Consider some complete metric space ${\displaystyle (M,d)}$  (generally ${\displaystyle \mathbb {R} }$ ² with the usual euclidean distance), and a pair of contracting maps on M:

${\displaystyle d_{0}:\ M\to M}$ 

${\displaystyle d_{1}:\ M\to M.}$ 

By the Banach fixed-point theorem, these have fixed points ${\displaystyle p_{0}}$  and ${\displaystyle p_{1}}$  respectively. Let x be a real number in the interval ${\displaystyle [0,1]}$ , having binary expansion

${\displaystyle x=\sum _{k=1}^{\infty }{\frac {b_{k}}{2^{k}}},}$ 

where each ${\displaystyle b_{k}}$  is 0 or 1. Consider the map

${\displaystyle c_{x}:\ M\to M}$ 

defined by

${\displaystyle c_{x}=d_{b_{1}}\circ d_{b_{2}}\circ \cdots \circ d_{b_{k}}\circ \cdots ,}$ 

where ${\displaystyle \circ }$  denotes function composition. It can be shown that each ${\displaystyle c_{x}}$  will map the common basin of attraction of ${\displaystyle d_{0}}$  and ${\displaystyle d_{1}}$  to a single point ${\displaystyle p_{x}}$  in ${\displaystyle M}$ . The collection of points ${\displaystyle p_{x}}$ , parameterized by a single real parameter x, is known as the de Rham curve.

The construction in terms of binary digits can be understood in two distinct ways. One way is as a mapping of Cantor space to distinct points in the plane. Cantor space is the set of all infinitely-long strings of binary digits. It is a discrete space, and is disconnected. Cantor space can be mapped onto the unit real interval by treating each string as a binary expansion of a real number. In this map, the dyadic rationals have two distinct representations as strings of binary digits. For example, the real number one-half has two equivalent binary expansions: ${\displaystyle h_{1}=0.1000\cdots }$  and ${\displaystyle h_{0}=0.01111\cdots }$  This is analogous to how one has 0.999...=1.000... in decimal expansions. The two points ${\displaystyle h_{0}}$  and ${\displaystyle h_{1}}$  are distinct points in Cantor space, but both are mapped to the real number one-half. In this way, the reals of the unit interval are a continuous image of Cantor space.

The same notion of continuity is applied to the de Rham curve by asking that the fixed points be paired, so that

${\displaystyle d_{0}(p_{1})=d_{1}(p_{0})}$ 

With this pairing, the binary expansions of the dyadic rationals always map to the same point, thus ensuring continuity at that point. Consider the behavior at one-half. For any point p in the plane, one has two distinct sequences:

${\displaystyle d_{0}\circ d_{1}\circ d_{1}\circ d_{1}\circ \cdots (p)}$ 

and

${\displaystyle d_{1}\circ d_{0}\circ d_{0}\circ d_{0}\circ \cdots (p)}$ 

corresponding to the two binary expansions ${\displaystyle 1/2=0.01111\cdots }$  and ${\displaystyle 1/2=0.1000\cdots }$ . Since the two maps are both contracting, the first sequence converges to ${\displaystyle d_{0}(p_{1})}$  and the second to ${\displaystyle d_{1}(p_{0})}$ . If these two are equal, then both binary expansions of 1/2 map to the same point. This argument can be repeated at any dyadic rational, thus ensuring continuity at those points. Real numbers that are not dyadic rationals have only one, unique binary representation, and from this it follows that the curve cannot be discontinuous at such points. The resulting de Rham curve ${\displaystyle p_{x}}$  is a continuous function of x, at all x.

In general, the de Rham curves are not differentiable.

De Rham curves are by construction self-similar, since

${\displaystyle p(x)=d_{0}(p(2x))}$  for ${\displaystyle x\in [0,1/2]}$  and

${\displaystyle p(x)=d_{1}(p(2x-1))}$  for ${\displaystyle x\in [1/2,1].}$ 

The self-symmetries of all of the de Rham curves are given by the monoid that describes the symmetries of the infinite binary tree or Cantor space. This so-called period-doubling monoid is a subset of the modular group.

The image of the curve, i.e. the set of points ${\displaystyle \{p(x),x\in [0,1]\}}$ , can be obtained by an Iterated function system using the set of contraction mappings ${\displaystyle \{d_{0},\ d_{1}\}}$ . But the result of an iterated function system with two contraction mappings is a de Rham curve if and only if the contraction mappings satisfy the continuity condition.

Detailed, worked examples of the self-similarities can be found in the articles on the Cantor function and on Minkowski's question-mark function. Precisely the same monoid of self-similarities, the dyadic monoid, apply to every de Rham curve.

The following systems generate continuous curves.

Cesàro curves edit

Cesàro curves, also known as Cesàro–Faber curves or Lévy C curves, are De Rham curves generated by affine transformations conserving orientation, with fixed points ${\displaystyle p_{0}=0}$  and ${\displaystyle p_{1}=1}$ .

