A 1-dimensional Peano continuum which is not an IFS attractor

In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS, herein). An attractor is a compact set which remains invariant for such a family. Thus, we consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which we will refer to as Banach and topological fractals, respectively. We present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.

show abstract

“…In [3], it was proved that the shark teeth constructed in the plane R 2 through the nondecreasing sequence n k = log 2 log 2 (k + 1) , k ∈ N,…”

Section: From (B) Both Sides Of This Inequality Become Positive Somentioning

confidence: 99%

Counterexamples for IFS-attractors

Nowak

Fernández–Martínez²

2016

Chaos, Solitons & Fractals

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following example constructed in [1] (see also [19]) indicates that this problem is not trivial even in the realm of compact metrizable spaces (cf. also examples from [13]), and shows that Theorem 6.8 cannot be strengthened by making F Banach contracting.…”

Section: By Theorem 22 the Conditionsmentioning

confidence: 99%

Contractive function systems, their attractors and metrization

Банах¹,

Kubiś²,

Novosad³

et al. 2015

TMNA

Self Cite

View full text Add to dashboard Cite

show abstract

“…Such a problem was considered for example in [2], [4], [5], [6], [10], [12], [13], [15], [16], [17]. In particular, M. Nowak [15] proved that a compact countable space X is homeomorphic to an IFS-attractor if and only if the scattered height (X) of X is a successor ordinal.…”

Section: Introductionmentioning

confidence: 99%

“…Following [2], we define a Hausdorff topological space X to be a topological fractal if X = f ∈F f (X) for a finite system of continuous self-maps of X, which is topologically contracting in the sense that for every open cover U of X, there is n ∈ N such that for any maps f 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Detecting topological and Banach fractals among zero-dimensional spaces

Банах

Nowak

Strobin

2015

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

A topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\mathcal F}f(X)$ for a finite system $\mathcal F$ of continuous self-maps of $X$, which is topologically contracting in the sense that for every open cover $\mathcal U$ of $X$ there is a number $n\in\mathbb N$ such that for any functions $f_1,\dots,f_n\in \mathcal F$, the set $f_1\circ\dots\circ f_n(X)$ is contained in some set $U\in\mathcal U$. If, in addition, all functions $f\in\mathcal F$ have Lipschitz constant $<1$ with respect to some metric generating the topology of $X$, then the space $X$ is called a Banach fractal. It is known that each topological fractal is compact and metrizable. We prove that a zero-dimensional compact metrizable space $X$ is a topological fractal if and only if $X$ is a Banach fractal if and only if $X$ is either uncountable or $X$ is countable and its scattered height $\hbar(X)$ is a successor ordinal. For countable compact spaces this classification was recently proved by M.Nowak.Comment: 7 page

show abstract

A 1-dimensional Peano continuum which is not an IFS attractor

Abstract: Answering an old question of M.Hata, we construct an example of a 1-dimensional Peano continuum which is not homeomorphic to an attractor of IFS.Comment: 4 pages, 2 figure

Cited by 16 publications

References 12 publications

Counterexamples for IFS-attractors

Counterexamples for IFS-attractors

Contractive function systems, their attractors and metrization

Detecting topological and Banach fractals among zero-dimensional spaces

Contact Info

Product

Resources

About