Is there a term for a function that is like an involutory function, but generalized to more recursions?

Question

So, an involutory function is a function that is its own inverse. In other words, a recursive call to this function would return the original input:

$$f(f(x)) \equiv x$$

Suppose we had a function with a similar property, but with more than additional recursions required:

e.g. $$f(f(f(x))) \equiv x$$

Is there any existing term to describe this (for the specific example given above, or generalized to n-recursions)? If not, how would you go about making up a new term to describe it?

Answer 1

This is sometimes called a functional ($n$-th) root of the identity.

See here.