Critical Hausdorff Measure for Higher-Dimensional Bedford-McMullen Subshifts

Abstract

Let $\Lambda=\operatorname{diag}(m_1,\ldots,m_d)$ be a diagonal expanding integer matrix with $m_1\ge\cdots\ge m_d\ge2$ and at least two distinct diagonal entries, and let $X\subset\mathcal A^{\mathbb N}$ be a subshift with weak specification over $\mathcal A=\prod_{r=1}^d\{0,1,\ldots,m_r-1\}$. Let $K=R(X)$ be the corresponding self-affine set. Writing $\gamma=\dim_{\mathrm H} K$, we prove a sharp critical-measure dichotomy: \begin{align*} \dim_{\mathrm H} K=\dim_{\mathrm B} K \Longleftrightarrow 0<\mathcal H^\gamma(K)<\infty,\quad \dim_{\mathrm H} K<\dim_{\mathrm B} K \Longleftrightarrow \mathcal H^\gamma(K)=\infty. \end{align*} In the second case, $\mathcal H^\gamma$ is not $\sigma$-finite on $K$. This resolves Peres'1994 conjecture for planar systems and extends to higher dimensions. For sofic subshifts, we prove the existence of the box dimension and show that dimension equality implies that the critical Hausdorff measure is positive and $\sigma$-finite, although it may be infinite. We also construct two examples with $\dim_{\mathrm H} K<\dim_{\mathrm B} K$: in one example the critical Hausdorff measure is positive and finite, whereas in the other it is infinite but $\sigma$-finite.