Let $Z$ be the the set of dyadic and ternary rationals in the interval $\left[\frac12,1\right)$ whose 3-adic valuation is either $-1$ or $0$, with the standard absolute value topology inherited from the real line.

Let $X=\{z\in Z:\nu_3(x)=0\}$

Let $Y=\{z\in Z:\nu_3(x)=-1\}$

Now define an equivalence relation $\sim$ which partitions $Z$ into equivalent pairs $\{x,y:y=f(x)\}$ where $f$ is given by the bijection $f:X\to Y$

$f(x)=\begin{cases}\frac{4x}3 &\text{if}& x<\frac34\\
\frac{2x}3& \text{if}& x>\frac34\end{cases}$

So for example $\frac5{8}\sim\frac56$ and $\frac{7}{8}\sim\frac{7}{12}$

The quotient (pseudo)metric $d_\sim$ on $Z,{\sim}$ is the infimum distance by which one can traverse from any equivalence class to another, by stepping on up to infinitely many equivalence classes in-between and summing only the distance *between* classes and not the distance travelled *within* classes.  More formally this is defined [here][1].

**Question**

I seek an explicit definition of $d_\sim$ in this instance.  Of course, in a sense Eric's answer does give that, but it leaves me with a requirement to somehow iterate over all possible sequences of equivalence classes and determine the shortest path, something well beyond my capabilities.

**Also Note**

While this question stands alone without reference to the Collatz conjecture, I feel more comfortable declaring that identifying this metric is a component of my study of the conjecture, partly in the spirit of full-disclosure, but also becase it may be material to the answer, to be mindful of the following observations:

- $g(x)=x+\frac132^{\nu_2(x)}$ is both a surjection $X\to Y$ and a surjection $Z/{\sim}\to Z/{\sim}$ and seen as a map $g:Z/{\sim}\to Z/{\sim}$ it is essentially the Collatz graph and its graph is connected if and only if the Collatz conjecture is true.  One should not be surprised therefore, if some proof that $d_\sim$ is the trivial metric $\forall [z_0],[z_1]:d_\sim([z_0],[z_1])=0$ were related to the claim that the graph of the orbit of $g$ through $Z/{\sim}$ is connected.

- The $n$-indexed sequences of the form $s_n(x)=x+(1-2^{-6n})\cdot2^{\nu_2(x)}\cdot3^{\nu_3(x)-1}$ form an exact cover of $X$ (up to subsequences) and $g$ is their infinite limit.  Moreover, for every $y\in Y$ there are precisely two $s_n$ (up to subsequences) whose union is the level set by $g$ of $y$.

- If $d_\sim$ is not the trivial (pseudo)metric, it seems likely a proof that some sequence gives the infimum for the metric may use the sequences given in the two bullets above.

- This question has a "dual", if you like, which is to enquire into the quotient pseudometric obtained when setting $x\sim' g(x)$ instead of $x\sim f(x)$ and moreover the conjecture is equivalent to the claim that $Z/({\sim}\cdot{\sim'})$ is a singleton having $Z$ as its only element.

  [1]: https://mathoverflow.net/a/123753/91341