SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Minkowski's question-mark function</title>

</head>

Minkowski's question-mark function

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

In <a href="Mathematics" title="Mathematics">mathematics</a>, Minkowski's question-mark function, denoted ?(x), is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> with unusual <a href="Fractal" title="Fractal">fractal</a> properties, defined by <a href="Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> in 1904.<a href="#cite_note-FOOTNOTEMinkowski1904171–172-1">[1]</a> It maps <a href="Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrational</a> numbers to <a href="Rational_number" title="Rational number">rational numbers</a> on the <a href="Unit_interval" title="Unit interval">unit interval</a>, via an expression relating the <a href="Continued_fraction" title="Continued fraction">continued fraction</a> expansions of the quadratics to the <a href="Binary_expansion" class="mw-redirect" title="Binary expansion">binary expansions</a> of the rationals, given by <a href="Arnaud_Denjoy" title="Arnaud Denjoy">Arnaud Denjoy</a> in 1938.<a href="#cite_note-FOOTNOTEDenjoy1938-2">[2]</a> It also maps rational numbers to <a href="Dyadic_rational" title="Dyadic rational">dyadic rationals</a>, as can be seen by a <a href="Recursive_definition" title="Recursive definition">recursive definition</a> closely related to the <a href="Stern%E2%80%93Brocot_tree" title="Stern–Brocot tree">Stern–Brocot tree</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition_and_intuition">Definition and intuition</h2></div>

One way to define the question-mark function involves the correspondence between two different ways of representing fractional numbers using finite or infinite <a href="Binary_sequence" class="mw-redirect" title="Binary sequence">binary sequences</a>. Most familiarly, a string of 0s and 1s with a single point mark ".", like "11.0010010000111111..." can be interpreted as the <a href="Binary_representation" class="mw-redirect" title="Binary representation">binary representation</a> of a number. In this case this number is

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2+1+{\frac {1}{8}}+{\frac {1}{64}}+\cdots =\pi .}</annotation>

</semantics>

</math>

There is a different way of interpreting the same sequence, however, using <a href="Continued_fraction" title="Continued fraction">continued fractions</a>.

Interpreting the <a href="Fractional_part" title="Fractional part">fractional part</a> "0.00100100001111110..." as a binary number in the same way, replace each consecutive block of 0s or 1s by its <a href="Run_length" class="mw-redirect" title="Run length">run length</a> (or, for the first block of zeroes, its run length plus one), in this case generating the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [3;3,1,2,1,4,6,\dots ]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [3;3,1,2,1,4,6,\dots ]}</annotation>

</semantics>

</math><img src="./5e257c9fe1e778cbb4e3f776f502c45e3bccc2c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.392ex; height:2.843ex;" alt="{\displaystyle [3;3,1,2,1,4,6,\dots ]}" loading="lazy">. Then, use this sequence as the coefficients of a continued fraction:<a href="#cite_note-FOOTNOTEFinch2003441–442-3">[3]</a><a href="#cite_note-FOOTNOTEPytheas_Fogg200295-4">[4]</a>

<mfrac>

<mfrac>

<mfrac>

<mfrac>

<mfrac>

<mfrac>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3+{\frac {1}{\displaystyle 3+{\frac {1}{\displaystyle 1+{\frac {1}{\displaystyle 2+{\frac {1}{\displaystyle 1+{\frac {1}{\displaystyle 4+{\frac {1}{\displaystyle 6+\dots }}}}}}}}}}}}\approx 3.2676}</annotation>

</semantics>

</math>

The question-mark function reverses this process: it translates the continued-fraction of a given <a href="Real_number" title="Real number">real number</a> into a run-length encoded binary sequence, and then reinterprets that sequence as a binary number.<a href="#cite_note-FOOTNOTEFinch2003441–442-3">[3]</a><a href="#cite_note-FOOTNOTEPytheas_Fogg200295-4">[4]</a> For instance, for the example above, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {?} (3.2676)\approx \pi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {?} (3.2676)\approx \pi }</annotation>

</semantics>

</math><img src="./16ca301cd6d429f43e1e018fa88294e6445ed588.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.796ex; height:2.843ex;" alt="{\displaystyle \operatorname {?} (3.2676)\approx \pi }" loading="lazy">. To define this formally, if an <a href="Irrational_number" title="Irrational number">irrational number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> has the (non-terminating) continued-fraction representation

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a_{0}+{\frac {1}{\displaystyle a_{1}+{\frac {1}{\displaystyle a_{2}+\cdots }}}}=[a_{0};a_{1},a_{2},\dots ]}</annotation>

</semantics>

</math>

then the value of the question-mark function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is defined as the value of the <a href="Series_(mathematics)" title="Series (mathematics)">infinite series</a>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {?} (x)=a_{0}+2\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n+1}}{2^{a_{1}+\cdots +a_{n}}}}.}</annotation>

</semantics>

</math>

A <a href="Rational_number" title="Rational number">rational number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> has a terminating continued-fraction representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a_{0};a_{1},a_{2},\dots ,a_{m}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a_{0};a_{1},a_{2},\dots ,a_{m}]}</annotation>

</semantics>

</math><img src="./b9665e5174aa4140ca93614d90c3e6a316700260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.297ex; height:2.843ex;" alt="{\displaystyle [a_{0};a_{1},a_{2},\dots ,a_{m}]}" loading="lazy">, so the value of the question-mark function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> reduces to the <a href="Dyadic_rational" title="Dyadic rational">dyadic rational</a> defined by a finite sum,

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {?} (x)=a_{0}+2\sum _{n=1}^{m}{\frac {\left(-1\right)^{n+1}}{2^{a_{1}+\cdots +a_{n}}}}.}</annotation>

</semantics>

</math>

A <a href="Quadratic_irrational_number" title="Quadratic irrational number">quadratic irrational number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is represented by a <a href="Periodic_continued_fraction" title="Periodic continued fraction">periodic continued fraction</a>, so the value of the question-mark function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is a periodic binary fraction and thus a non-dyadic rational number.

