Authors:
- Describes how a compact metric space may be associated with an infinite graph whose boundary is the original space
- Explores an approach to metrics and measures from an integrated point of view
- Shows a relation between geometry (Ahlfors regular conformal dimension) and analysis (critical index of p-energies)
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2265)
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Table of contents (4 chapters)
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Front Matter
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Back Matter
About this book
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.
Reviews
“The monograph is well-written and concerns a novel idea which has great potential to become a major concept in areas such as fractal geometry and dynamical systems theory. It is written at the level of graduate students and for researchers interested in the aforementioned areas.” (Peter Massopust, zbMATH 1455.28001, 2021)
Authors and Affiliations
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Graduate School of Informatics, Kyoto University, Kyoto, Japan
Jun Kigami
Bibliographic Information
Book Title: Geometry and Analysis of Metric Spaces via Weighted Partitions
Authors: Jun Kigami
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-030-54154-5
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
Softcover ISBN: 978-3-030-54153-8Published: 17 November 2020
eBook ISBN: 978-3-030-54154-5Published: 16 November 2020
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: VIII, 164
Number of Illustrations: 10 b/w illustrations
Topics: Geometry, Analysis, Hyperbolic Geometry, Measure and Integration, Topology