Where can I learn about explicit homeomorphisms from $X=\Bbb Z[1/6]^+$ to $Y=\Bbb Z[1/2]\cap(0,1]$ in the 2-adic metric?
To be clear, I'm referring to $X$ the products of the positive dyadic and ternary rationals into $Y$ the dyadic rationals in the right-closed unit interval.
Moreover, if possible, a continuous transform from one to the other (NOT necessarily with one as a subspace of the other) whose infinite limit is the homeomorphism. But that would be a luxury and not required to be considered an answer, or a pointer to a suitable source.
I'm a lot more sure now than I was when I wrote the question that I've treated the endpoints $\{0,1\}$ correctly but there's a little scope for error on my part there so modulo a single point is still an answer.
I haven't investigated whether I need group and monoid operations preserved but I assume it unlikely as one is torsion and the other not.
What I have tried:
I can make a cover of the dyadic rationals but find it much more tricky to get a cover of the larger set $X$ straight in my mind. I have an inkling what the homeomorphism I have in mind looks like - it writes a 2-adic number from the right in $\Bbb Q_2$ ending with $-1/3$ immediately to the left of the decimal point, and a binary sequence to the right, and then you need to transform that as a whole, either by adding a third or the function $3x+1$ or even $3x+2^{\nu_2(x)}$ to it to get the dyadic rationals in the interval. It's still a work in progress but seeing other, similar examples would help me.
Here's the proposed homeomorphism $X\to Y$:
$T(x)=1+3\sum_{n=0}^\infty (c^n(x)\pmod 2)$
Where $c^n(x)$ is the $n^{th}$ composition of $c(x)=3x+2^{\nu_2(x)}$ and $2^{\nu_2(x)}$ is the highest power of $2$ that divides $x$.
A couple of things about that which look like they may be significant...
- $(T-1)/3$ is definitely a 2-adic isometry and therefore $T:\text{preimage}(Y)\to Y$ is definitely a homeomorphism
- What I lack is to show that $X$ is the preimage of $Y$.
- The interval of preimage of $Y$ coincides with the interval of $Y$.
- I'm satisfied the preimage of $Y$ has a certain degree of density in $X$
- I can prove that all the integers which are contained in the preimage of $X$ are final when we order the numbers by truncating $\frac13(Y-1)$ if odd, and $2Y$ if even