The Hausdorff measure and uniform fibre conditions for Bara\'nski carpet

TL;DR

This paper investigates the Hausdorff measure of Barański carpets, establishing a dichotomy for its finiteness and introducing uniform fibre conditions that characterize dimension coincidences and regularity.

Contribution

It introduces four uniform fibre conditions for Barański carpets, linking them to measure finiteness, dimension equality, and regularity, and provides a criterion for the measure dichotomy.

Findings

Dichotomy: Hausdorff measure is either zero or infinite.

Uniform fibre conditions characterize dimension coincidences.

Lower fibre condition implies Ahlfors regularity.

Abstract

For a self-affine carpet $K$ of Bara\'{n}ski, we establish a dichotomy: $ \text{either }\quad 0<\mathcal{H}^{\dim_{\text{H}} K}(K)<+\infty \quad\text{ or } \quad\mathcal{H}^{\dim_{\text{H}} K}(K)=+\infty. $ We introduce four types of uniform fibre condition for $K$: Hausdorff ($\textbf{u.f.H}$), Box ($\textbf{u.f.B}$), Assouad ($\textbf{u.f.A}$), and Lower ($\textbf{u.f.L}$), which are progressively stronger, with $ \textbf{u.f.L} \Longrightarrow \textbf{u.f.A} \Longrightarrow \textbf{u.f.B} \Longrightarrow \textbf{u.f.H}, $ and each implication is strict. The condition $\textbf{u.f.H}$ serves as a criterion for the dichotomy. The remaining three conditions provide an equivalent characterization for the coincidence of any two distinct dimensions. The condition $\textbf{u.f.L}$ is also equivalent to the Ahlfors regularity of $K$. As a corollary, $\dim_{\text{H}} K=\dim_{\text{B}} K$ is sufficient but not necessary for $0<\mathcal{H}^{\dim_{\text{H}} K}(K)<+\infty$.