Dimensions of Bedford-McMullen type sets in ℤ2
Abstract
We study forward orbits in ℤ2 generated by the expanding affine maps (x,y)↦(mx+i,ny+j), where m≥ n≥2 are integers and (i,j) ranges over a nonempty digit set Λ⊆{0,…,m−1}×{0,…,n−1}. We obtain explicit formulae for the mass, Beurling, discrete packing, and Assouad dimensions, the discrete Hausdorff dimension and its lower variant, and the lower entropy index. When m=n, all these dimensions equal logm#Λ. When m>n, they can differ, and several depend on the starting point through the sizes of the endpoint columns of Λ. The two Hausdorff dimensions coincide and admit a pressure variational formula. Our proofs use finite symbolic models and approximate squares to relate digit counts to coverings in centered windows, translated windows, and annuli. We also characterize invariant lattice sets and determine their dimensions. Finally, we compare the orbit dimensions with semigroup growth and the dimensions of the compact dual attractor. In particular, the mass and Beurling dimensions need not equal the semigroup growth exponent, in contrast to the corresponding one-dimensional theory.
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BibTeX
@misc{miao2026dimensions,
title = {{Dimensions of Bedford-McMullen type sets in $\mathbb Z^2$}},
author = {Junjie Miao and Minghui Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.35181},
eprint = {2609.35181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35181},
}
RIS
TY - UNPB
TI - Dimensions of Bedford-McMullen type sets in ℤ^2
AU - Junjie Miao
AU - Minghui Xu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.35181
UR - https://arxiv.org/abs/2609.35181
AB - We study forward orbits in ℤ^2 generated by the expanding affine maps (x,y)↦(mx+i,ny+j), where m≥ n≥2 are integers and (i,j) ranges over a nonempty digit set Λ⊆{0,…,m−1}×{0,…,n−1}. We obtain explicit formulae for the mass, Beurling, discrete packing, and Assouad dimensions, the discrete Hausdorff dimension and its lower variant, and the lower entropy index. When m=n, all these dimensions equal log_m#Λ. When m>n, they can differ, and several depend on the starting point through the sizes of the endpoint columns of Λ. The two Hausdorff dimensions coincide and admit a pressure variational formula. Our proofs use finite symbolic models and approximate squares to relate digit counts to coverings in centered windows, translated windows, and annuli. We also characterize invariant lattice sets and determine their dimensions. Finally, we compare the orbit dimensions with semigroup growth and the dimensions of the compact dual attractor. In particular, the mass and Beurling dimensions need not equal the semigroup growth exponent, in contrast to the corresponding one-dimensional theory.
ER -