A non-injective Assouad-type theorem with sharp dimension

TL;DR

This paper proves that metric spaces with Nagata dimension n can be mapped via Lipschitz light maps from their snowflake versions to Euclidean space, extending Assouad's theorem and impacting conformal dimension theory.

Contribution

It establishes a non-injective Assouad-type theorem for snowflake metrics with sharp dimension bounds and introduces applications to conformal dimension.

Findings

Lipschitz light maps exist from snowflakes of Nagata dimension n to f^n

The theorem extends Assouad's classical result to non-injective maps

Applications to new variants of conformal dimension

Abstract

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" $(X,d^\epsilon)$ admit Lipschitz light maps to $\mathbb{R}^n$ for all $0<\epsilon<1$. 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.