Peano continua with self regenerating fractals

Abstract

We deal with the question of Masayoshi Hata: is every Peano continuum a topological fractal? A compact space X is a topological fractal if there exists $\mathcal{F}$ a finite family of self-maps on X such that $X={\bigcup }_{f\in \mathcal{F}}f(X)$ and for every open cover $\mathcal{U}$ of X there is $n\in \mathbb{N}$ such that for all maps $f_1,\dots ,f_n\in \mathcal{F}$ the set $f_1\circ \dots \circ f_n(X)$ is contained in some set $U\in \mathcal{U}$.

In the paper we present some idea how to extend a topological fractal and we show that a Peano continuum is a topological fractal if it contains so-called self regenerating fractal with nonempty interior. A Hausdorff topological space A is a self regenerating fractal if for every non-empty open subset U, A is a topological fractal for some family of maps constant on $A\setminus U$.

The notion of self regenerating fractal much better reflects the intuitive perception of self-similarity. We present some classical fractals which are self regenerating.

Introduction

We deal with the topological generalization of IFS-attractors - a compact set invariant under the finite family of contractions. Let us recall that a function f between two metric spaces is called contraction if its Lipschitz constant $\mathrm{Lip}f<1$. By an iterated function system (IFS) on a metric space X we understand a finite family of contractions $X\to X$. A function $X\to X$ is called a self-map on X. For a given family $\mathcal{F}$ of self-maps on X we define the following families of maps:

$${\mathcal{F}}^0=\{i{d}_X\},{\mathcal{F}}^n=\{f_1\circ \dots \circ f_n;f_1,\dots ,f_n\in \mathcal{F}\}.$$

Moreover, let $\mathcal{F}(Y)={\bigcup }_{f\in \mathcal{F}}f(Y)$ for $Y\subset X$.

Let $\mathcal{F}$ be an iterated function system on a metric space X. A compact set $A\subset X$ is an IFS-attractor (deterministic fractal, self-similar set) for $\mathcal{F}$ if $A=\mathcal{F}(A)$. A simple example of such space is the unit interval $[0,1]$ - the IFS-attractor for family $\{\frac{x}{2},\frac{x+1}{2}\}$. Other known examples are the ternary Cantor set, Koch curve, Sierpiński triangle, Sierpiński carpet, etc. It can be also shown that every arc of finite length is an IFS-attractor (see [7]).

A topological fractal is a topological version of IFS-attractor. It is a pair $(X,\mathcal{F})$ where X is a compact space, $\mathcal{F}$ is a finite family of continuous self-maps which has some topological contractive property on X (see Definition 2.1) and $X=\mathcal{F}(X)$. Topological fractals have been studied in various contexts, e.g. among countable spaces [6] or zero-dimensional spaces [2]. Now we are interested in topological fractals in the class of Peano continua. By Peano continuum we understand a metrizable, locally connected continuum or, equivalently (thanks to the Hahn–Mazurkiewicz theorem), a continuous image of the unit interval. Masayoshi Hata proved in [5] that for every topological fractal $(X,\mathcal{F})$ if X is connected, then it is locally connected, so it is a Peano continuum. It is still an open question: is every Peano continuum a topological fractal?

Looking for the conditions when a Peano continuum P becomes a topological fractal, we discover that the existence of a free arc (an open subset of P homeomorphic to the interval) implies this result (see [4]). This leads us to the notion of self regenerating fractal. A Hausdorff topological space X is called a self regenerating fractal if for every nonempty, open subset U there exists $\mathcal{F}$, a family of continuous functions constant outside U such that $(X,\mathcal{F})$ is a topological fractal. Having such a self regenerating fractal as a subset with nonempty interior, guarantees that a Peano continuum is a topological fractal together with some family of maps. This is our main result presented in the chapter 4:

Main theorem

For every Peano continuum X which has $A\subset X$ self regenerating fractal with nonempty interior, X is an underlying space for some topological fractal.

This statement and definition of self regenerating fractal are proposed by Taras Banakh and developed by the author.