Integer-base maps and b-adic circle intervals

On the circle of The circle, rotations and the doubling map, for an integer $b\ge 2$ put $D_b(x)=\{bx\}$. For every $n\ge 0$ its level-n b-adic intervals are

$I_{n,k}=[k\mathrm{/}{b}^n,(k+1)\mathrm{/}{b}^n),0\le k<b^n\mathrm{.}$

These intervals partition $[0,1)$; at level zero the only interval is the whole circle. On the branch $I_{1,j}$ the map is $D_b(x)=bx-j$. Iterating fractional-part arithmetic gives $D_b^n(x)=\{b^{n}x\}$, and on $I_{n,k}$ this is $b^{n}x-k$. Each branch maps bijectively onto $[0,1)$ with inverse $y\mapsto (y+k)\mathrm{/}{b}^n$. Half-open endpoints ensure that every x belongs to exactly one branch, including x=0. Multiplication by the integer b respects congruence modulo one, and the distance formula gives $d(D_{b}x,D_{b}y)\le bd(x,y)$. Thus these are continuous circle maps. The previously defined doubling map is exactly $D=D_2$. None of these algebraic or metric facts uses a measure or a choice axiom.