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Ergodic × p-invariant measures on 𝕋2 with no dimension dropping projections

  • Amir Algom
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Abstract

Fix an integer p≥ 2 and 0<s<1. We construct an ergodic × p-invariant measure μ on 𝕋2 of dimension s, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass O(ws) to every planar tube of width w. The result is sharp at the endpoint s=1 as shown by Pyörälä, Shmerkin, Suomala and Wu (2025). In contrast, we show that every quasi-Bernoulli Tp-invariant measure with positive dimension admits a dimension-dropping projection.

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Cite

BibTeXRIS
BibTeX
@misc{algom2026ergodic,
  title = {{Ergodic $\times p$-invariant measures on $\mathbbT^2$ with no dimension dropping projections}},
  author = {Amir Algom},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29569},
  eprint = {2608.29569},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29569},
}
RIS
TY  - UNPB
TI  - Ergodic × p-invariant measures on 𝕋^2 with no dimension dropping projections
AU  - Amir Algom
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29569
UR  - https://arxiv.org/abs/2608.29569
AB  - Fix an integer p≥ 2 and 0<s<1. We construct an ergodic × p-invariant measure μ on 𝕋^2 of dimension s, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass O(w^s) to every planar tube of width w. The result is sharp at the endpoint s=1 as shown by Pyörälä, Shmerkin, Suomala and Wu (2025). In contrast, we show that every quasi-Bernoulli T_p-invariant measure with positive dimension admits a dimension-dropping projection.
ER  -