  Let $I=[0,1)$, $b\in \{2,3,\ldots\}$ and $f:I\to I$ be an injective piecewise
$\frac{1}{b}$-affine map, that is, assume that there exists a partition of $I$
into intervals $I_1,\ldots,I_n$ such that $\vert f(x)-f(y)\vert\le\frac1b \vert
x-y\vert$ for all $x,y\in I_i$ and $1\le i\le n$. In this note, we study the
$\delta$-parameter family of maps $f_{\delta}=R_{\delta}\circ f$, where
$R_\delta:x\mapsto \{x+\delta\}$. More precisely, we show that the set
$\mathcal{N}$ of parameters $\delta$ for which $f_{\delta}$ has only natural
codings with maximal complexity is a non-empty set with Hausdorff
\mbox{dimension $0$}. We also show that for all $\delta\in\mathcal{N}$, the map
$f_{\delta}$ is topologically semiconjugate to a minimal $n$-interval exchange
transformation satisfying Keane's i.d.o.c. condition. The main result turns out
to be a concrete application of the result by Mauduit and Moreira that the set
of numbers having $b$-adic expansion with entropy $0$ has Hausdorff dimension
$0$.
