Embedding fractals in Banach, Hilbert or Euclidean spaces

TL;DR

This paper demonstrates that metric fractals can be embedded into Banach, Hilbert, and Euclidean spaces with various equivalences, and shows that certain finite-dimensional spaces containing a Cantor set can be realized as fractals in Euclidean spaces.

Contribution

It establishes new embedding results for metric fractals into classical Banach, Hilbert, and Euclidean spaces, including ultrametric cases and fractals with the doubling property.

Findings

Metric fractals are isometrically equivalent to fractals in $C[0,1]$ and $ ext{ell}_

Metric fractals are bi-Lipschitz equivalent to fractals in $c_0$

Fractals with the doubling property can be embedded into Euclidean space $$

Abstract

By a metric fractal we understand a compact metric space $K$ endowed with a finite family $\mathcal F$ of contracting self-maps of $K$ such that $K=\bigcup_{f\in\mathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each $f\in\mathcal F$ extends to a contracting self-map of $X$, then we say that $(K,\mathcal F)$ is a fractal in $X$. We prove that each metric fractal $(K,\mathcal F)$ is $\bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $\ell_\infty$; $\bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; $\bullet$ isometrically equivalent to a fractal in the Hilbert space $\ell_2$ if $K$ is an ultrametric space. We prove that for a metric fractal $(K,\mathcal F)$ with the doubling property there exists $k\in\mathbb N$ such that the metric fractal $(K,\mathcal F^{\circ k})$ endowed with the fractal structure $\mathcal F^{\circ k}=\{f_1\circ\dots\circ f_k:f_1,\dots,f_k\in\mathcal F\}$ is equi-H\"older equivalent to a fractal in a Euclidean space $\mathbb R^d$. This result is used to prove our main result saying that each finite-dimensional compact metrizable space $K$ containing an open uncountable zero-dimensional space $Z$ is homeomorphic to a fractal in a Euclidean space $\mathbb R^d$. For $Z$, being a copy of the Cantor set, this embedding result was proved by Duvall and Husch in 1992.