度量测度空间上的爱因斯坦关系
The Einstein Relation on Metric Measure Spaces
AI总结：
本文在度量测度空间上回顾了联系Hausdorff、局部游走和谱维数的爱因斯坦关系，证明了三种分形维数在双Lipschitz同构下的不变性，并探讨了非Lipschitz映射对维数的扭曲影响。
摘要：
本笔记基于F. Burghart于2018年7月在斯图加特大学完成的、由Freiberg教授指导的硕士论文。我们在配备了合适算子的度量测度空间这一抽象设定下，回顾了联系空间上Hausdorff维数、局部游走维数和谱维数的爱因斯坦关系。与分形几何中的通常定义相比，这需要一些调整。主要结果确立了这三种分形维数概念在mm-空间之间的双Lipschitz连续同构下的不变性，并更一般地解释了爱因斯坦关系背后的解析与随机结构的转移是如何运作的。尽管任何同胚都足以实现这种结构的转移，但非Lipschitz映射会以不同方式扭曲Hausdorff维数和局部游走维数。为了说明这一点，我们考察了Hölder正则变换及其对局部游走维数的影响，并描述了分数布朗运动图上的爱因斯坦关系。最后，我们给出了可能有助于建立爱因斯坦关系一般理论的进一步问题的简短列表。
英文摘要：
This note is based on F. Burghart's master thesis at Stuttgart university from July 2018, supervised by Prof. Freiberg. We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at Hölder regular transformations and how they influence the local walk dimension and describe the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation.
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