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arXiv · 2412.06579

Fibre stability for dominated self-affine sets

Roope Anttila·Alex Rutar
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Abstract

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 $$\max_{E\in\operatorname{Tan}(K)} \max_{x\in \pi_{V^\bot}(E)} \operatorname{dim_H} (\pi_{V^\bot}^{-1}(x)\cap E) = \operatorname{dim_A} K - \operatorname{dim_A} \pi_{V^\bot}(K).$$ 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 $$\max_{x\in \pi_{V^\bot}(K)} \operatorname{dim_H} (\pi_{V^\bot}^{-1}(x)\cap K) = \operatorname{dim_A} K - \operatorname{dim_A} \pi_{V^\bot}(K).$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments.

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Roope Anttila, Alex Rutar. 2024-12-09. Fibre stability for dominated self-affine sets. https://arxiv.org/abs/2412.06579

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