RO-NMO 2007 Problem 94

Given a set $A$ and a function $f: A\rightarrow A$ , denote by $f_{1}(A)=f(A)$ , $f_{2}(A)=f(f_{1}(A))$ , $f_{3}(A)=f(f_{2}(A))$ , and so on, ( $f_{n}(A)=f(f_{n-1}(A))$ , where the notation $f(B)$ means the set $\{ f(x) \ : \ x\in B\}$ of images of points from $B$ ). Denote also by $f_{\infty}(A)=f_{1}(A)\cap f_{2}(A)\cap \ldots = \bigcap_{n\geq 1}f_{n}(A)$ .

a) Show that if $A$ is finite, then $f(f_{\infty}(A))=f_{\infty}(A)$ .

b) Determine if the above is true for $A=\mathbb{N}\times \mathbb{N}$ and the function $$f\big((m,n)\big)=\begin{cases}(m+1,n) & \text{if }n\geq m\geq 1 (0,0) & \text{if }m>n (0,n+1) & \text{if }n=0. \end{cases}$$