Let's we have finite field $F_q$ and we have couple of involutions $g$,$f$ with exactly one fixed point (zero).

Let's take any element $alpha \in F_q$

Let's start applying involutions to this element in the following way:

$ ... f(g(alpha)$ ... $g(alpha)$ ... $alpha$ ... $f(alpha)$ ... $g(f(alpha)) ... $

Let's imagine that on some step the element on the left side become equal to the element on the right side for any $alpha$.

What is the name for such couple of involutions with the following property ?

Are there any theorems regarding existence of such couple of involutions ?

The example of such couple of involutions is here: Chains of numbers generated by 2 involutions.