论文标题
Bi-lipschitz的形式的几何形状
Bi-Lipschitz geometry of quasiconformal trees
论文作者
论文摘要
一棵准形式树是一棵翻倍的公制树，其中每个弧的直径在其端点之间的距离的固定倍数上方界定。我们在两个方向上研究这些树木的几何形状。首先，我们以纯粹的组合方式构建了公制树的目录，并表明每条准文献都是Bi-lipschitz，等同于我们目录中的一棵树。这是受Herron-Meyer和Rohde的准杂志的启发。其次，我们表明，当且仅当它的一组叶子集才能嵌入这种嵌入时，准形式树bi-lipschitz嵌入了欧几里得空间中。特别是，所有的准arcs bi-lipschitz都嵌入了一些欧几里得空间。
A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry of these trees in two directions. First, we construct a catalog of metric trees in a purely combinatorial way, and show that every quasiconformal tree is bi-Lipschitz equivalent to one of the trees in our catalog. This is inspired by results of Herron-Meyer and Rohde for quasi-arcs. Second, we show that a quasiconformal tree bi-Lipschitz embeds in a Euclidean space if and only if its set of leaves admits such an embedding. In particular, all quasi-arcs bi-Lipschitz embed into some Euclidean space.
