{"title":"在巴拿赫、希尔伯特或欧几里得空间中嵌入分形","authors":"T. Banakh, M. Nowak, F. Strobin","doi":"10.4171/JFG/94","DOIUrl":null,"url":null,"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 \n$\\bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $\\ell_\\infty$; \n$\\bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; \n$\\bullet$ isometrically equivalent to a fractal in the Hilbert space $\\ell_2$ if $K$ is an ultrametric space. \nWe 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.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2018-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Embedding fractals in Banach, Hilbert or Euclidean spaces\",\"authors\":\"T. Banakh, M. Nowak, F. Strobin\",\"doi\":\"10.4171/JFG/94\",\"DOIUrl\":null,\"url\":null,\"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 \\n$\\\\bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $\\\\ell_\\\\infty$; \\n$\\\\bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; \\n$\\\\bullet$ isometrically equivalent to a fractal in the Hilbert space $\\\\ell_2$ if $K$ is an ultrametric space. \\nWe 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.\",\"PeriodicalId\":48484,\"journal\":{\"name\":\"Journal of Fractal Geometry\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2018-06-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Fractal Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/JFG/94\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Fractal Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/JFG/94","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Embedding fractals in Banach, Hilbert or Euclidean spaces
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.