Typical self-affine sets with non-empty interior

Let $T_1, \dotsc , T_m$ be a family of $d \times d$ invertible real matrices with $\rVert T_i \rvert \lt 1/2$ for $1 \leq i \leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system $\lbrace T_i x + a_i \rbrace$ on $\mathbb{R}^d$ has non-empty interior for almost all $(a_1 , \dotsc , a_m) \in \mathbb{R}^{md}$.

Keywords

self-affine sets, interior, Sobolev dimension

2010 Mathematics Subject Classification

28A78, 28A80, 37C45, 37C70

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Dedicated to the memory of Professor Ka-Sing Lau

Received 21 September 2022

Accepted 27 July 2023

Published 12 July 2024