Hausdorff dimension of the exceptional set of interval piecewise affine contractions

TL;DR

This paper proves that the set of parameters for which a certain class of interval piecewise affine contractions, when composed with rotations, are not asymptotically periodic, has zero Hausdorff dimension.

Contribution

It establishes that the exceptional set for these contractions has zero Hausdorff dimension, extending to a more general class of piecewise Lipschitz contractions.

Findings

The exceptional set $\\mathcal{E}_f$ has zero Hausdorff dimension.

The result applies to a broad class of piecewise Lipschitz contractions.

The proof involves a general theorem on Lipschitz 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.