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Let $f:\mathbb R\to\mathbb R$ be a continuous function such that $f(f(f(x)))=-8x$. Must we have $f(x)=-2x$? I can prove this if I assume that $f$ is continuously differentiable everywhere, but is that condition necessary? I was inspired to ask this question after I saw the thread a continuous function satisfying $f(f(f(x)))=-x$ other than f(x)=-x .

Some facts I've discovered, which you may use without proof:

  • $f$ is bijective, decreasing and therefore $f$ is a homeomorphism and differentiable almost everywhere

  • $f(0)=0$ because if $f(0)=a$, then $f(f(a))=0$, so $-8a=f(f(f(a)))=f(0)=a\implies a=0$.

  • If $x\neq 0$, then the list $\ldots,f^{-2}(x),f^{-1}(x),x,f(x),f^2(x),\ldots$ has no duplicate members, where $f^n$ is an $n$th functional power of $f$.

  • If $f$ is differentiable at $0$, then $f'(0)=-2$ because $f'(0)f'(0)f'(0)=-8$.

  • $x$ and $f(x)$ have opposite signs if $x\neq 0$, because $f(0)=0$ and $f$ is decreasing.

  • $f\circ f$ restricted to the positive numbers is an increasing function.

  • If we assume $f$ is continuously differentiable everywhere, then $f(x)=-2x$. Proof: Let $a_n=f^n(x)$ for some $x\neq 0$. Then $f'(a_1)f'(a_2)f'(a_3)=-8$, and $f'(a_2)f'(a_3)f'(a_4)=-8$ which gives us $f'(a_k)=f'(a_{k+3})$ for all integers $k$. Since $a_{3k}=(-8)^kx\to 0$ when $k\to-\infty$, we get $f'(x)=\lim_{k\to-\infty}f'(f^{-3k}(x))=f'(0)=-2$ and therefore $f(x)=-2x$ is the only solution.

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2 Answers

up vote 6 down vote accepted

I will set $f(-x)=-f(x)$ and $g(x)=-f(x)$.

Now $g(g(g(x)))=-f(-f(-f(x)))=-f(f(f(x)))=8x$.

So, I am searching a continuous $g$ from $\mathbb R^+$ to $\mathbb R^+$.

Now, I am regarding $h(x)=\log(g(\exp(x)))$ where $h(x)$ is a function from $\mathbb R$ to $\mathbb R$ that is continuous.

Then $$\begin{multline} h(h(h(x)))=\log(g(\exp(\log(g(\exp(\log(g(\exp x))))))))\\ =\log(g(g(g(\exp(x))))=\log(8\exp(x))=\log 8 + x=x+3\log 2.\end{multline}$$

Now, set $i(x)=h(x\log 2)/\log(2)$. Clearly, $i(i(i(x)))=x +3$.

Now, it is easy to see that it is enough to find three continuous increasing bijections of intervals, so that their concatenation is the identity.

For example, let $a(x)=x$, $b(x)=x^2$, and $c(x)=\sqrt x$ be such continuous increasing bijections for the interval $(0,1)$.

Now, take

$$i(x)=\begin{cases} a(\{x\})+\lfloor x\rfloor +1 \text{ if $3n \le \lfloor x \rfloor < 3n+1$ }\\ b(\{x\})+\lfloor x\rfloor +1 \text{ if $3n+1 \le \lfloor x \rfloor < 3n+2$ }\\ c(\{x\})+\lfloor x\rfloor +1 \text{ if $3n+2 \le \lfloor x \rfloor < 3n+3$}\\ \end{cases}$$

where $\{x\}$ denotes the fractional part of $x$.

$i(x)$ is certainly continuous, increasing and has the right concatenation property.

This gives one of many continuous solutions for the original equation.


In case you are wondering why this is substantially different from $f(f(f(x))))=-x$, note that for these equations the important object is the structure of the graph of real numbers with arrows from $x$ to $g(x)$ where $g$ is the function on the right-hand-side.

For $-8x$, the real numbers are split in 0 and doubly infinite chains, while for $-x$, the real numbers are split in 0 and circles of length 2.

To go to $f$ from the infinite chains, one has to take three of the chains and cycle through them to have a chain of $f$. There are so many ways of doing that that it is quite hopeful that some of them are continuous.

If you try to combine three of the circles of length 2 in a similar way, this will not give a monotone function, however, $f$ is bijective and continuous, so it has to be monotone. So, the only possibility is that $f$ maps each circle to itself which gives the unique solution.

So, in summary, chains are easier to reconcile with monotony than circles.

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Nice answer. Can you comment on the equation $f(f(f(x))) = -\frac{x}{8}$? Will this behave similarly to the explanation in your answer, or in a different way? –  Srivatsan Nov 25 '11 at 10:12
    
@Srivatsan Just take the inverse of $f$ in my solution, it is a bijection after all. But you can replace $8$ by any constant that is neither zero nor $\pm 1$. –  Phira Nov 25 '11 at 10:17
    
Yes, you're right. Thanks for your comment, and your answer. –  Srivatsan Nov 25 '11 at 10:20
    
Cool! I have two questions: 1. Do you think there exist weird solutions where $f$ isn't odd? 2. I don't really understand your comment about $f(f(f(x)))=-x$. A priori, this splits the real numbers into $0$ and cycles of length at most 6; how do you deduce that the cycles have length 2? The proof I posted in the other thread I just stumbled upon by accident. My intuition was basically that if intervals are mapped to intervals, then something bad should happen when an interval around the origin "collides" with another interval. –  Samuel Nov 25 '11 at 13:59
    
@Samuel You are right, my comment on the circles of length 2 is flawed, but I will reflect for a moment before I correct it. –  Phira Nov 25 '11 at 14:11
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First, I'll use the notation $ f \circ f \circ f $.

Note that $ \frac {d} {dx} f\circ f\circ f = (f' \circ f \circ f)(f'\circ f)f'=C$.

Denote $ A=(f' \circ f \circ f), B=(f'\circ f)$ you get that $ A,B,f'\ne 0$ everywhere. $ \frac {d^2} {dx^2} f\circ f\circ f = (f' \circ f \circ f)(f'\circ f)f'=(f''\circ f\circ f)Bf'+A(f''\circ f)f'\cdot f' + ABf''=0$. Since $A,B,f'\ne 0$ this could hold everywhere iff $f''=0$ everywhere, and you get that the solution has to be a linear function. From what you already achieved it's easy to see that the only such polynomial is $-2x$.

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I just realized that my assumption that f is twice differentiable isn't justified. However, I think this still has some value, as it doesn't require f to be continuously differentiable. –  Shai Deshe Nov 25 '11 at 9:57
    
Twice differentiable implies continuously differentiable. How do you deduce $f''=0$? –  Samuel Nov 25 '11 at 14:01
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