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Geometry and analysis of metric spac...

Kigami, Jun.

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Geometry and analysis of metric spaces via weighted partitions

Record Type:	Electronic resources : Monograph/item
Title/Author:	Geometry and analysis of metric spaces via weighted partitions/ by Jun Kigami.
Author:	Kigami, Jun.
Published:	Cham :Springer International Publishing : : 2020.,
Description:	viii, 164 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Metric spaces. -
Online resource:	https://doi.org/10.1007/978-3-030-54154-5
ISBN:	9783030541545

Geometry and analysis of metric spaces via weighted partitions
Kigami, Jun.

Geometry and analysis of metric spaces via weighted partitions[electronic resource] /by Jun Kigami. - Cham :Springer International Publishing :2020. - viii, 164 p. :ill., digital ;24 cm. - Lecture notes in mathematics,v.22650075-8434 ;. - Lecture notes in mathematics ;v.2265..

The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an 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 text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.

ISBN: 9783030541545

Standard No.: 10.1007/978-3-030-54154-5doiSubjects--Topical Terms:

546825
Metric spaces.

LC Class. No.: QA611.28 / .K54 2020

Dewey Class. No.: 514.325

Geometry and analysis of metric spaces via weighted partitions
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520 $a The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an 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 text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.
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