On the structures of generating iterated function systems of Cantor sets
On the structures of generating iterated function systems of Cantor sets
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DOI：
10.1016/j.aim.2009.06.022
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发表时间：
2009-12
影响因子：
1.7
通讯作者：
De-Jun Feng;Yang Wang
中科院分区：
文献类型：
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作者：
De-Jun Feng;Yang Wang
康托集 F 的生成 IFS 是吸引子为 F 的 IFS。对于给定的康托集（例如中三康托集），我们考虑其生成 IFS 的集合。我们检查最小生成 IFS 的存在性，即 F 的每个其他生成 IFS 都是该 IFS 的迭代。我们还研究了开集条件（OSC）下R中康托集F的齐次生成IFS半群的结构。如果 dimHF<1，我们证明该集合的所有生成 IFS 必须具有对数可通约的收缩因子。从这个对数通约性定理，我们推导出在 OSC 下生成 F 的 IFS 的半群的结构定理。我们还研究了几何对半群结构的影响。下面将举几个例子来说明我们研究问题的难度。
A generating IFS of a Cantor set F is an IFS whose attractor is F. For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under the open set condition (OSC). If dimHF<1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study.