Critical Hausdorff Measure for Higher-Dimensional Bedford-McMullen Subshifts
Abstract
Let Λ=diag(m1,…,md) be a diagonal expanding integer matrix with m1≥⋯≥ md≥2 and at least two distinct diagonal entries, and let X⊂𝒜ℕ be a subshift with weak specification over 𝒜=∏r=1d{0,1,…,mr−1}. Let K=R(X) be the corresponding self-affine set. Writing γ=dimH K, we prove a sharp critical-measure dichotomy: dimH K=dimB K Longleftrightarrow 0<ℋγ(K)<∞, dimH K<dimB K Longleftrightarrow ℋγ(K)=∞. In the second case, ℋγ is not σ-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 σ-finite, although it may be infinite. We also construct two examples with dimH K<dimB K: in one example the critical Hausdorff measure is positive and finite, whereas in the other it is infinite but σ-finite.
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BibTeX
@misc{qiu2026critical,
title = {{Critical Hausdorff Measure for Higher-Dimensional Bedford-McMullen Subshifts}},
author = {Hua Qiu and Qi Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29642},
eprint = {2609.29642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29642},
}
RIS
TY - UNPB
TI - Critical Hausdorff Measure for Higher-Dimensional Bedford-McMullen Subshifts
AU - Hua Qiu
AU - Qi Wang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29642
UR - https://arxiv.org/abs/2609.29642
AB - Let Λ=diag(m_1,…,m_d) be a diagonal expanding integer matrix with m_1≥⋯≥ m_d≥2 and at least two distinct diagonal entries, and let X⊂𝒜^ℕ be a subshift with weak specification over 𝒜=∏_r=1^d{0,1,…,m_r−1}. Let K=R(X) be the corresponding self-affine set. Writing γ=dim_H K, we prove a sharp critical-measure dichotomy: dim_H K=dim_B K Longleftrightarrow 0<ℋ^γ(K)<∞, dim_H K<dim_B K Longleftrightarrow ℋ^γ(K)=∞. In the second case, ℋ^γ is not σ-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 σ-finite, although it may be infinite. We also construct two examples with dim_H K<dim_B K: in one example the critical Hausdorff measure is positive and finite, whereas in the other it is infinite but σ-finite.
ER -