Assuring Lipschitzian and contraction of a mapping

Question

Let $M=\{A\in P(\mathbb{N}):card(A)\geq 3\}$. Now define a mapping $T:M\rightarrow M$ by $T(A)=B$, where $B=\{\varphi(n):n\in A\}$ and $\varphi:\mathbb{N}\rightarrow\mathbb{N}$ is defined by $$\varphi(n)= \begin{cases} n/2 & \text{when n is even} \\ 3n+1 & \text{when n is odd} \end{cases} $$ Is there any suitable metric on $M$ such that the map $T$ is both Lipschitz and contraction? Here $P(\mathbb{N})$ is the power set of natural number.

Answer 1

Unfortunately, the map $M$ cannot be a contraction because it has more than one fixed point, for instance, $\Bbb N$ and $\{1,2,4\}$.

Answer 2

$T$ is not a well-defined map. Because, the image of $\{1,2,8\}$ is $\{1,4\}$ under the map $T$.