Question

3. Suppose that (M, ρ) is a compact metric space and f : (M, p)-+ (M,p) is a function such that (Vz, y E M) ρ (z, y) ρ (f (x), f (y)). a. Let x E (M, ρ) and consider the sequence of points {f(n) (X)}n 1 . (Remember: fn) denotes the composition of f with itself, n times, so for each n, f+() rn, k E N) such that ρ (f(m) (x) ,f(n +k) (r)) < ε . Prove that (Ve > 0) (

Answer 1

Let $x\in M$ We need to prove that for all $\epsilon>0$ there exist
m,h e
such that
(m+k)
.

Consider the sequence $x=f^{(0)}(x), f^{(1)}(x),f^{(2)}(x),\cdots$ . Because $(M,\rho)$ is compact, this sequence has a convergent subsequence, say $f^{(i_1)}(x),f^{(i_2)}(x),\cdots$ , converging to
УЕМ
:

lim f(in )(2) y =

Thus, there is some $n_\epsilon\in\mathbb N$ such that $n>n_\epsilon$ implies $\rho(y,f^{(i_n)}(x))<0.25(\epsilon)$ . Thus, for all $m,n>n_\epsilon$ we have

$\begin{align*}\rho(f^{(i_m)}(x),f^{(i_n)}(x))&\leq \rho(y,f^{(i_n)}(x))+\rho(y,f^{(i_m)}(x))\\ &<0.25(\epsilon)+0.25(\epsilon)\\ &=0.5\epsilon\end{align*}$

In particular, we have

$\begin{align*}\rho(f^{(i_{n_\epsilon+1})}(x),f^{(i_{n_\epsilon+2})}(x))&\leq 0.5\epsilon\end{align*}$

Let $\begin{align*}m={i_{n_\epsilon+1}}\end{align*}$ and $\begin{align*}k={i_{n_\epsilon+2}}-{i_{n_\epsilon+1}}\end{align*}$ ; then we have

$\begin{align*}\rho(f^{(m)}(x), f^{m+k)}(x))=\rho(f^{(i_{n_\epsilon+1})}(x),f^{(i_{n_\epsilon+2})}(x))&\leq 0.5\epsilon<\epsilon\end{align*}$

This proves the desired statement.