Piecewise contractions and b-adic expansions

Abstract

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 $0$.