<div class="mw-heading mw-heading2"><h2 id="Self-symmetry">Self-symmetry</h2></div>

The question mark is clearly visually self-similar. A <a href="Monoid" title="Monoid">monoid</a> of self-similarities may be generated by two operators S and R acting on the <a href="Unit_square" title="Unit square">unit square</a> and defined as follows:

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S(x,y)&=\left({\frac {x}{x+1}},{\frac {y}{2}}\right),\\[5px]R(x,y)&=(1-x,1-y).\end{aligned}}}</annotation>

</semantics>

</math>

Visually, S shrinks the unit square to its bottom-left quarter, while R performs a <a href="Point_reflection" title="Point reflection">point reflection</a> through its center.

A point on the <a href="Function_graph" class="mw-redirect" title="Function graph">graph</a> of ? has coordinates (x, ?(x)) for some x in the unit interval. Such a point is transformed by S and R into another point of the graph, because ? satisfies the following identities for all x ∈ [0, 1]:

<mtr>

<mtd>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {?} \left({\frac {x}{x+1}}\right)&={\frac {\operatorname {?} (x)}{2}},\\[5px]\operatorname {?} (1-x)&=1-\operatorname {?} (x).\end{aligned}}}</annotation>

</semantics>

</math>

These two operators may be repeatedly combined, forming a monoid. A general element of the monoid is then

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{a_{1}}RS^{a_{2}}RS^{a_{3}}\cdots }</annotation>

</semantics>

</math>

for positive integers a1, a2, a3, …. Each such element describes a <a href="Self-similarity" title="Self-similarity">self-similarity</a> of the question-mark function. This monoid is sometimes called the <a href="Period-doubling_monoid" class="mw-redirect" title="Period-doubling monoid">period-doubling monoid</a>, and all period-doubling fractal curves have a self-symmetry described by it (the <a href="De_Rham_curve" title="De Rham curve">de Rham curve</a>, of which the question mark is a special case, is a category of such curves). The elements of the monoid are in correspondence with the rationals, by means of the identification of a1, a2, a3, … with the continued fraction [0; a1, a2, a3,…]. Since both

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S:x\mapsto {\frac {x}{x+1}}}</annotation>

</semantics>

</math>

and

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T:x\mapsto 1-x}</annotation>

</semantics>

</math>

are <a href="Linear_fractional_transformation" title="Linear fractional transformation">linear fractional transformations</a> with integer coefficients, the monoid may be regarded as a subset of the <a href="Modular_group" title="Modular group">modular group</a> PSL(2, Z).

<div class="mw-heading mw-heading2"><h2 id="Quadratic_irrationals">Quadratic irrationals</h2></div>

The question mark function provides a one-to-one mapping from the non-dyadic rationals to the <a href="Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrationals</a>, thus allowing an explicit proof of countability of the latter. These can, in fact, be understood to correspond to the <a href="Periodic_orbit" class="mw-redirect" title="Periodic orbit">periodic orbits</a> for the <a href="Dyadic_transformation" title="Dyadic transformation">dyadic transformation</a>. This can be explicitly demonstrated in just a few steps.

<div class="mw-heading mw-heading3"><h3 id="Dyadic_symmetry">Dyadic symmetry</h3></div>

Define two moves: a left move and a right move, valid on the <a href="Unit_interval" title="Unit interval">unit interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x\leq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1}</annotation>

</semantics>

</math><img src="./30810e06ad49f3a837bd2193d4392eda1f74e7ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.852ex; height:2.343ex;" alt="{\displaystyle 0\leq x\leq 1}" loading="lazy"> as

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{D}(x)={\frac {x}{2}}}</annotation>

</semantics>

</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{C}(x)={\frac {x}{1+x}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{C}(x)={\frac {x}{1+x}}}</annotation>

</semantics>

</math><img src="./845ba722616689739333d0e2225f954ee7b2ec94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.47ex; height:4.843ex;" alt="{\displaystyle L_{C}(x)={\frac {x}{1+x}}}" loading="lazy">

and

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{D}(x)={\frac {1+x}{2}}}</annotation>

</semantics>

</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{C}(x)={\frac {1}{2-x}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{C}(x)={\frac {1}{2-x}}}</annotation>

</semantics>

</math><img src="./05dba2a991912ebadafb41d47a71e47e918c3ef1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.651ex; height:5.343ex;" alt="{\displaystyle R_{C}(x)={\frac {1}{2-x}}}" loading="lazy">

The question mark function then obeys a left-move symmetry

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{D}\circ {\text{?}}={\text{?}}\circ L_{C}}</annotation>

</semantics>

</math>

and a right-move symmetry

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{D}\circ {\text{?}}={\text{?}}\circ R_{C}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \circ }</annotation>

</semantics>

</math><img src="./99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy"> denotes <a href="Function_composition" title="Function composition">function composition</a>. These can be arbitrarily concatenated. Consider, for example, the sequence of left-right moves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle LRLLR.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle LRLLR.}</annotation>

</semantics>

</math><img src="./cd9a7e192e435f0622d170c44cb9d4cf4f677271.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.923ex; height:2.176ex;" alt="{\displaystyle LRLLR.}" loading="lazy"> Adding the subscripts C and D, and, for clarity, dropping the composition operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \circ }</annotation>

</semantics>

</math><img src="./99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy"> in all but a few places, one has:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{D}R_{D}L_{D}L_{D}R_{D}\circ {\text{?}}={\text{?}}\circ L_{C}R_{C}L_{C}L_{C}R_{C}}</annotation>

