Dimension of affine iterated function systems on ℝd
Abstract
Let {fi(x)=Mix+ai}i∈ I be an affine iterated function system on ℝd with attractor Λ, and let μ be an associated self-affine measure. Assume that the Mi generate a Zariski dense subgroup of GLd(ℝ). Under the strong open set condition, we show that dimHΛ equals the affinity dimension and that dimHμ equals the Lyapunov dimension. The above results are consequences of the following projection theorem, which requires no separation condition. For every 1≤ n≤ d and for Furstenberg measure a.e. (d−n)-dimensional subspace V, dimH (PV)*μ=min{n, dimHμ}. This projection theorem in fact holds under hypotheses weaker than Zariski density: simplicity of the Lyapunov spectrum and a mild transversality condition (the Oseledets non-concentration property). The source of dimension increment in our proof is the positive dimension of the fiberwise conditional measures, exploited through Wu's one-dimensional entropy increase theorem. A key ingredient in reducing the higher-dimensional problem to this setting is a new structural result: a slab-filling property of the measure.
Related
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- Upper box dimension of inhomogeneous attractors of C1 iterated function systems
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BibTeX
@misc{li2026dimension,
title = {{Dimension of affine iterated function systems on $\mathbbR^d$}},
author = {Jialun Li and Wenyu Pan and Zhiyuan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15485},
eprint = {2610.15485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15485},
}
RIS
TY - UNPB
TI - Dimension of affine iterated function systems on ℝ^d
AU - Jialun Li
AU - Wenyu Pan
AU - Zhiyuan Zhang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15485
UR - https://arxiv.org/abs/2610.15485
AB - Let {f_i(x)=M_ix+a_i}_i∈ I be an affine iterated function system on ℝ^d with attractor Λ, and let μ be an associated self-affine measure. Assume that the M_i generate a Zariski dense subgroup of GL_d(ℝ). Under the strong open set condition, we show that dim_HΛ equals the affinity dimension and that dim_Hμ equals the Lyapunov dimension. The above results are consequences of the following projection theorem, which requires no separation condition. For every 1≤ n≤ d and for Furstenberg measure a.e. (d−n)-dimensional subspace V, dim_H (P^V)_*μ=min{n, dim_Hμ}. This projection theorem in fact holds under hypotheses weaker than Zariski density: simplicity of the Lyapunov spectrum and a mild transversality condition (the Oseledets non-concentration property). The source of dimension increment in our proof is the positive dimension of the fiberwise conditional measures, exploited through Wu's one-dimensional entropy increase theorem. A key ingredient in reducing the higher-dimensional problem to this setting is a new structural result: a slab-filling property of the measure.
ER -