Is there a name for a class of functions whose compositions are "periodic"? e.g., $f(g(h(x))) = g(h(f(x))) = h(f(g(x))) = x$ I am interested in whether a specific class of functions has been studied; namely, functions whose composition is "periodic", in the sense that:
$f(g(h(x))) = h(f(g(x))) = g(h(f(x))) = x$
A trivial example is any invertible function and its inverse, such as $f(f^{-1}(x)) = f^{-1}(f(x)) = x$. But I am looking for the general case that are characterized a specific "sequential" structure (which functions and their inverses satisfy):
$f_n(f_{n-1}(...(f_1(x)))...) = f_{n-1}(f_{n-2}(...f_1(f_n(x))...) = ... = f_1(f_n(f_{n-1}(...f_2(x)...)$
Have such functions and their properties been studied? Do they have a name? I am vaguely reminded of finite groups...
 A: For ease of notation, I limit what I am writing here to the $n=3$ case. But the same ideas apply to greater $n$ as well. I also assume the domain of each function is $\mathbb{R}$.
As this is described, it follows that each function is invertible:

*

*$f$ is surjective on $\mathbb{R}$ because for any $x$, $f(\cdots)=x$.

*if $f(a)=f(b)$, then $a=g(h(f(a)))=g(h(f(b)))=b$, so $f$ is one-to-one.

And the same holds for $g$ and $h$.
So you may as well limit your attention to invertible functions in the first place. Now we want three invertible functions such that <OP conditions here>.
But if we are only looking at invertible functions, then having $f(g(h(x)))=x$ alone will imply that other permutations in the cycle also work out to equal $x$. For example letting $x=f(y)$ implies $f(g(h(f(y))))=f(y)$, and now apply $f^{-1}$ to get  $g(h(f(y)))=y$.
So ultimately this would boil down to having three invertible functions that happen to compose in one order to the identity function. To think of it group theoretically, the collection of invertible functions with full domain is a group with composition as the operation. And we just want three group members that multiply to the identity.
In this light, it doesn't seem likely that there is group theoretic vocabulary for some chain of elements that happen to multiply to the identity (except of course for when $n=2$). But maybe there is a name for this phenomenon in functional analysis.
