A class of continua that are not attractors of any IFS

Topology and its Applications

Strobin

2015

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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

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Detecting topological and Banach fractals among zero-dimensional spaces

Банах

Topology and its Applications

Strobin

2015

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“…In [12], it was proved that such space cannot be a weak IFS-attractor, though it is easy to show that it is a Banach fractal. In fact, we can topologically transform the space P in a way such that every arc L n is a straight line segment whose length is equal to 2 −n .…”

mentioning

confidence: 99%

Counterexamples for IFS-attractors

Chaos, Solitons & Fractals

Fernández–Martínez²

2016

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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.

“…In other words, the unit interval (which is obviously an IFS-attractor) has a compatible metric (taken from the plane) such that it fails being an IFS-attractor. In this direction, Kulczycki and the author [5] gave a general condition on a connected compact space which implies that it has a compatible metric making it a non-IFS-attractor. Finally, [3] showed that the Cantor set has a metric such that it fails to be the attractor of even a countable system of contractions.…”

mentioning

confidence: 99%

Topological classification of scattered IFS-attractors

Topology and its Applications

2013

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Abstract. We study countable compact spaces as potential attractors of iterated function systems. We give an example of a convergent sequence in the real line which is not an IFS-attractor and for each countable ordinal δ we show that a countable compact space of height δ + 1 can be embedded in the real line so that it becomes the attractor of an IFS. On the other hand, we show that a scattered compact metric space of limit height is never an IFS-attractor.