Consider an isometric homeomorphism $T$ on the 2-adic integers $\Bbb Z_2$, which fixes the numbers $-1,0$ and exchanges $(-1/3,1)$, and obeys $2T(x)=T(2x)$.
I'm aware that, in general, p-adic isometries continue to enjoy a lot of freedom even after many points are determined, but I can also see scope for these rules to be quite restrictive.
Is it possible that these rules give rise to a single, canonical isometry, particularly on the natural number $\Bbb N$ subset of $\Bbb Z_2$?