Let $f,g$ be functions from $\Bbb Z_2\to\Bbb Z_2^\times$
$f(x)=2x+1$ and $g(x)=\frac{2x-1}3$
Then I have the notion that $f,g$ are topologically conjugate to each other in the 2-adic metric. I'd like to verify that but I find it offputting that the domains of the functions don't coincide with the range.
Anyway, proceding regardless, I can ask myself "is there a homeomorphism that conjugates one to the other?"
So I guess the homeomorphism would be $x\mapsto\frac{x-2}3$ or $x\mapsto3x+2$
So to my tiny brain, that homeomorphism looks good because it's one to one and onto (because the maps to and from zero fall outside of $\Bbb Z_2^\times$, preserves the 2-adic value $\lvert x\rvert_2=1$ and $d(f(x),f(y))=d(x,y)$. Is that correct?
I'm concerned I don't have any conjugation going on there, just a transformation from one to the other. But how could I, as the codomain is not within the range?
EDIT:
I've moved my thinking along. If I let $h=\frac{x-2}3$ and conjugate $f$ I seem to get $hfh^{-1}=f$ rather than $g$ so I'm definitely doing something wrong here. Note, I have no doubt they are topological conjugates, but I want to derive the homeomorphism.
The fact $hfh^{-1}=f$ suggests $f,h$ commute - so i checked whether $ax+b$ and $cx+d$ commute in general, and I got $hfh^{-1}=ax+(1-a)d+bc$, confirming they DON'T in general commute, but I substituted in this special case and got $ax+(1-a)d+bc=2x+1$. I found this which looks relevant but I seem to be going down a rabbit hole not relevant to the original question.