This function is a…

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…

The Minkowski question mark function is a rich object which can be explored from the perspective of dynamical systems, complex dynamics, metric number theory, multifractal analysis, transfer operators, integral transforms, and as a function…

We prove the recent conjecture that Minkowski's question mark measure is regular, in the sense of Ullman-Stahl-Totik.

The Minkowski question mark function, maping the unit interval to itself, is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of…

We study analogues of Minkowski's question mark function $?(x)$ related to continued fractions with even or odd partial quotients. We prove that these functions are H\"older continuous with precise exponents, and that they linearize the…

The present article deals with properties of a certain function of the Minkowski type with arguments defined by Engel series. Differential, integral, and other properties of the function were considered.

The Minkowski's Question-Mark function is a singular homeomorphism of the unit interval that maps the set of quadratic surds into the rationals. This function has deserved the attention of several authors since the beginning of the…

We prove new results on the derivative of the Minkowski question mark function. Some of our theorems are non-improvable.

The Minkowski question-mark function $?(x)$ is a continuous strictly increasing function defined on $[0,1]$ interval. It is well known fact that the derivative of this function, if exists, can take only two values: $0$ and $+\infty$. It is…

Minkowski's question mark function is strictly increasing on $[0, 1]$ and hence defines a Stieltjes measure on $[0, 1]$. A problem originating from Salem in 1943, is to determine whether the Fourier series of this measure decay to zero or…

In this paper we consider a discrete-time dynamical system on the real line by random iteration of two functions. These functions are assumed to satisfy appropriate monotonicity conditions; optionally, a symmetry condition may be imposed.…

A one-to-one continuous function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski ?(x) function. It is…

1 : We use properties of the Stern Sequence for numerical computations of moments $\int^1_0 t^n d?(t)$ associated to Minkowski's Question Mark function.

The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period…

We consider regularity for solutions of a class of de Rham's functional equations. Under some smoothness conditions of functions consisting the equation, we improve some results in Hata (Japan J. Appl. Math. 1985). Our results are…

We give criteria on the existence of a so-called mark function in the context of marked metric measure spaces (mmm-spaces). If an mmm-space admits a mark function, we call it functionally-marked metric measure space (fmm-space). This is not…

To every log-concave function $f$ one may associate a pair of measures $(\mu_{f},\nu_{f})$ which are the surface area measures of $f$. These are a functional extension of the classical surface area measure of a convex body, and measure how…

We introduce the concept of Minkowski normality, a different type of normality for the regular continued fraction expansion. We use the ordering \[ \frac{1}{2},\quad \frac{1}{3}, \frac{2}{3},\quad \frac{1}{4}, \frac{3}{4},\frac{2}{5},…

A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence…