Question

The function `f(x) = 1/(1-x)` on `RR` \ `{0, 1}` has the (rather nice) property that `f(f(f(x))) = x`. Is there a simple example of a function `g(x)` such that `g(g(g(g(x)))) = x` but `g(g(x)) != x`?

Answer 1

The function:

`g(x) = 1/x` when `x in (0, 1) uu (-oo, -1)`

`g(x) = -x` when `x in (-1, 0) uu (1, oo)`

works, but is not as simple as `f(x) = 1/(1-x)`

Explanation:

We can split `RR` \ `{ -1, 0, 1 }` into four open intervals `(-oo, -1)`, `(-1, 0)`, `(0, 1)` and `(1, oo)` and define `g(x)` to map between the intervals cyclically.

This is a solution, but are there any simpler ones?