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Geometry and Analysis of Metric Spaces via Weighted Partitions 1st ed. 2020 [Minkštas viršelis]

  • Formatas: Paperback / softback, 164 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 10 Illustrations, black and white; VIII, 164 p. 10 illus., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2265
  • Išleidimo metai: 17-Nov-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030541533
  • ISBN-13: 9783030541538
Kitos knygos pagal šią temą:
Geometry and Analysis of Metric Spaces via Weighted Partitions 1st ed. 2020Didesnis paveikslėlis
  • Formatas: Paperback / softback, 164 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 10 Illustrations, black and white; VIII, 164 p. 10 illus., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2265
  • Išleidimo metai: 17-Nov-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030541533
  • ISBN-13: 9783030541538
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030541538.html
Raktažodžiai:

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:

  1. It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic.
  2. 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.
  3. 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.


Recenzijos

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)

1 Introduction and a Showcase
1(16)
1.1 Introduction
1(7)
1.2 Summary of the Main Results; the Case of 2-dim. Sphere
8(9)
2 Partitions, Weight Functions and Their Hyperbolicity
17(38)
2.1 Tree with a Reference Point
17(3)
2.2 Partition
20(8)
2.3 Weight Function and Associated "Visual Pre-metric"
28(5)
2.4 Metrics Adapted to Weight Function
33(8)
2.5 Hyperbolicity of Resolutions and the Existence of Adapted Metrics
41(14)
3 Relations of Weight Functions
55(42)
3.1 Bi-Lipschitz Equivalence
55(8)
3.2 Thickness of Weight Functions
63(3)
3.3 Volume Doubling Property
66(9)
3.4 Example: Subsets of the Square
75(10)
3.5 Gentleness and Exponentiality
85(5)
3.6 Quasisymmetry
90(7)
4 Characterization of Ahlfors Regular Conformal Dimension
97(56)
4.1 Construction of Adapted Metric I
97(8)
4.2 Construction of Ahlfors Regular Metric 1
105(3)
4.3 Basic Framework
108(4)
4.4 Construction of Adapted Metric II
112(4)
4.5 Construction of Ahlfors Regular Metric II
116(7)
4.6 Critical Index of p-Energies and the Ahlfors Regular Conformal Dimension
123(12)
4.7 Relation with p-Spectral Dimensions
135(7)
4.8 Combinatorial Modulus of Curves
142(4)
4.9 Positivity at the Critical Value
146(7)
A Fact from Measure Theory 153(2)
B List of Definitions, Notations and Conditions 155(6)
Bibliography 161