Titlebook: Geometry and Analysis of Metric Spaces via Weighted Partitions; Jun Kigami Book 2020 The Editor(s) (if applicable) and The Author(s), unde

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Partitions, Weight Functions and Their Hyperbolicity,In this section, we review basic notions and notations on a tree with a reference point.

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Relations of Weight Functions,In this section, we define the notion of bi-Lipschitz equivalence of weight functions. Originally the definition, Definition 3.1.1, only concerns the tree structure . and has nothing to do with a partition of a space.

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Characterization of Ahlfors Regular Conformal Dimension,In this section, we present a sufficient condition for the existence of an adapted metric to a given weight function. The sufficient condition obtained in this section will be used to construct an Ahlfors regular metric later.

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CLAM

persistence

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Contracture

Book 2020 iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the

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Introduction and a Showcase,nit interval [0, 1] shown in Fig. 1.1. Let ..?=?[0, 1] and divide .. in half as . and .. Next, .. and .. are divided in half again and yield .. for each (., .)?∈{0, 1}.. Repeating this procedure, we obtain . satisfying . for any .?≥?0 and ..…..?∈{0, 1}.. In this example, there are two notable proper

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