Exercises
[093] Let \(A\) be a set and let \(g:A→ A\) be injective. We define the relation \(x∼ y\) which is true when an \(n≥ 0\) exists such that \(x=g^ n (y)\) or \(x= g^ n(y)\); where
\[ g^ n=\overbrace{g◦ \cdots ◦ g}^ n \]is the \(n\)-th iterate of the composition. (We decide that \(g^ 0\) is identity). Show that \(x∼ y\) is an equivalence relation. Study equivalence classes. Let \(U=⋂_{n=1}^∞ g^ n(A)\) be the intersection of repeated images. Show that each class is entirely contained in \(U\) or is external to it.
Solution 1