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

A compact metric space X is called an IFS-attractor if X = n i=1 f i (X) for some contracting self-maps f 1 , . . . , f n : X → X. In this case the family {f 1 , . . . , f n } is called an iterated function system (briefly, an IFS), see [2]. We recall that a map f : X → X is contracting if its Lipschitz constant

is less than 1. Attractors of IFS appear naturally in the Theory of Fractals, see [2], [3]. Topological properties of IFS-attractors were studied by M.Hata in [4]. In particular, he observed that each connected IFS-attractor X is locally connected. The reason is that X has property S. We recall [6, 8.2] that a metric space X has property S if for every ε > 0 the space X can be covered by finite number of connected subsets of diameter < ε. It is well-known [6, 8.4] that a connected compact metric space X is locally connected if and only if it has property S if and only if X is a Peano continuum (which means that X is the continuous image of the interval [0, 1]). Therefore, a compact space X is not homeomorphic to an IFS-attractor whenever X is connected but not locally connected. Now it is natural to ask if there is a Peano continuum homeomorphic to no IFS-attractor. An easy answer is “Yes” as every IFS-attractor has finite topological dimension, see [3]. Consequently, no infinitedimensional compact topological space is homeomorphic to an IFS-attractor. In such a way we arrive to the following question posed by M. Hata in [4].

In this paper we shall give a negative answer to this question. Our counterexample is a rim-finite plane Peano continuum. A topological space X is called rim-finite if it has a base of the topology consisting of open sets with finite boundaries. It follows that each compact rim-finite space X has dimension dim(X) ≤ 1.

It should be mentioned that an example of a Peano continuum K ⊂ R 2 , which is not isometric to an IFS-attractor was constructed by M.Kwieciński in [5]. However the continuum of Kwieciński is homeomorphic to an IFS-attractor, so it does not give an answer to Problem 1.1.

In order to prove Theorem 1.2 we shall observe that each connected IFS-attractor has finite S-dimension. This dimension was introduced and studied in [1].

The S-dimension S-Dim(X) is for each metric space X with property S. For each ε > 0 denote by S ε (X) the smallest number of connected subsets of diameter < ε that cover the space X and let S-Dim(X) = lim ε→+0 -ln S ε (X) ln ε .

For each Peano continuum X we can also consider a topological invariant S-dim(X) = inf{S-Dim(X, d) : d is a metric generating the topology of X}.

By [1, 5.1], S-dim(X) ≥ dim(X), where dim(X) stands for the covering topological dimension of X.

Theorem 2.1. Assume that a connected compact metric space X is an attractor

Then X has finite S-dimensions S-dim(X) ≤ S-Dim(X) ≤ -ln(n) ln(λ) .

Proof. The inequality S-dim(X) ≤ S-Dim(X) follows from the definition of the S-dimension S-dim(X). The inequality S-Dim(X) ≤ -ln(n) ln(λ) will follow as soon as for every δ > 0 we find ε 0 > 0 such that for every ε ∈ (0, ε 0 ] we get

Let D = diam(X) be the diameter of the metric space X. Since

We claim that the number ε 0 = λ k0-1 D has the required property. Indeed, given any ε ∈ (0, ε 0 ] we can find k ≥ k 0 with λ k D < ε ≤ λ k-1 D and observe that

In the next section we shall construct an example of a rim-finite plane Peano continuum M with infinite S-dimension S-dim(M ). Theorem 2.1 implies that the space M is not homeomorphic to an IFS-attractor and this proves Theorem 1.2.

Our M is a partial case of the spaces constructed in [1] and called “shark teeth”. Consider the piecewise linear periodic function

, n] for some n ∈ Z, whose graph looks as follows:

For every n ∈ N consider the function

which is a homothetic copy of the function ϕ(t).

Consider the non-decreasing sequence

where ⌊x⌋ is the integer part of x. Our example is the continuum

in the plane R 2 , which looks as follows:

This content is AI-processed based on open access ArXiv data.