Limit formulas for metric measure invariants and phase transition property

2014/02/27 by Ryunosuke Ozawa, Ozawa, Ryunosuke, Takashi Shioya +1 · 2 citations
Mathematics · #53C23 #Advanced Topology and Set Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1402.6831

openalex publication_date 2014/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of our limit formulas as follows. We have a criterion of the phase transition property for a sequence of metric measure spaces or pyramids, and find some examples of symmetric spaces of noncompact type with the phase transition property. We also give a simple proof of a theorem by Funano-Shioya on the limit of an N-Lévy family.

Cited by

• Poincaré Beta Balls: Radial Laws, Shell Transforms, and Phase Diagram
• Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions

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• Convergence of metric transformed spaces
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