</semantics>

</math>

Arbitrary finite-length strings in the letters L and R correspond to the <a href="Dyadic_rationals" class="mw-redirect" title="Dyadic rationals">dyadic rationals</a>, in that every dyadic rational can be written as both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=n/2^{m}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=n/2^{m}}</annotation>

</semantics>

</math><img src="./b93dfeeadce8a57f173d5e15fc970b513d362ff9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.649ex; height:2.843ex;" alt="{\displaystyle y=n/2^{m}}" loading="lazy"> for integer n and m and as finite length of bits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}</annotation>

</semantics>

</math><img src="./a04a56b38079089ee5cff10482be90a326869cfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.389ex; height:2.509ex;" alt="{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}\in \{0,1\}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{k}\in \{0,1\}.}</annotation>

</semantics>

</math><img src="./5a216bac798702ebe1a2dba84c088ca2a2dd6128.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.258ex; height:2.843ex;" alt="{\displaystyle b_{k}\in \{0,1\}.}" loading="lazy"> Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the question mark function.

Some notational rearrangements can make the above slightly easier to express. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{0}}</annotation>

</semantics>

</math><img src="./32d13273b9af4564fa2c421c96d039c414db8628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>

</semantics>

</math><img src="./3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"> stand for L and R. Function composition extends this to a <a href="Monoid" title="Monoid">monoid</a>, in that one can write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{010}=g_{0}g_{1}g_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{010}=g_{0}g_{1}g_{0}}</annotation>

</semantics>

</math><img src="./c6e259c8a6d532d13783e5d6f423ad022b1738c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.396ex; height:2.009ex;" alt="{\displaystyle g_{010}=g_{0}g_{1}g_{0}}" loading="lazy"> and generally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{A}g_{B}=g_{AB}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{A}g_{B}=g_{AB}}</annotation>

</semantics>

</math><img src="./f75be0a170d2924de25755e6132645c6d48e07d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.082ex; height:2.009ex;" alt="{\displaystyle g_{A}g_{B}=g_{AB}}" loading="lazy"> for some binary strings of digits A, B, where AB is just the ordinary <a href="Concatenation" title="Concatenation">concatenation</a> of such strings. The dyadic monoid M is then the monoid of all such finite-length left-right moves. Writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \in M}</annotation>

</semantics>

</math><img src="./c9e4c574bcce05b99e8f917a3c0d50c9ca733922.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.545ex; height:2.676ex;" alt="{\displaystyle \gamma \in M}" loading="lazy"> as a general element of the monoid, there is a corresponding self-symmetry of the question mark function:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{D}\circ {\text{?}}={\text{?}}\circ \gamma _{C}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Isomorphism">Isomorphism</h3></div>

An explicit mapping between the rationals and the dyadic rationals can be obtained providing a reflection operator

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r(x)=1-x}</annotation>

</semantics>

</math>

and noting that both

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\circ R_{D}\circ r=L_{D}}</annotation>

</semantics>

</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\circ R_{C}\circ r=L_{C}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\circ R_{C}\circ r=L_{C}}</annotation>

</semantics>

</math><img src="./9795cb811ccecba84ab09cf430df9eaa41ba8d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.895ex; height:2.509ex;" alt="{\displaystyle r\circ R_{C}\circ r=L_{C}}" loading="lazy">

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r^{2}=1}</annotation>

</semantics>

</math><img src="./a49c775370fb37a15f3816d31c407d02d6b42967.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.364ex; height:2.676ex;" alt="{\displaystyle r^{2}=1}" loading="lazy"> is the <a href="Identity_element" title="Identity element">identity</a>, an arbitrary string of left-right moves can be re-written as a string of left moves only, followed by a reflection, followed by more left moves, a reflection, and so on, that is, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{a_{1}}rL^{a_{2}}rL^{a_{3}}\cdots }">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{a_{1}}rL^{a_{2}}rL^{a_{3}}\cdots }</annotation>

</semantics>

</math><img src="./0e142fa1e26c3ca776a9f8610b51ba6c1ad67afc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.756ex; height:2.343ex;" alt="{\displaystyle L^{a_{1}}rL^{a_{2}}rL^{a_{3}}\cdots }" loading="lazy"> which is clearly isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{a_{1}}TS^{a_{2}}TS^{a_{3}}\cdots }">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{a_{1}}TS^{a_{2}}TS^{a_{3}}\cdots }</annotation>

</semantics>

</math><img src="./fa3c14d42f28c177df2c4072c988774c8909fc0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.748ex; height:2.343ex;" alt="{\displaystyle S^{a_{1}}TS^{a_{2}}TS^{a_{3}}\cdots }" loading="lazy"> from above. Evaluating some explicit sequence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{D},R_{D}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{D},R_{D}}</annotation>

</semantics>

</math><img src="./cfa01280bcf6f1a4b663ba3d0306ed6b096ead1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.567ex; height:2.509ex;" alt="{\displaystyle L_{D},R_{D}}" loading="lazy"> at the function argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=1}</annotation>

</semantics>

</math><img src="./ee42176e76ae6b56d68c42ced807e08b962a2b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}" loading="lazy"> gives a dyadic rational; explicitly, it is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}</annotation>

</semantics>

</math><img src="./a04a56b38079089ee5cff10482be90a326869cfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.389ex; height:2.509ex;" alt="{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}" loading="lazy"> where each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k}\in \{0,1\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{k}\in \{0,1\}}</annotation>

</semantics>

</math><img src="./29912b00b5cbf50fcb0b762808bcbe7fd45c80e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.611ex; height:2.843ex;" alt="{\displaystyle b_{k}\in \{0,1\}}" loading="lazy"> is a binary bit, zero corresponding to a left move and one corresponding to a right move. The equivalent sequence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{C},R_{C}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{C},R_{C}}</annotation>

