Fibre stability for dominated self-affine sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912149474902016 |
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| author | Anttila, Roope Rutar, Alex |
| author_facet | Anttila, Roope Rutar, Alex |
| contents | Let $K$ be a planar self-affine set. Assuming a weak domination condition on the matrix parts, we prove for all backward Furstenberg directions $V$ that $$\max_{E\in\operatorname{Tan}(K)} \max_{x\in π_{V^\bot}(E)} \operatorname{dim_H} (π_{V^\bot}^{-1}(x)\cap E) = \operatorname{dim_A} K - \operatorname{dim_A} π_{V^\bot}(K).$$ Here, $\operatorname{Tan}(K)$ denotes the space of weak tangents of $K$. Unlike previous work on this topic, we require no separation or irreducibility assumptions. However, if in addition the strong separation condition holds, then there exists a $V\in X_F$ so that $$\max_{x\in π_{V^\bot}(K)} \operatorname{dim_H} (π_{V^\bot}^{-1}(x)\cap K) = \operatorname{dim_A} K - \operatorname{dim_A} π_{V^\bot}(K).$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06579 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fibre stability for dominated self-affine sets Anttila, Roope Rutar, Alex Dynamical Systems Classical Analysis and ODEs Metric Geometry 28A80 (Primary) 37C45, 30L10 (Secondary) Let $K$ be a planar self-affine set. Assuming a weak domination condition on the matrix parts, we prove for all backward Furstenberg directions $V$ that $$\max_{E\in\operatorname{Tan}(K)} \max_{x\in π_{V^\bot}(E)} \operatorname{dim_H} (π_{V^\bot}^{-1}(x)\cap E) = \operatorname{dim_A} K - \operatorname{dim_A} π_{V^\bot}(K).$$ Here, $\operatorname{Tan}(K)$ denotes the space of weak tangents of $K$. Unlike previous work on this topic, we require no separation or irreducibility assumptions. However, if in addition the strong separation condition holds, then there exists a $V\in X_F$ so that $$\max_{x\in π_{V^\bot}(K)} \operatorname{dim_H} (π_{V^\bot}^{-1}(x)\cap K) = \operatorname{dim_A} K - \operatorname{dim_A} π_{V^\bot}(K).$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments. |
| title | Fibre stability for dominated self-affine sets |
| topic | Dynamical Systems Classical Analysis and ODEs Metric Geometry 28A80 (Primary) 37C45, 30L10 (Secondary) |
| url | https://arxiv.org/abs/2412.06579 |