Because of these constraints, Cesàro curves are uniquely determined by a complex number ${\displaystyle a}$  such that ${\displaystyle |a|<1}$  and ${\displaystyle |1-a|<1}$ .

The contraction mappings ${\displaystyle d_{0}}$  and ${\displaystyle d_{1}}$  are then defined as complex functions in the complex plane by:

${\displaystyle d_{0}(z)=az}$ 

${\displaystyle d_{1}(z)=a+(1-a)z.}$ 

For the value of ${\displaystyle a=(1+i)/2}$ , the resulting curve is the Lévy C curve.

Koch–Peano curves edit

In a similar way, we can define the Koch–Peano family of curves as the set of De Rham curves generated by affine transformations reversing orientation, with fixed points ${\displaystyle p_{0}=0}$  and ${\displaystyle p_{1}=1}$ .

These mappings are expressed in the complex plane as a function of ${\displaystyle {\overline {z}}}$ , the complex conjugate of ${\displaystyle z}$ :

${\displaystyle d_{0}(z)=a{\overline {z}}}$ 

${\displaystyle d_{1}(z)=a+(1-a){\overline {z}}.}$ 

The name of the family comes from its two most famous members. The Koch curve is obtained by setting:

${\displaystyle a_{\text{Koch}}={\frac {1}{2}}+i{\frac {\sqrt {3}}{6}},}$ 

while the Peano curve corresponds to:

${\displaystyle a_{\text{Peano}}={\frac {(1+i)}{2}}.}$ 

The de Rham curve for ${\displaystyle a=(1+ib)/2}$  for values of ${\displaystyle b}$  just less than one visually resembles the Osgood curve. These two curves are closely related, but are not the same. The Osgood curve is obtained by repeated set subtraction, and thus is a perfect set, much like the Cantor set itself. The construction of the Osgood set asks that progressively smaller triangles to be subtracted, leaving behind a "fat" set of non-zero measure; the construction is analogous to the fat Cantor set, which has a non-zero measure. By contrast, the de Rham curve is not "fat"; the construction does not offer a way to "fatten up" the "line segments" that run "in between" the dyadtic rationals.

General affine maps edit

The Cesàro–Faber and Peano–Koch curves are both special cases of the general case of a pair of affine linear transformations on the complex plane. By fixing one endpoint of the curve at 0 and the other at one, the general case is obtained by iterating on the two transforms

${\displaystyle d_{0}={\begin{pmatrix}1&0&0\\0&\alpha &\delta \\0&\beta &\varepsilon \end{pmatrix}}}$ 

and

${\displaystyle d_{1}={\begin{pmatrix}1&0&0\\\alpha &1-\alpha &\zeta \\\beta &-\beta &\eta \end{pmatrix}}.}$ 

Being affine transforms, these transforms act on a point ${\displaystyle (u,v)}$  of the 2-D plane by acting on the vector

${\displaystyle {\begin{pmatrix}1\\u\\v\end{pmatrix}}.}$ 

The midpoint of the curve can be seen to be located at ${\displaystyle (u,v)=(\alpha ,\beta )}$ ; the other four parameters may be varied to create a large variety of curves.

The blancmange curve of parameter ${\displaystyle w}$  can be obtained by setting ${\displaystyle \alpha =\beta =1/2}$ , ${\displaystyle \delta =\zeta =0}$  and ${\displaystyle \varepsilon =\eta =w}$ . That is:

${\displaystyle d_{0}={\begin{pmatrix}1&0&0\\0&1/2&0\\0&1/2&w\end{pmatrix}}}$ 

and

${\displaystyle d_{1}={\begin{pmatrix}1&0&0\\1/2&1/2&0\\1/2&-1/2&w\end{pmatrix}}.}$ 

Since the blancmange curve for parameter ${\displaystyle w=1/4}$  is a parabola of the equation ${\displaystyle f(x)=4x(1-x)}$ , this illustrates the fact that on some occasions, de Rham curves can be smooth.

Minkowski's question mark function edit

Minkowski's question mark function is generated by the pair of maps

${\displaystyle d_{0}(z)={\frac {z}{z+1}}}$ 

and

${\displaystyle d_{1}(z)={\frac {1}{2-z}}.}$ 

Given any two functions ${\displaystyle d_{0}}$  and ${\displaystyle d_{1}}$ , one can define a mapping from Cantor space, by repeated iteration of the digits, exactly the same way as for the de Rham curves. In general, the result will not be a de Rham curve, when the terms of the continuity condition are not met. Thus, there are many sets that might be in one-to-one correspondence with Cantor space, whose points can be uniquely labelled by points in the Cantor space; however, these are not de Rham curves, when the dyadic rationals do not map to the same point.

Julia set of the Mandelbrot set edit

The Mandelbrot set is generated by a period-doubling iterated equation ${\displaystyle z_{n+1}=z_{n}^{2}+c.}$  The corresponding Julia set is obtained by iterating the opposite direction. This is done by writing ${\displaystyle z_{n}=\pm {\sqrt {z_{n+1}-c}}}$ , which gives two distinct roots that the forward iterate ${\displaystyle z_{n+1}}$  "came from". These two roots can be distinguished as

${\displaystyle d_{0}(z)=+{\sqrt {z-c}}}$ 

and

${\displaystyle d_{1}(z)=-{\sqrt {z-c}}.}$ 

Fixing the complex number ${\displaystyle c}$ , the result is the Julia set for that value of ${\displaystyle c}$ . This curve is continuous when ${\displaystyle c}$  is inside the Mandelbrot set; otherwise, it is a disconnected dust of points. However, the reason for continuity is not due to the de Rham condition, as, in general, the points corresponding to the dyadic rationals are far away from one-another. In fact, this property can be used to define a notion of "polar opposites", of conjugate points in the Julia set.

It is easy to generalize the definition by using more than two contraction mappings. If one uses n mappings, then the n-ary decomposition of x has to be used instead of the binary expansion of real numbers. The continuity condition has to be generalized in:

${\displaystyle d_{i}(p_{n-1})=d_{i+1}(p_{0})}$ , for ${\displaystyle i=0\ldots n-2.}$ 

This continuity condition can be understood with the following example. Suppose one is working in base-10. Then one has (famously) that 0.999...= 1.000... which is a continuity equation that must be enforced at every such gap. That is, given the decimal digits ${\displaystyle b_{1},b_{2},\cdots ,b_{k}}$  with ${\displaystyle b_{k}\neq 9}$ , one has

${\displaystyle b_{1},b_{2},\cdots ,b_{k},9,9,9,\cdots =b_{1},b_{2},\cdots ,b_{k}+1,0,0,0,\cdots }$ 

Such a generalization allows, for example, to produce the Sierpiński arrowhead curve (whose image is the Sierpiński triangle), by using the contraction mappings of an iterated function system that produces the Sierpiński triangle.

Ornstein and others describe a multifractal system, where instead of working in a fixed base, one works in a variable base.

Consider the product space of variable base-${\displaystyle m_{n}}$  discrete spaces

${\displaystyle \Omega =\prod _{n\in \mathbb {N} }A_{n}}$ 

for ${\displaystyle A_{n}=\mathbb {Z} /m_{n}\mathbb {Z} =\{0,1,\cdots ,m_{n}-1\}}$  the cyclic group, for ${\displaystyle m_{n}\geq 2}$  an integer. Any real number in the unit interval can be expanded in a sequence ${\displaystyle (a_{1},a_{2},a_{3},\cdots )}$  such that each ${\displaystyle a_{n}\in A_{n}}$ . More precisely, a real number ${\displaystyle 0\leq x\leq 1}$  is written as

${\displaystyle x=\sum _{n=1}^{\infty }{\frac {a_{n}}{\prod _{k=1}^{n}m_{k}}}}$ 

This expansion is not unique, if all ${\displaystyle a_{n}=0}$  past some point ${\displaystyle K<n}$ . In this case, one has that

${\displaystyle a_{1},a_{2},\cdots ,a_{K},0,0,\cdots =a_{1},a_{2},\cdots ,a_{K}-1,m_{K+1}-1,m_{K+2}-1,\cdots }$ 

Such points are analogous to the dyadic rationals in the dyadic expansion, and the continuity equations on the curve must be applied at these points.

For each ${\displaystyle A_{n}}$ , one must specify two things: a set of two points ${\displaystyle p_{0}^{(n)}}$  and ${\displaystyle p_{1}^{(n)}}$  and a set of ${\displaystyle m_{n}}$  functions ${\displaystyle d_{j}^{(n)}(z)}$  (with ${\displaystyle j\in A_{n}}$ ). The continuity condition is then as above,

${\displaystyle d_{j}^{(n)}(p_{1}^{(n+1)})=d_{j+1}^{(n)}(p_{0}^{(n+1)})}$ , for ${\displaystyle j=0,\cdots ,m_{n}-2.}$ 

Ornstein's original example used

${\displaystyle \Omega =\left(\mathbb {Z} /2\mathbb {Z} \right)\times \left(\mathbb {Z} /3\mathbb {Z} \right)\times \left(\mathbb {Z} /4\mathbb {Z} \right)\times \cdots }$ 

• Iterated function system
• Refinable function
• Modular group
• Fuchsian group

• Georges de Rham, On Some Curves Defined by Functional Equations (1957), reprinted in Classics on Fractals, ed. Gerald A. Edgar (Addison-Wesley, 1993), pp. 285–298.
• Linas Vepstas, A Gallery of de Rham curves, (2006).
• Linas Vepstas, Symmetries of Period-Doubling Maps, (2006). (A general exploration of the modular group symmetry in fractal curves.)