</semantics>

</math><img src="./e6fe0bf2213439120699c2943a84902c190ff8b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.343ex; height:2.509ex;" alt="{\displaystyle L_{C},R_{C}}" loading="lazy"> moves, evaluated at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=1}</annotation>

</semantics>

</math><img src="./ee42176e76ae6b56d68c42ced807e08b962a2b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}" loading="lazy"> gives a rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p/q.}</annotation>

</semantics>

</math><img src="./859645f5823d17a5abfa11c70bb975118fccebe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.138ex; height:2.843ex;" alt="{\displaystyle p/q.}" loading="lazy"> It is explicitly the one provided by the continued fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q=[a_{1},a_{2},a_{3},\ldots ,a_{j}]}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p/q=[a_{1},a_{2},a_{3},\ldots ,a_{j}]}</annotation>

</semantics>

</math><img src="./6319ec435eb3944942d45abfc0c50719fe2084a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:24.121ex; height:3.009ex;" alt="{\displaystyle p/q=[a_{1},a_{2},a_{3},\ldots ,a_{j}]}" loading="lazy"> keeping in mind that it is a rational because the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{j})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{j})}</annotation>

</semantics>

</math><img src="./4b50dfa36063ee028ee7df7689a3ff2811ab9370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.047ex; height:3.009ex;" alt="{\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{j})}" loading="lazy"> was of finite length. This establishes a one-to-one correspondence between the dyadic rationals and the rationals.

<div class="mw-heading mw-heading3"><h3 id="Periodic_orbits_of_the_dyadic_transform">Periodic orbits of the dyadic transform</h3></div>

Consider now the <a href="Periodic_orbit" class="mw-redirect" title="Periodic orbit">periodic orbits</a> of the <a href="Dyadic_transformation" title="Dyadic transformation">dyadic transformation</a>. These correspond to bit-sequences consisting of a finite initial "chaotic" sequence of bits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0},b_{1},b_{2},\ldots ,b_{k-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0},b_{1},b_{2},\ldots ,b_{k-1}}</annotation>

</semantics>

</math><img src="./ba4c718044c2e91a6caf795a30686cfc673ee894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.588ex; height:2.509ex;" alt="{\displaystyle b_{0},b_{1},b_{2},\ldots ,b_{k-1}}" loading="lazy">, followed by a repeating string <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{k},b_{k+1},b_{k+2},\ldots ,b_{k+m-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{k},b_{k+1},b_{k+2},\ldots ,b_{k+m-1}}</annotation>

</semantics>

</math><img src="./17abaf9f6312edf7c9ac9ab3e540bba4ccb547ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.614ex; height:2.509ex;" alt="{\displaystyle b_{k},b_{k+1},b_{k+2},\ldots ,b_{k+m-1}}" loading="lazy"> of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">. Such repeating strings correspond to a rational number. This is easily made explicit. Write

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=\sum _{j=0}^{m-1}b_{k+j}2^{-j-1}}</annotation>

</semantics>

</math>

one then clearly has

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{\infty }b_{k+j}2^{-j-1}=y\sum _{j=0}^{\infty }2^{-jm}={\frac {y}{1-2^{m}}}}</annotation>

</semantics>

</math>

Tacking on the initial non-repeating sequence, one clearly has a rational number. In fact, every rational number can be expressed in this way: an initial "random" sequence, followed by a cycling repeat. That is, the periodic orbits of the map are in one-to-one correspondence with the rationals.

<div class="mw-heading mw-heading3"><h3 id="Periodic_orbits_as_continued_fractions">Periodic orbits as continued fractions</h3></div>

Such periodic orbits have an equivalent periodic continued fraction, per the isomorphism established above. There is an initial "chaotic" orbit, of some finite length, followed by a repeating sequence. The repeating sequence generates a <a href="Periodic_continued_fraction" title="Periodic continued fraction">periodic continued fraction</a> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=[a_{n},a_{n+1},a_{n+2},\ldots ,a_{n+r},x].}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=[a_{n},a_{n+1},a_{n+2},\ldots ,a_{n+r},x].}</annotation>

</semantics>

</math><img src="./10a89ef810b932a64d608ac6058ba0b03b2107fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.993ex; height:2.843ex;" alt="{\displaystyle x=[a_{n},a_{n+1},a_{n+2},\ldots ,a_{n+r},x].}" loading="lazy"> This continued fraction has the form<a href="#cite_note-FOOTNOTEKhinchin1997-5">[5]</a>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\frac {\alpha x+\beta }{\gamma x+\delta }}}</annotation>

</semantics>

</math>

with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma ,\delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma ,\delta }</annotation>

</semantics>

</math><img src="./2eb7ee31949dcce33b443132ac97f42927549e43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.232ex; height:2.843ex;" alt="{\displaystyle \alpha ,\beta ,\gamma ,\delta }" loading="lazy"> being integers, and satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \delta -\beta \gamma =\pm 1.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \delta -\beta \gamma =\pm 1.}</annotation>

</semantics>

</math><img src="./747ed5e22262ea113a3fdda51cf4fff3b99e30ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.687ex; height:2.843ex;" alt="{\displaystyle \alpha \delta -\beta \gamma =\pm 1.}" loading="lazy"> Explicit values can be obtained by writing

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\mapsto {\begin{pmatrix}1&0\\1&1\end{pmatrix}}}</annotation>

</semantics>

</math>

for the shift, so that

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{n}\mapsto {\begin{pmatrix}1&0\\n&1\end{pmatrix}}}</annotation>

</semantics>

</math>

while the reflection is given by

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\mapsto {\begin{pmatrix}-1&1\\0&1\end{pmatrix}}}</annotation>

</semantics>

</math>

so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{2}=I}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T^{2}=I}</annotation>

</semantics>

</math><img src="./f458c4c76d9cd7b68270141de61bf02df6c5a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.044ex; height:2.676ex;" alt="{\displaystyle T^{2}=I}" loading="lazy">. Both of these matrices are <a href="Unimodular_matrix" title="Unimodular matrix">unimodular</a>, arbitrary products remain unimodular, and result in a matrix of the form

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{a_{n}}TS^{a_{n+1}}T\cdots TS^{a_{n+r}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}}</annotation>

</semantics>

</math>

giving the precise value of the continued fraction. As all of the matrix entries are integers, this matrix belongs to the projective <a href="Modular_group" title="Modular group">modular group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PSL(2,\mathbb {Z} ).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PSL(2,\mathbb {Z} ).}</annotation>

</semantics>

</math><img src="./b3cc1f19e8147798d3f8d723e279bb2fb78943ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.03ex; height:2.843ex;" alt="{\displaystyle PSL(2,\mathbb {Z} ).}" loading="lazy">

Solving explicitly, one has that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma x^{2}+(\delta -\alpha )x-\beta =0.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma x^{2}+(\delta -\alpha )x-\beta =0.}</annotation>

</semantics>

</math><img src="./5be62b2fd1ef0604e6764f406bed5737c0a8fdcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.082ex; height:3.176ex;" alt="{\displaystyle \gamma x^{2}+(\delta -\alpha )x-\beta =0.}" loading="lazy"> It is not hard to verify that the solutions to this meet the definition of quadratic irrationals. In fact, every quadratic irrational can be expressed in this way. Thus the quadratic irrationals are in one-to-one correspondence with the periodic orbits of the dyadic transform, which are in one-to-one correspondence with the (non-dyadic) rationals, which are in one-to-one correspondence with the dyadic rationals. The question mark function provides the correspondence in each case.

<div class="mw-heading mw-heading2"><h2 id="Properties_of_?(x)">Properties of ?(x)</h2></div>

The question-mark function is a <a href="Strictly_increasing" class="mw-redirect" title="Strictly increasing">strictly increasing</a> and continuous,<a href="#cite_note-FOOTNOTEFinch2003442-6">[6]</a> but not <a href="Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> function. The <a href="Derivative" title="Derivative">derivative</a> is defined <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>, and can take on only two values, 0 (its value almost everywhere, including at all <a href="Rational_number" title="Rational number">rational numbers</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation>

</semantics>

</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy">.<a href="#cite_note-FOOTNOTEDushistovaMoshchevitin2012-7">[7]</a> There are several constructions for a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> that, when integrated, yields the question-mark function. One such construction is obtained by measuring the density of the <a href="Farey_sequence" title="Farey sequence">Farey numbers</a> on the real number line. The question-mark measure is the prototypical example of what are sometimes referred to as <a href="Multifractal" class="mw-redirect" title="Multifractal">multi-fractal measures</a>.

The question-mark function maps rational numbers to <a href="Dyadic_rational" title="Dyadic rational">dyadic rational numbers</a>, meaning those whose <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">base two</a> representation terminates, as may be proven by induction from the recursive construction outlined above. It maps <a href="Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrationals</a> to non-dyadic rational numbers. In both cases it provides an <a href="Order_isomorphism" title="Order isomorphism">order isomorphism</a> between these sets,<a href="#cite_note-FOOTNOTEGirgensohn1996-8">[8]</a> making concrete <a href="Cantor's_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a> according to which every two unbounded countable dense linear orders are order-isomorphic.<a href="#cite_note-FOOTNOTEBhattacharjeeMacphersonMöllerNeumann1997-9">[9]</a> It is an <a href="Odd_function" class="mw-redirect" title="Odd function">odd function</a>, and satisfies the functional equation ?(x + 1) = ?(x) + 1; consequently x ↦ ?(x) − x is an odd <a href="Periodic_function" title="Periodic function">periodic function</a> with period one. If ?(x) is irrational, then x is either <a href="Algebraic_number" title="Algebraic number">algebraic</a> of degree greater than two, or <a href="Transcendental_number" title="Transcendental number">transcendental</a>.

The question-mark function has <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a> at 0, <style data-mw-deduplicate="TemplateStyles:r1214402035">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

/* end https://en.wikipedia.org/ */

</style>⁠1/2⁠ and 1, and at least two more, symmetric about the midpoint. One is approximately 0.42037.<a href="#cite_note-FOOTNOTEFinch2003442-6">[6]</a>

It was conjectured by Moshchevitin that they were the only 5 fixed points.<a href="#cite_note-FOOTNOTEMoshchevitin2020-10">[10]</a>

In 1943, <a href="Rapha%C3%ABl_Salem" title="Raphaël Salem">Raphaël Salem</a> raised the question of whether the Fourier–Stieltjes coefficients of the question-mark function <a href="Vanish_at_infinity" title="Vanish at infinity">vanish at infinity</a>.<a href="#cite_note-FOOTNOTESalem1943-11">[11]</a> In other words, he wanted to know whether or not

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\int _{0}^{1}e^{2\pi inx}\,\operatorname {d?} (x)=0.}</annotation>

</semantics>

</math>

This was answered affirmatively by Jordan and Sahlsten, as a special case of a result on <a href="Gibbs_measure" title="Gibbs measure">Gibbs measures</a>.<a href="#cite_note-FOOTNOTEJordanSahlsten2016-12">[12]</a>

The graph of Minkowski question mark function is a special case of fractal curves known as <a href="De_Rham_curve" title="De Rham curve">de Rham curves</a>.

<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

The recursive definition naturally lends itself to an <a href="Algorithm" title="Algorithm">algorithm</a> for computing the function to any desired degree of accuracy for any real number, as the following <a href="C_(programming_language)" title="C (programming language)">C</a> function demonstrates. The algorithm descends the <a href="Stern%E2%80%93Brocot_tree" title="Stern–Brocot tree">Stern–Brocot tree</a> in search of the input x, and sums the terms of the binary expansion of y = ?(x) on the way. As long as the <a href="Loop_invariant" title="Loop invariant">loop invariant</a> qr − ps = 1 remains satisfied there is no need to reduce the fraction ⁠m/n⁠ = ⁠p + r/q + s⁠, since it is already in lowest terms. Another invariant is ⁠p/q⁠ ≤ x < ⁠r/s⁠. The <code>for</code> loop in this program may be analyzed somewhat like a <code>while</code> loop, with the conditional break statements in the first three lines making out the condition. The only statements in the loop that can possibly affect the invariants are in the last two lines, and these can be shown to preserve the truth of both invariants as long as the first three lines have executed successfully without breaking out of the loop. A third invariant for the body of the loop (up to floating point precision) is y ≤ ?(x) < y + d, but since d is <a href="Division_by_two" title="Division by two">halved</a> at the beginning of the loop before any conditions are tested, our conclusion is only that y ≤ ?(x) < y + 2d at the termination of the loop.

To <a href="Loop_variant" title="Loop variant">prove termination</a>, it is sufficient to note that the sum <code>q + s</code> increases by at least 1 with every iteration of the loop, and that the loop will terminate when this sum is too large to be represented in the primitive C data type <code>long</code>. However, in practice, the conditional break when <code>y + d == y</code> is what ensures the termination of the loop in a reasonable amount of time.

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>/* Minkowski's question-mark function */

double minkowski(double x) {

long p = x;

long q = 1, r = p + 1, s = 1, m, n;

double d = 1, y = p;

if (x < p || (p < 0) ^ (r <= 0))

return x; /* out of range ?(x) =~ x */

while (true) { /* invariants: q * r - p * s == 1 && p / q <= x && x < r / s */

d /= 2;

if (y + d == y)

break; /* reached max possible precision */

m = p + r;

if ((m < 0) ^ (p < 0))

break; /* sum overflowed */

n = q + s;

if (n < 0)

break; /* sum overflowed */

if (x < (double)m / n) {

r = m;

s = n;

} else {

y += d;

p = m;

q = n;

}

return y + d; /* final round-off */

}

</pre></div>

<div class="mw-heading mw-heading2"><h2 id="Probability_distribution">Probability distribution</h2></div>

Restricting the Minkowski question mark function to  ?:[0,1] → [0,1], it can be used as the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of a <a href="Singular_distribution" title="Singular distribution">singular distribution</a> 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,<a href="#cite_note-FOOTNOTEAlkauskas2010-13">[13]</a> and so a mean and <a href="Median" title="Median">median</a> of 0.5, a <a href="Standard_deviation" title="Standard deviation">standard deviation</a> of about 0.2023, a <a href="Skewness" title="Skewness">skewness</a> of 0, and an excess kurtosis about −1.147.

<ul><li><a href="Cantor_function" title="Cantor function">Cantor function</a>, which can be understood as reinterpreting <a href="Ternary_numeral_system" title="Ternary numeral system">ternary numbers</a> as binary numbers, analogously to the way the question-mark function reinterprets continued fractions as binary numbers.</li>

<li><a href="Hermite's_problem" title="Hermite's problem">Hermite's problem</a>, to which one of the approaches uses generalization of Minkowski's question-mark function.<a href="#cite_note-14">[14]</a></li>

<li><a href="Pompeiu_derivative" title="Pompeiu derivative">Pompeiu derivative</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

<li id="cite_note-FOOTNOTEMinkowski1904171–172-1"><a href="#cite_ref-FOOTNOTEMinkowski1904171–172_1-0">^</a> <a href="#CITEREFMinkowski1904">Minkowski (1904)</a>, pp. 171–172.

</li>

<li id="cite_note-FOOTNOTEDenjoy1938-2"><a href="#cite_ref-FOOTNOTEDenjoy1938_2-0">^</a> <a href="#CITEREFDenjoy1938">Denjoy (1938)</a>.

</li>

<li id="cite_note-FOOTNOTEFinch2003441–442-3">^ <a href="#cite_ref-FOOTNOTEFinch2003441–442_3-0">a</a> <a href="#cite_ref-FOOTNOTEFinch2003441–442_3-1">b</a> <a href="#CITEREFFinch2003">Finch (2003)</a>, pp. 441–442.

</li>

<li id="cite_note-FOOTNOTEPytheas_Fogg200295-4">^ <a href="#cite_ref-FOOTNOTEPytheas_Fogg200295_4-0">a</a> <a href="#cite_ref-FOOTNOTEPytheas_Fogg200295_4-1">b</a> <a href="#CITEREFPytheas_Fogg2002">Pytheas Fogg (2002)</a>, p. 95.

</li>

<li id="cite_note-FOOTNOTEKhinchin1997-5"><a href="#cite_ref-FOOTNOTEKhinchin1997_5-0">^</a> <a href="#CITEREFKhinchin1997">Khinchin (1997)</a>.

</li>

<li id="cite_note-FOOTNOTEFinch2003442-6">^ <a href="#cite_ref-FOOTNOTEFinch2003442_6-0">a</a> <a href="#cite_ref-FOOTNOTEFinch2003442_6-1">b</a> <a href="#CITEREFFinch2003">Finch (2003)</a>, p. 442.

</li>

<li id="cite_note-FOOTNOTEDushistovaMoshchevitin2012-7"><a href="#cite_ref-FOOTNOTEDushistovaMoshchevitin2012_7-0">^</a> <a href="#CITEREFDushistovaMoshchevitin2012">Dushistova & Moshchevitin (2012)</a>.

</li>

<li id="cite_note-FOOTNOTEGirgensohn1996-8"><a href="#cite_ref-FOOTNOTEGirgensohn1996_8-0">^</a> <a href="#CITEREFGirgensohn1996">Girgensohn (1996)</a>.

</li>

<li id="cite_note-FOOTNOTEBhattacharjeeMacphersonMöllerNeumann1997-9"><a href="#cite_ref-FOOTNOTEBhattacharjeeMacphersonMöllerNeumann1997_9-0">^</a> <a href="#CITEREFBhattacharjeeMacphersonMöllerNeumann1997">Bhattacharjee et al. (1997)</a>.

</li>

<li id="cite_note-FOOTNOTEMoshchevitin2020-10"><a href="#cite_ref-FOOTNOTEMoshchevitin2020_10-0">^</a> <a href="#CITEREFMoshchevitin2020">Moshchevitin (2020)</a>.

</li>

<li id="cite_note-FOOTNOTESalem1943-11"><a href="#cite_ref-FOOTNOTESalem1943_11-0">^</a> <a href="#CITEREFSalem1943">Salem (1943)</a>.

</li>

<li id="cite_note-FOOTNOTEJordanSahlsten2016-12"><a href="#cite_ref-FOOTNOTEJordanSahlsten2016_12-0">^</a> <a href="#CITEREFJordanSahlsten2016">Jordan & Sahlsten (2016)</a>.

</li>

<li id="cite_note-FOOTNOTEAlkauskas2010-13"><a href="#cite_ref-FOOTNOTEAlkauskas2010_13-0">^</a> <a href="#CITEREFAlkauskas2010">Alkauskas (2010)</a>.

</li>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBeaverGarrity2004" class="citation cs2"><a href="Olga_Beaver" title="Olga Beaver">Beaver, Olga R.</a>; <a href="Thomas_A._Garrity" title="Thomas A. Garrity">Garrity, Thomas</a> (2004), "A two-dimensional Minkowski ?(x) function", Journal of Number Theory, 107 (1): 105–134, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0210480">math/0210480</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2004.01.008">10.1016/j.jnt.2004.01.008</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2059953">2059953</a></cite>

</li>

</ol></div>

<div class="mw-heading mw-heading3"><h3 id="Historical_sources">Historical sources</h3></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}

/* end https://en.wikipedia.org/ */

</style><div class="refbegin" style="">

<ul><li><cite id="CITEREFMinkowski1904" class="citation cs2"><a href="Hermann_Minkowski" title="Hermann Minkowski">Minkowski, Hermann</a> (1904), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150104205306/http://ada00.math.uni-bielefeld.de/ICM/ICM1904/">"Zur Geometrie der Zahlen"</a>, Verhandlungen des III. internationalen Mathematiker-Kongresses in Heidelberg, pp. 164–173, <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:36.0281.01">36.0281.01</a>, archived from <a rel="nofollow" class="external text" href="http://ada00.math.uni-bielefeld.de/ICM/ICM1904/">the original</a> on 4 January 2015</cite></li>

<li><cite id="CITEREFDenjoy1938" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Arnaud_Denjoy" title="Arnaud Denjoy">Denjoy, Arnaud</a> (1938), "Sur une fonction réelle de Minkowski", J. Math. Pures Appl., Série IX (in French), 17: 105–151, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0018.34602">0018.34602</a></cite></li></ul>

</div>

<div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3></div>

<ul><li><cite id="CITEREFAlkauskas2010" class="citation cs2">Alkauskas, Giedrius (2010), "The moments of Minkowski question mark function: the dyadic period function", Glasgow Mathematical Journal, 52 (1): 41–64, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0801.0051">0801.0051</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0017089509990152">10.1017/S0017089509990152</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2587817">2587817</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115167042">115167042</a></cite></li>

<li><cite id="CITEREFBhattacharjeeMacphersonMöllerNeumann1997" class="citation cs2">Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; <a href="Peter_M._Neumann" title="Peter M. Neumann">Neumann, Peter M.</a> (1997), "Rational numbers", Notes on Infinite Permutation Groups, Texts and Readings in Mathematics, vol. 12, Berlin: Springer-Verlag, pp. 77–86, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-93-80250-91-5_9">10.1007/978-93-80250-91-5_9</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-85931-13-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1632579">1632579</a></cite></li>

<li><cite id="CITEREFDushistovaMoshchevitin2012" class="citation cs2">Dushistova, Anna A.; Moshchevitin, Nikolai G. (March 2012), "On the derivative of the Minkowski question mark function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ?(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ?(x)}</annotation>

</semantics>

</math><img src="./d4a74bc67f24aab1e92eeb351c97210ab530811f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.236ex; height:2.843ex;" alt="{\displaystyle ?(x)}" loading="lazy">", Journal of Mathematical Sciences, 182 (4): 463–471, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.2219">0706.2219</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10958-012-0750-2">10.1007/s10958-012-0750-2</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2825515">2825515</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115156022">115156022</a></cite></li>

<li><cite id="CITEREFFinch2003" class="citation cs2">Finch, Steven R. (2003), <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalcons0000finc">Mathematical constants</a>, Encyclopedia of Mathematics and Its Applications, vol. 94, <a href="Cambridge" title="Cambridge">Cambridge</a>: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-81805-6</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1054.00001">1054.00001</a></cite></li>

<li><cite id="CITEREFGirgensohn1996" class="citation cs2">Girgensohn, Roland (1996), "Constructing singular functions via Farey fractions", <a href="Journal_of_Mathematical_Analysis_and_Applications" title="Journal of Mathematical Analysis and Applications">Journal of Mathematical Analysis and Applications</a>, 203 (1): 127–141, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.1996.0370">10.1006/jmaa.1996.0370</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1412484">1412484</a></cite></li>

<li><cite id="CITEREFJordanSahlsten2016" class="citation cs2">Jordan, Thomas; Sahlsten, Tuomas (2016), "Fourier transforms of Gibbs measures for the Gauss map", <a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a>, 364 (3–4): 983–1023, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1312.3619">1312.3619</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013arXiv1312.3619J">2013arXiv1312.3619J</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00208-015-1241-9">10.1007/s00208-015-1241-9</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56046793">56046793</a></cite></li>

<li><cite id="CITEREFKhinchin1997" class="citation cs2"><a href="Aleksandr_Khinchin" title="Aleksandr Khinchin">Khinchin, A. Ya.</a> (1997) [Originally published in Russian, 1935], "10: Quadratic irrational numbers and periodic continued fractions", Continued Fractions, <a href="University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>, pp. 47–50, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-69630-8</bdi></cite>; reprinted by Dover Publications, 1997</li>

<li><cite id="CITEREFMoshchevitin2020" class="citation cs2 cs1-prop-interwiki-linked-name"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D1%89%D0%B5%D0%B2%D0%B8%D1%82%D0%B8%D0%BD,_%D0%9D%D0%B8%D0%BA%D0%BE%D0%BB%D0%B0%D0%B9_%D0%93%D0%B5%D1%80%D0%BC%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87" class="extiw external" title="ru:Мощевитин, Николай Германович">Moshchevitin, Nikolay</a> [in Russian] (25 November 2020), "Open problems session", <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ClkddCrWZus">Diophantine Problems, Determinism and Randomness</a>, CIRM, 29:03 – via YouTube</cite></li>

<li><cite id="CITEREFPytheas_Fogg2002" class="citation cs2">Pytheas Fogg, N. (2002), <a href="Val%C3%A9rie_Berth%C3%A9" title="Valérie Berthé">Berthé, Valérie</a>; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.), Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, vol. 1794, Berlin: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-44141-0</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1014.11015">1014.11015</a></cite></li>

<li><cite id="CITEREFSalem1943" class="citation cs2"><a href="Rapha%C3%ABl_Salem" title="Raphaël Salem">Salem, Raphaël</a> (1943), <a rel="nofollow" class="external text" href="https://www.ams.org/journals/tran/1943-053-03/S0002-9947-1943-0007929-6/S0002-9947-1943-0007929-6.pdf">"On some singular monotonic functions which are strictly increasing"</a> (PDF), <a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>, 53 (3): 427–439, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990210">10.2307/1990210</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990210">1990210</a></cite></li></ul>

</div>

<div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3></div>

<ul><li><cite id="CITEREFAlkauskas2008" class="citation cs2">Alkauskas, Giedrius (2008), <a rel="nofollow" class="external text" href="http://eprints.nottingham.ac.uk/10641/">Integral transforms of the Minkowski question mark function</a>, PhD thesis, <a href="University_of_Nottingham" title="University of Nottingham">University of Nottingham</a></cite></li>

<li><cite id="CITEREFBibiloniParadisViader1998" class="citation cs2">Bibiloni, L.; Paradis, J.; Viader, P. (1998), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150622194657/http://www.econ.upf.es/en/research/onepaper.php?id=226">"A new light on Minkowski's ?(x) function"</a>, Journal of Number Theory, 73 (2): 212–227, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjnth.1998.2294">10.1006/jnth.1998.2294</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10230%2F843">10230/843</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0928.11006">0928.11006</a>, archived from <a rel="nofollow" class="external text" href="http://www.econ.upf.es/en/research/onepaper.php?id=226">the original</a> on 22 June 2015</cite></li>

<li><cite id="CITEREFBibiloniParadisViader2001" class="citation cs2">Bibiloni, L.; Paradis, J.; Viader, P. (2001), "The derivative of Minkowski's singular function", Journal of Mathematical Analysis and Applications, 253 (1): 107–125, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.2000.7064">10.1006/jmaa.2000.7064</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0995.26005">0995.26005</a></cite></li>

<li><cite id="CITEREFConley2003" class="citation cs2">Conley, R. M. (2003), A Survey of the Minkowski ?(x) Function, Masters thesis, <a href="West_Virginia_University" title="West Virginia University">West Virginia University</a></cite></li>

<li><cite id="CITEREFConway2000" class="citation cs2"><a href="John_Horton_Conway" title="John Horton Conway">Conway, J. H.</a> (2000), "Contorted fractions", On Numbers and Games (2nd ed.), Wellesley, Mass.: A K Peters, pp. 82–86</cite></li>

<li><cite id="CITEREFVepstas2004" class="citation cs2">Vepstas, L. (2004), <a rel="nofollow" class="external text" href="http://www.linas.org/math/chap-minkowski.pdf">The Minkowski Question Mark and the Modular Group SL(2,Z)</a> (PDF)</cite></li>

<li><cite id="CITEREFVepstas2008" class="citation arxiv cs2">Vepstas, L. (2008), "On the Minkowski Measure", <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0810.1265">0810.1265</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.DS">math.DS</a>]</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://uosis.mif.vu.lt/~alkauskas/minkowski.htm">An extensive bibliography list</a></li>

<li><cite id="CITEREFWeisstein" class="citation web cs2"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html">"Minkowski's Question Mark Function"</a>, <a href="MathWorld" title="MathWorld">MathWorld</a></cite></li>

<li><a rel="nofollow" class="external text" href="https://gist.github.com/pallas/5565556">Simple IEEE 754 implementation in C++</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-05" href="https://en.wikipedia.org/wiki/?title=Minkowski's_question-mark_function&oldid=1304261765">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>