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Consider the mapping $f:x\to\frac{1}{x}, (x\ne0)$. It is trivial to see that $f(f(x))=x$.
My question is whether or not there exists a continuous map $g$ such that $g(g(g(x)))\equiv g^{3}(x)=x$? Furthermore, is there a way to find out if there is such a function that $g^{p}(x)=x$ for a prime $p$?

Edit: I realise I was a little unclear - I meant to specify that it was apart from the identity map. The other 'condition' I wanted to impose isn't very precise; I was hoping for a function that didn't seem defined for the purpose. However, the ones that are work perfectly well and they certainly answer the question.

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So, not 3 other maps, but of 1 map composed 3 times? –  Graphth Sep 14 '12 at 19:23
    
Yes, that's what I meant. –  Daniel Littlewood Sep 14 '12 at 19:34
    
Presumably you want $g$ to be continuous? –  Michael Joyce Sep 14 '12 at 19:36

4 Answers 4

up vote 11 down vote accepted

Consider the class of Möbius functions: $$f(x) = {ax+b\over cx+d}$$ for some constants $a,b,c,d$. If we represent Möbius function $f$ by its matrix of coefficients: $$\hat f = \pmatrix{ a&b \\ c&d }$$ then it turns out that to compose two Möbius functions $f$ and $g$ we just multiply their corresponding matrices.

In particular, if $f$ is a Möbius function with matrix $\hat f$, then $f(f(f(x)))$ is also a Möbius function, with matrix ${\hat f}{}^3$.

So it suffices to find a $2\times 2$ matrix $M$ with $M^3 = I$. There are numerous examples, but one such is $$\def\ang{{\frac{2\pi}3}}\pmatrix{\cos\ang & \sin\ang \\ -\sin\ang & \cos\ang} = \pmatrix{-\frac12 & \frac{\sqrt3}2 \\ -\frac{\sqrt3}2 & -\frac12 }.$$ This is just the matrix for the linear transformation of the plane that rotates the plane by a one-third turn about the origin.

So the corresponding Möbius function with period 3 is $$f(x) = {x-\sqrt 3\over x\sqrt3 + 1}.$$

By replacing $2\pi\over 3$ with $2\pi\over n$, you can construct a function with any period you want.

Note that you are not restricted to $M$ with $M^3 = I$. Since $I$ and $kI$ are the same when considered as Möbius functions, $M^3 = kI$ will work for any $k$.

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This looks like exactly what I want, thank you! –  Daniel Littlewood Sep 14 '12 at 19:40
    
I'm glad I could help. –  MJD Sep 14 '12 at 19:43

The function $$f(x) = {1\over 1-x}$$ has $f^3(x) = x$ for all $x$ where $f^3(x)$ is defined. (All reals except for 0 and 1.)

Other than the identity, there is no continuous function $\Bbb R\to \Bbb R$ having $f^3(x) = x$ for all $x$, by Sharkovskii's theorem.

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Of what type? ${}{}$ –  Mariano Suárez-Alvarez Sep 14 '12 at 19:54
    
@MarianoSuárez-Alvarez Better? –  MJD Sep 14 '12 at 19:58
    
What about any finite period for a continuous function? Period 2 is easy... –  user641 Sep 15 '12 at 0:06
    
@SteveD You really want to take a look at Sharkovskii's theorem, which has a lot to say about finite periods of continuous functions on R. –  MJD Sep 15 '12 at 0:53

Consider the function $f:\mathbb R\to\mathbb R$ such that $f(i)=i+1$ for all $i\in\{1,\dots,p-1\}$, $f(p)=1$ and $f(x)=x$ for all $x\in\mathbb R\setminus\{1,\dots,p\}$.

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This example will probably only help in showing that you want the function to satisfy some other conditions which you did not make explicit... –  Mariano Suárez-Alvarez Sep 14 '12 at 19:24

If you don’t impose any other conditions on $g$, it’s certainly possible. For $n\in\Bbb Z$ let $I_n=[n,n+1)$, and define

$$g:\Bbb R\to\Bbb R:x\mapsto\begin{cases}x+1,&\text{if }x\in I_n\text{ and }3\not\mid n\\ x-2,&\text{if }n\in I_n\text{ and }3\mid n\;. \end{cases}$$

This map translates $I_{3k+1}$ to $I_{3k+2}$, $I_{3k+2}$ to $I_{3k+3}$, and $I_{3k+3}$ back down to $I_{3k+1}$; it fixes no points of $\Bbb R$.

This generalizes: you can replace $3$ by any integer $m\ge 2$.

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