The Hausdorff measure and uniform fibre conditions for Barański carpet

Abstract

For a self-affine carpet $K$ of Barański, we establish a dichotomy: $\text{either }0<{\mathcal{H}}^{{\dim }_{\text{H}}K}(K)<+\infty \text{ or }{\mathcal{H}}^{{\dim }_{\text{H}}K}(K)=+\infty .$ We introduce four types of uniform fibre condition for $K$: Hausdorff ($\text{u.f.H}$), Box ($\text{u.f.B}$), Assouad ($\text{u.f.A}$), and Lower ($\text{u.f.L}$), which are progressively stronger, with $\text{u.f.L}⟹\text{u.f.A}⟹\text{u.f.B}⟹\text{u.f.H},$ and each implication is strict. The condition $\text{u.f.H}$ 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 $\text{u.f.L}$ 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 $0<{\mathcal{H}}^{{\dim }_{\text{H}}K}(K)<+\infty$.

40 pages, 2 figures