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

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.


Introduction
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].

Problem 1.1. Is each finite-dimensional Peano continuum homeomorphic to an IFS-attractor?
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.

Theorem 1.2. There is a rim-finite plane Peano continuum homeomorphic to no IFS-attractor.
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.

S-dimension of IFS-attractors
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 defined 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 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}.
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.

The space M
Our space M is a partial case of the spaces constructed in [1] and called "shark teeth". Consider the piecewise linear periodic function 1 2 , 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: (1) M is a rim-finite plane Peano continuum; (2) dim(M ) = 1 and S-dim(X) = ∞; (3) M is not homeomorphic to an IFS attractor.
Proof. It is easy to see that X is a rim-finite plane Peano continuum. The rimfiniteness of M implies that dim(M ) = 1. To show that S-dim(M ) = ∞, consider the number sequence m = (2 n k ) ∞ k=1 and observe that the space M is homeomorphic to the "shark teeth" space W m considered in [1]. Taking into account that lim k→∞ 2 n k k α = 0 for any α > 0 and applying Theorem 7.3(6) of [1], we conclude that S-dimM = S-dimW m = ∞. By Theorem 2.1, the space M is not homeomorphic to an IFS-attractor.

Some Open Questions
We shall say that a compact topological space X is a topological IFS-attractor if X = n i=1 f i (X) for some continuous maps f 1 , . . . , f n : X → X such for any open cover U of X there is m ∈ N such that for any functions g 1 , . . . , g m ∈ {f 1 , . . . , f n } the set g 1 •· · ·•g m (X) lies in some set U ∈ U. It is easy to see that each IFS-attractor is a topological IFS-attractor and each connected topological IFS-attractor is metrizable and locally connected.