De Rham curve

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In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.

The Cantor function, Minkowski's question mark function, the Lévy C curve, the blancmange curve and the Koch curve are all special cases of the general de Rham curve.

1 Construction
2 Properties
3 Classification and examples
4 See also
5 References

[edit] Construction

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

$d_0:\ M \to M$

$d_1:\ M \to M$

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

$x = \sum_{k=1}^\infty \frac{b_k}{2^k}$

where each $b k$ is 0 or 1. Consider the map

$c_x:\ M \to M$

defined by

$c_x = d_{b_1} \circ d_{b_2} \circ \cdots \circ d_{b_k} \circ \cdots$

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

[edit] Properties

When the fixed points are paired such that

d 0 (p 1) = d 1 (p 0)

then it may be shown that the resulting curve $p x$ is a continuous function of x. When the curve is continuous, it is not in general differentiable. 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 set. This so-called period-doubling monoid is a subset of the modular group.

[edit] Classification and examples

[edit] Césaro curves

Cesaro curve for a=0.3+i0.3

Cesaro curve for a=0.5+i0.5

Césaro curves (or Césaro-Faber curves) are De Rham curves generated by affine transformations conserving orientation, with fixed points $p 0 = 0$ and $p 1 = 1$ .

Because of these constraints, Césaro curves are uniquely determined by a complex number $a$ such that $| a | < 1$ and $| 1 - a | < 1$ .

The contraction mappings $d 0$ and $d 1$ are then defined as complex functions in the complex plane by:

d 0 (z) = a z

d 1 (z) = a + (1 - a) z

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

[edit] Koch-Peano curves

Koch-Peano curve for a=0.6+i0.37

Koch-Peano curve for a=0.6+i0.45

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 $p 0 = 0$ and $p 1 = 1$ .

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

$d_0(z) = a\overline{z}$

$d_1(z) = a + (1-a)\overline{z}$

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

$a_{\text{Kock}}=\frac{1}{2} + i\frac{\sqrt{3}}{6}$

while the Peano curve corresponds to:

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

[edit] General affine maps

Generic affine de Rham curve

The Cesaro-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

$d_0=\begin{pmatrix} 1 & 0 & 0 \\ 0 & \alpha &\delta \\ 0 & \beta & \epsilon \end{pmatrix}$

and

$d_1=\begin{pmatrix} 1&0&0 \\ \alpha & 1-\alpha&\zeta \\ \beta&-\beta&\eta \end{pmatrix}$

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

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

The midpoint of the curve can be seen to be located at $(u, v) = (α,β)$ ; the other four parameters may be varied to create a large variety of curves.

The blancmange curve of parameter $w$ can be obtained by setting $α = β = ε = 1 / 2$ , $δ = ζ = 0$ and $η = w$ . That is:

$d_0=\begin{pmatrix} 1&0&0 \\ 0 & 1/2&0 \\ 0&1/2&w \end{pmatrix}$

and

$d_1=\begin{pmatrix} 1&0&0 \\ 1/2 & 1/2&0 \\ 1/2&-1/2&w \end{pmatrix}$

Since the blancmange curve of parameter $w = 1 / 4$ is the parabola of equation $f (x) = 4 x (1 - x)$ , this illustrate the fact that in some occasion, de Rham curves can be smooth.

[edit] Minkowski's question mark function

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

$d_0(z) = \frac{z}{z+1}$

and

$d_1(z)= \frac{1}{z+1}$

[edit] See also

[edit] References

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, Symmetries of Period-Doubling Maps, (2004). (A general exploration of the modular group symmetry in fractal curves.)

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