Hausdorff dimension of the exceptional set of interval piecewise affine contractions

Abstract

Let $I=[0,1)$, $-1<\lambda<1$ and $f\colon I\to I$ 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 $k\geq2$ subintervals and $b_1,\ldots, b_k\in\mathbb{R}$ such that $f(x)=\lambda x+ b_i$ for every $x\in[a_{i-1},a_{i})$ and $i=1,\ldots,k$. 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 $R_\delta\colon I\to I$ 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.