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

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 $0<\mathcal{H}^{\dim_{\text{H}} K}(K)<+\infty$.