标题： 从Lipschitz嵌入到尘埃型自相似集之间的Lipschitz等价 显示英文标题

标题： From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets

摘要： 设$K,F\subset\mathbb{R}^d$为两个具有相同豪斯多夫维数的尘埃型自相似集。我们考虑当$K$到$F$存在利普希茨嵌入时，这是否已经意味着它们的利普希茨等价。 我们的主要结果有三个方面：(1) 如果$K$的 Lipschitz 映像在具有正 Hausdorff 测度的集合中与$F$相交，那么$K$可以 Lipschitz 满射到$F$；(2) 如果$F$进一步为齐次的，那么$K, F$的生成迭代函数系统的比值应为代数相关的，因此，$K$和$F$是 Lipschitz 等价的；(3) 在没有齐次性假设的情况下，Lipschitz 等价可能不成立。 这回答了Balka和Keleti在[Adv. Math. 446 (2024), 109669]中的两个问题。 显示英文摘要

摘要： Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result is threefold: (1) if the Lipschitz image of $K$ intersects $F$ in a set of positive Hausdorff measure, then $K$ admits a Lipschitz surjection onto $F$; (2) if $F$ is in addition homogeneous, then the generating iterated function systems of $K, F$ should have algebraically dependent ratios and consequently, $K$ and $F$ are Lipschitz equivalent; (3) the Lipschitz equivalence can fail without the homogeneity assumption. This answers two questions in Balka and Keleti [Adv. Math. 446 (2024), 109669].