Must T be a contraction?

Question

Let $X$ be a complete metric space . Suppose $T:X\to X$ is a function and $T^n$ is a contraction for some positive integer n. Here $T^n$ is the composition of $T$ with itself $n$ times. Must $T$ have a fixed point? Must $T$ be a contraction ?

Answer 1

$T$ doesn't have to have fixed point. Let's consider space $X=\{1\} \times I \cup \{2\} \times I$, where $I$ is a unit compact interval, and $T \colon X \rightarrow X$, which acts on $X$ as follows: it takes each interval, shrinks it to $[0,\frac{1}{2}]$ and moves to the other interval, such that $T(\{1\} \times I)=\{2\} \times[0,\frac{1}{2}]$ and $T(\{2\} \times I)=\{1\} \times[0,\frac{1}{2}]$, it does not have fixed point, but $T^2$ does have one, for example $x=(1,0)$.