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 $\bullet$ 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.