Finding $f$ such that $f(f(f(f…(x)))) = x$

I would like to find the set of continuous functions $$f_n(x)$$, where $$f_n(x):\mathbb{R}\to \mathbb{R}$$ satisfies $$f_n(f_n(f_n(f_n...(x)))) = x$$ where there are $$n$$ iterations of $$f(x)$$. For example $$f_1(x)$$ would be the solution to $$f_1(x)=x$$. $$f_2(x)$$ would be the solution to $$f_2(f_2(x)) = x$$.

For $$f_1(x)$$, the only solution is $$f_1(x)=x$$. For $$f_2(x)$$, the solutions are involutions.

For $$f_3(x)$$, the only answer is $$f_3(x)=x$$. For all other $$f_n(x)$$, one solution is $$f_n(x) = x$$.

My question: For $$n \ge 3$$, is $$f_n(x) = x$$ the only solution? If not, what are the solutions?

Edit: @MattSamuel said that any involution works for an even $$n$$. This is because $$f_n(f_n(x))$$ can be replaced with $$x$$. For example, $$f_2(f_2(f_2(f_2(f_2(f_2(x)))))) = f_2(f_2(f_2(f_2(x)))) = f_2(f_2(x)) = x$$ However, this does not necessarily mean that involutions are the only set of solutions for $$f_{2k}(x)$$.

• Involutions work for any even number of compositions. – Matt Samuel Jul 27 '19 at 19:48
• @MattSamuel Wow, that is true. Can functions that aren't involutions work for an even number of compositions? – Varun Vejalla Jul 27 '19 at 19:51
• Probably, but I don't know any off hand. – Matt Samuel Jul 27 '19 at 19:52
• @PackSciences, only real functions. – Varun Vejalla Jul 27 '19 at 19:58
• Ok, good you edited the question – PackSciences Jul 27 '19 at 19:58

There are no monotone solution for odd $$n$$ other then $$f_n(x) = x$$. There are also no continuous solutions as any continuous injection is monotone.

There are many discontinuous solutions for any $$n$$. Represent any real number $$x$$ as $$x = \lfloor x \rfloor + \{x\}$$ (integer and floor part), so we now have bijection $$\mathbb{R} \leftrightarrow [0, 1) \times \mathbb{Z}$$. Choose your favorite bijection $$g_n: \mathbb Z \leftrightarrow \mathbb Z$$ of order $$n$$ - for example, $$g_n(i) = (i + 1) \mod n + \lfloor\frac i n\rfloor$$ - split $$\mathbb{Z}$$ into segments of length $$n$$ and rotate any segment. Now define $$f_n(x) = \{x\} + g_n(\lfloor x \rfloor)$$. It's even continuous everywhere but in points $$n - 1 + kn$$.

(for simplicity, I'll denote $$f^n$$ to be $$n$$-th iteration of $$f$$ - we will not need powers here)

The only continuous solutions are involutions - answer you linked can be extended to proof it. $$f$$ have to be monotonic - if it isn't - say we have $$f(x) > f(y) > f(z)$$ while $$x > z > y$$ - then it's not injective, as there is point in $$q \in [z, x]$$ s.t. $$f(q) = f(y)$$, so we will have $$f^n(q) = f^n(y)$$ but $$q \neq y$$.

If $$f$$ is strictly increasing, then $$f(x) = x$$ by @Najib's argument.

If $$f$$ is strictly decreasing, then $$f$$ has a single fixed point $$x_0$$. We have $$f(x_0 + a) < f(x_0) = x_0$$ and $$f(x_0 - a) > f(x_0) = x_0$$ for positive $$a$$. $$g =f\circ f$$ is continuous and injective - so monotonic. As $$f(x_0 + 1) < f(x_0)$$, we have $$f(f(x_0 + 1)) > x_0$$, so $$g$$ is increasing. If $$g(x) \neq x$$ for some $$x$$, we will again have $$f^n(x) \neq x$$. So $$g(x) = x$$. And thus $$f$$ is involution.

• Added part about continuous functions - involutions are the only answer. – mihaild Jul 27 '19 at 21:50
• Just to make sure I understand correctly, there are no other solutions for odd $n$ other than $f_n(x) = x$ and no other even solutions for even $n$ other than involutions? – Varun Vejalla Jul 28 '19 at 17:46
• Yes. Also, the only involutions on $\mathbb{R}$ are $f(x) = x$ and $f(x) = a - x$. – mihaild Jul 28 '19 at 18:53
• That doesn't sound right. Isn't $1/x$ an involution too? – Varun Vejalla Jul 28 '19 at 19:48
• Sorry, the only continuous involutions. $1 / x$ (defined in $0$ as $0$) isn't continuous. – mihaild Jul 28 '19 at 20:06