A Non-injective Assouad-Type Theorem with Sharp Dimension

Lipschitz light maps, defined by Cheeger and Kleiner, are a class of non-injective “foldings” between metric spaces that preserve some geometric information. We prove that if a metric space ( X ,  d ) has Nagata dimension n , then its “snowflakes” ( X , d ϵ ) admit Lipschitz light maps to R n for al...

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Bibliographic Details
Published in:The Journal of geometric analysis 2024-02, Vol.34 (2), Article 45
Main Author: David, Guy C.
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Metric space
Theorems
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Summary:Lipschitz light maps, defined by Cheeger and Kleiner, are a class of non-injective “foldings” between metric spaces that preserve some geometric information. We prove that if a metric space ( X ,  d ) has Nagata dimension n , then its “snowflakes” ( X , d ϵ ) admit Lipschitz light maps to R n for all 0 < ϵ < 1 . This can be seen as an analog of a well-known theorem of Assouad. We also provide an application to a new variant of conformal dimension.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-023-01503-7