Abstract. In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches.

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

We show that the space called shark teeth is a topological IFSattractor, that is for every open cover of X = n i=1 f i (X), its image under every suitable large composition from the family of continuous functions {f 1 , ..., fn} lies in some set from the cover. In particular, there exists a space which is not homeomorphic to any IFS-attractor but is a topological IFSattractor.2010 Mathematics Subject Classification. Primary 28A80; 54D05; 54F50; 54F45.

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.

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

This paper presents a sufficient condition for a continuum in R to be embeddable in R in such a way that its image is not an attractor of any iterated function system. An example of a continuum in R 2 that is not an attractor of any weak iterated function system is also given.
MSC:28A80, 54F15, 37B25, 54H20

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.

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