关于使用替换构造分形空间及组合 Loewner 性质
On Constructions of Fractal Spaces Using Replacement and the Combinatorial Loewner Property
AI总结：
本文通过引入迭代图系统（IGS）构造新的自相似分形空间，否定了 Kleiner 关于自相似分形中组合 Loewner 性质蕴含拟对称于 Loewner 空间的猜想，并展示了这些空间在扩散、Sobolev 空间、能量测度与共形维数上的丰富分析行为。
摘要：
组合 Loewner 性质由 Bourdon 和 Kleiner 引入，作为一般分形和双曲群边界的 Loewner 性质的拟对称不变替代。虽然 Loewner 性质有些限制性，但组合 Loewner 性质非常普遍——Bourdon 和 Kleiner 证明了许多熟悉的分形和群边界满足该性质。如果 $X$ 拟对称于一个 Loewner 空间，则它具有组合 Loewner 性质。Kleiner 在 2006 年猜想，对于自相似分形，该命题的逆命题成立——希望这将导致许多奇异 Loewner 空间的存在。我们否定了这一猜想，并给出了第一批自相似、组合 Loewner 但不拟对称于 Loewner 空间的例子。在此过程中，我们引入了一种自相似替换规则，称为迭代图系统（IGS），其灵感来自 Laakso 的工作。这产生了一类新的丰富的分形空间，其中势能及其共形维数的闭式计算成为可能。这些空间在分形分析中关于扩散、Sobolev 空间、能量测度和共形维数表现出丰富的行为类别。这些行为扩展了已知的 Cantor 集、gasket、Vicsek 集以及通常过于困难的类地毯空间的例子。尤其是由该构造产生的 Kleiner 猜想的反例很有趣，因为它们开辟了研究不拟对称于 Loewner 空间的组合 Loewner 空间新领域的可能性。
英文摘要：
The combinatorial Loewner property was introduced by Bourdon and Kleiner as a quasisymmetrically invariant substitute for the Loewner property for general fractals and boundaries of hyperbolic groups. While the Loewner property is somewhat restrictive, the combinatorial Loewner property is very generic -- Bourdon and Kleiner showed that many familiar fractals and group boundaries satisfy it. If $X$ is quasisymmetric to a Loewner space, it has the combinatorial Loewner property. Kleiner conjectured in 2006 that the converse to this holds for self-similar fractals -- the hope being that this would lead to the existence of many exotic Loewner spaces. We disprove this conjecture and give the first examples of spaces which are self-similar, combinatorially Loewner and which are not quasisymmetric to Loewner spaces. In the process we introduce a self-similar replacement rule, called iterated graph systems (IGS), which is inspired by the work of Laakso. This produces a new rich class of fractal spaces, where closed form computations of potentials and their conformal dimensions are possible. These spaces exhibit a rich class of behaviors from analysis on fractals in regards to diffusions, Sobolev spaces, energy measures and conformal dimensions. These behaviors expand on the known examples of Cantor sets, gaskets, Vicsek sets, and the often too difficult carpet-like spaces. Especially the counterexamples to Kleiner's conjecture that arise from this construction are interesting, since they open up the possibility to study the new realm of combinatorially Loewner spaces that are not quasisymmetric to Loewner spaces.
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