# When deRham curve is bijection?

Motivation: Suppose we have deRham curve. From wikipedia:

Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings 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 $c_x = d_{b_1} \circ d_{b_2} \circ ... \circ d_{b_k} \circ ...$ 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.

Such curve may be continuous, and we know when it happened. One of the examples of deRham curves is the Minkowski question mark function $?(x)$, which may be defined as function from Stern-Brocot tree to dyadic rationals. As a deRham curve it is defined by

$$d_0(z) = \frac{z}{z+1} \quad {\rm and } \quad d_1(z)= \frac{1}{z+1}$$

$?(x)$ is continuous function of x, and even has explicit formula for inverse function, so it is bijection!

Question: Let's define deRham curve for domain $M=[0,1]\times[0,1]$ with standard metric ( as a subset of $R^2$). What requirements have to be set on functions $d_0(x)$ and $d_0(x)$ defining deRham curve (or on other objects) in order to obtain deRham curve which is function and bijection from $[0,1]$ to $[0,1]$?

Remarks:

Indeed: as new user I was able to add only one link, so I decide to omit deRham and insert $?(x)$ function.

$[0,1]$ here is set of real numbers $a$ such that $0 \leq a \leq 1$. Yes, it was a mistake, thank You for pointing!

@Yemon Choi - As You may see, Minkowski question mark function $?(x)$ is deRham curve and normal function as well. So some deRham curves are both continuous curve defined in $M=[0,1]\times[0,1]$ ( and we know when - see wikipedia definition) and good functions from $[0,1]$ to $[0,1]$. Probably other example is Cantor curve which is $M \to M$ and normal function simultaneously - and its even continuous. Also Blancmange curve is deRham curve and simultaneously normal function but not has inverse since it take the same values for different argument. If You consider function definition then You may see that function F between X and Y is a relation on XxY which is functional, which means: "functional (also called right-definite or right-unique1): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z." So I don't see here any conceptual problems but rather only formal ones.

Maybe I should describe my motivation: General picture is that: some deRham curves on $M=[0,1]\times[0,1]$ gives a functions from $[0,1] \to [0,1]$ ( when?) and some of them are even bijections. It is interesting which are bijections because such kind of curves defines in natural ( even if complicated) way some coding for representing real numbers by simple alphabet - You may see wikipedia discussion about relation between $?(x)$ and coding of the rationals on Stern-Brockot tree. Every number in Stern-Brockot binary tree may be represented by sequence of letters $LRRRLRRLLL...$ depicting the path from the root of the tree to the number. So as in this way You may wrote every rational number, and even every real if You include infinite sequences. This way You may have some "non-positional system" for representing numbers - in this place in fact You have system based on continuous fractions. Then deRham curves seems to me as natural way of generalization of such structure to other kind of "non-positional systems".It is interesting whether it is structure with many examples of such bijections or maybe $?(x)$ and Cantor curve - are the only ones. How big is that space of bijections? Are there any which allows faster computations? Or maybe more accurate or with less memory requirements to represent some kind of rationals or real numbers??

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Not providing a link to the relevant Wikipedia page when you refer to it is so 80s... –  Mariano Suárez-Alvarez Feb 5 '10 at 23:48
I've already cleaned up one wikipedia link and added the blockquote. If anyone, preferably the original poster, could clarify the "question" being asked -- what is $[1,0]$ meant to be, anyway? -- that would be nice. –  Yemon Choi Feb 6 '10 at 0:52
Your de Rham curve is supposed to take values in $M=[0,1]\times [0,1]$. So I still do not understand how you want it to be a bijection from $[0,1]$ to $[0,1]$. –  Yemon Choi Feb 6 '10 at 10:18
I think kakaz is asking whether the de Rham curve (or, to be technically correct, its image in $[0,1]^2$) is the graph of a bijection from $[0,1]$ to $[0,1]$. –  Pete L. Clark Feb 22 '10 at 13:05
@Pete: Yes You are completely right: asking in a way You propose is much more clear then my own. Thank You! –  kakaz Feb 22 '10 at 13:23