Mapping Analytic sets onto cubes by little Lipschitz functions

TL;DR

The paper proves that metric spaces with packing dimension greater than n can be mapped onto n-dimensional cubes using little Lipschitz functions, revealing new connections between dimension theory and Lipschitz mappings.

Contribution

It establishes that analytic metric spaces with high packing dimension admit little Lipschitz maps onto cubes, and introduces new embedding and extension results for such functions.

Findings

Spaces with packing dimension > n map onto n-cubes via little Lipschitz functions

Any analytic metric space contains compact subsets that embed into ultrametric spaces with nearly full packing dimension

Little Lipschitz functions on closed sets can be extended to the whole space

Abstract

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$ \operatorname{lip}(f(x)=\liminf_{r\to0}\frac{\operatorname{diam} f(B(x,r))}{r} $$ is finite for every $x\in X$. We prove that if a compact (or, more generally, analytic) metric space has packing dimension greater than $n$, then $X$ can be mapped onto an $n$-dimensional cube by a little Lipschitz function. The result requires two facts that are interesing in their own right. First, an analytic metric space $X$ contains, for any $\varepsilon>0$, a compact subset $S$ that embeds into an ultrametric space by a Lipschitz map, and $\dim_P S\geq\dim_P X-\varepsilon$. Second, a little Lipschitz function on a closed subset admits a little Lipschitz extension.