具有非空内部的典型自阿芬集合

Pub Date : 2024-07-12 DOI:10.4310/ajm.2023.v27.n5.a1

De-Jun Feng, Zhou Feng

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引用次数: 0

摘要

让 $T_1, \dotsc , T_m$ 是一个 $d \times d$ 可逆实矩阵族，其中 $\rVert T_i \rvert \lt 1/2$ for $1 \leq i \leq m$。我们在这些矩阵上提供了一些充分条件，使得迭代函数系统 $\lbrace T_i x + a_i \rbrace$ 在 $\mathbb{R}^d$ 上产生的自链集对于几乎所有 $(a_1 , \dotsc , a_m) \in \mathbb{R}^{md}$ 都具有非空的内部。

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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 $(a_1 , \dotsc , a_m) \in \mathbb{R}^{md}$.

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