Hausdorff dimension of the exceptional set of interval piecewise affine contractions

Let $I=[0,1)$, $-1<\lambda <1$ and $f\text{\hspace{0.17em}⁣}:I\to I$ be a piecewise $\lambda$-affine map of the interval $I$, i.e., there exist a partition $0=a_0<a_1<\cdots <a_{k-1}<a_k=1$ of the interval $I$ into $k\ge 2$ subintervals and $b_1,\dots ,b_k\in \mathbb{R}$ such that $f(x)=\lambda x+b_i$ for every $x\in [a_{i-1},a_i)$ and $i=1,\dots ,k$. The exceptional set ${\mathcal{E}}_f$ of $f$ is the set of parameters $\delta \in \mathbb{R}$ such that $R_{\delta }\circ f$ is not asymptotically periodic, where $R_{\delta }\text{\hspace{0.17em}⁣}:I\to I$ is the rotation of angle $\delta$. In this paper we prove that ${\mathcal{E}}_f$ has zero Hausdorff dimension. We derive this result from a more general theorem concerning piecewise Lipschitz contractions on $\mathbb{R}$ that has independent interest.