Geometry and Analysis of Metric Spaces via Weighted Partitions
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Beschreibung
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.
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.
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)
Autor*in
Jun Kigami
Themen in »Geometry and Analysis of Metric Spaces via Weighted Partitions«
Ahlfors Regular Conformal Dimension Gromov Hyperbolicity Infinite Graph Metrics Partition
Stimmen zu »Geometry and Analysis of Metric Spaces via Weighted Partitions«
“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)
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