Recently there are several works devoted to the study of self-similar subsets of a given self-similar set, which turns out to be a difficult problem. Let [Formula: see text] be an integer and let [Formula: see text]. Let [Formula: see text] be the uniform Cantor set defined by the following set equation: [Formula: see text] We show that for any [Formula: see text], [Formula: see text] and [Formula: see text] essentially have the same self-similar subsets. Precisely, [Formula: see text] is a self-similar subset of [Formula: see text] if and only if [Formula: see text] is a self-similar subset of [Formula: see text], where [Formula: see text] (similarly [Formula: see text]) is the coding map from the symbolic space [Formula: see text] to [Formula: see text].

中文翻译：

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关于具有小HAUSDORFF维数的均匀康托集的自相似子集的评述

最近有几项工作致力于研究给定自相似集的自相似子集，结果证明这是一个难题。令 [Formula: see text] 为整数并令 [Formula: see text]。令 [Formula: see text] 是由以下集合方程定义的统一 Cantor 集： [Formula: see text] 我们证明对于任何 [Formula: see text]、[Formula: see text] 和 [Formula: see text]本质上具有相同的自相似子集。准确地说，[Formula: see text] 是 [Formula: see text] 的自相似子集当且仅当 [Formula: see text] 是 [Formula: see text] 的自相似子集，其中 [Formula: see text] text]（类似 [Formula: see text]）是从符号空间 [Formula: see text] 到 [Formula: see text] 的编码映射。