Embedding fractals in Banach, Hilbert or Euclidean spaces

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

• isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and ${\ell }_{\infty }$;

• bi-Lipschitz equivalent to a fractal in the Banach space $c_0$;

• 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 ${\mathcal{F}}^{\circ k}=\{f_1\circ \cdots \circ f_k:f_1,\dots ,f_k\in \mathcal{F}\}$ is equi-Hölder 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.