Redundant Functions?

Consider the function $f:\mathcal P(\Bbb N)\to\mathcal P(\Bbb N)$ such that $f(A)=\{a:a\in A,a\text{ prime}\}$. This has the property of not having any effect upon being iterated, i.e. $f(A)=f(f(A))=\cdots$. I was wondering what this property would be called and why/whether it is important to anything.

Such functions are called idempotent and are characterised by $f(x)=f(f(x))$. As the Wikipedia article says, this notion comes up in abstract algebra.
• Great answer! What would it be called when $f^n(x) = f^{n+1}(x)$ for some finite $n$ (rather than just $f^1(x) = f^2(x)$)? – Austin Weaver Sep 19 '17 at 14:57