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arXiv 1504.07396math.DS
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关于自仿射分形的凸包

On the Convex Hulls of Self-Affine Fractals

Ibrahim Kirat, Ilker Kocyigit

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摘要：

假设实$n\times n$矩阵的集合${\mathcal{T}}= \{T_1, T_2,...,T_q \} $具有小于$1$的联合谱半径。则对任意的数集$ D= \{d_1, \cdots, d_q\} \subset {\Bbb R}^n$，存在唯一的满足$ F = \bigcup _{j =1}^q T_j(F + d_j)$的非空紧集$F=F({\mathcal{T}},D)$，称之为自仿射分形.我们认为现有的标准的凸包$F$是一个多面体，这是由于Kirat和Kocyigit。在本文中，我们加强了对$T_1=T_2=\cdots =T_q $情形的判别准则。更具体地说，我们给出了一个上界的步骤所需的决定是否凸包的$F$是一个多面体或没有。这改进了我们关于这个主题的早期结果。

英文摘要：

Suppose that the set ${\mathcal{T}}= \{T_1, T_2,...,T_q \} $ of real $n\times n$ matrices has joint spectral radius less than $1$. Then for any digit set $ D= \{d_1, \cdots, d_q\} \subset {\Bbb R}^n$, there exists a unique nonempty compact set $F=F({\mathcal{T}},D)$ satisfying $ F = \bigcup _{j =1}^q T_j(F + d_j)$, which is called a self-affine fractal. We consider an existing criterion for the convex hull of $F$ to be a polytope, which is due to Kirat and Kocyigit. In this note, we strengthen our criterion for the case $T_1=T_2=\cdots =T_q $. More specifically, we give an upper bound for the number of steps needed for deciding whether the convex hull of $F$ is a polytope or not. This improves our earlier result on the topic.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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