Let $f_{a,b}(x)=ax-b\cdot2^{\nu_2(x)}$ be a function on the field of 2-adic numbers $\Bbb Q_2$ parametrized by the pair of 2-adic units $a,b\in\Bbb Z_2^\times$
Then quantifying over $a,b\in\Bbb Z_2^\times$ all $f_{a,b}(x)$ are topologically conjugate to each other over $\Bbb Q_2$, and are in fact topologically conjugate to the tent map with parameter $\mu=2$ and the logistic map with parameter $r=4$.
Let $T_{a,b}(x)$ be the unique function that topologically conjugates $f_{a,b}(x)$ to $f_{1,1}(x)$
Then $T_{1,1}(x)$ is the identity of all such $T$ under composition - it composes with any given $T_{a,b}(x)$ to leave it unchanged. Let $(T,\circ)$ be the group of all such $T$ under composition.
Now restrict $a,b$ to 2-adic units which are eventually periodic. I believe that restriction forms a subgroup $S$ although I do not yet have a proof that it is closed under composition.
The action of $S$ is what I refer to in the question title as "rational affine topological conjugation of two-adics."
I believe that the action of $S$ is closed to periodic 2-adic numbers, i.e. for all eventually periodic 2-adic numbers $x$, the action of $S$ on $x$ generates only 2-adic numbers which are themselves also eventually periodic.
I'm sure this must be eminently provable. Can you prove it, or provide a counterexample?
A proof of this would prove and significantly generalise Lagarias' periodicity conjecture.