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.

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 two facts:

• $g(x)=x+\frac132^{\nu_2(x)}$ 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.