Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Detecting topological and Banach fractals among zero-dimensional spaces   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.

28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs