Should I care if some system forms a group?

Question

I am currently studying a discrete dynamical system where the function composition on the set of periodic points forms a finite group.

From a dynamical systems perspective, is there any reason to care about this property? Would it be worth continuing to study the properties of this group?

Answer 1

Assuming that you have checked that each element does have an inverse, note that:

The set if iterates on the set of periodic points is finite if and only if there is a maximal period.

So it is fair to say that nothing else can be learned.