Attractor of Cantor Type with Positive Measure

We construct an iterated function system consisting of strictly increasing contractions f,g:[0,1]→[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g:[0,1]\rightarrow [0,1]$$\end{document} with f([0,1])∩g([0,1])=∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f([0,1])\cap g([0,1])=\emptyset $$\end{document} and such that its attractor has positive Lebesgue measure.


Introduction
Self-similar measures associated with iterated function systems (shortly IFS) have many significant and interesting applications in various areas of science, including mathematics, and in particular, the theory of functional equations (see e.g. [9,7,1]). Studying a functional equation connected with the problem posed in [6,8], we come to the following question.

Preliminaries
We say that a function f : Any L-Lipschitz function f : [a, b] → R with L < 1 is said to be a contraction.
The following fact will be used frequently in the announced construction; its proof is simple, so we omit it. In this paper, whenever we are given a finite collection of contractions defined on the interval [0, 1] into itself, we refer to it as iterated function system.
The following fact is well-known (see [5, From the moreover part of Theorem 2.2 we see that the attractor of the IFS considered in Question 1.1 looks like a set of Cantor type; in fact, (1.1) and the strict monotonicity of f 1 , . . . , f N imply A k+1 A k for every k ∈ N. Let us mention here that not every set of Cantor type can be an attractor of some IFS (see [2]), and moreover, that typical closed sets in [0, 1] can not be attractors of any IFS (see [3]). It is also known that the family of all attractors is dense, path connected and an F σ set in the space of all nonempty and compact subsets of [0, 1] endowed with the Hausdorff metric (see [4]).
Note that the strict monotonicity in Question 1.1 is crucial. Indeed, if we omit the word "strictly", then there is no problem to give an example of an IFS whose attractor is of positive Lebesgue measure. and . By Theorem 2.2 the set [0, 1 3 ] ∪ [ 2 3 , 1] is the attractor of the considered IFS, which clearly is not of Cantor type.

The similitudes case
Now we prove that if the considered IFS consists of similitudes, then the answer to the posed question is positive.
From now on, we denote by L 1 the Lebesgue measure on the real line. Proof. For every n ∈ {1, . . . , N }, let the similitude f n be of the form = 0, and the proof is complete.

Construction of the example
We begin with an explanation of the idea how we construct the announced example. Consider the IFS consisting of the contractions f 0 , g 0 : It is well-known that the attractor of this IFS is the Cantor set (see e.g. [5,Chapter 9]), which has Lebesgue measure zero. The problem is that the gap ( 1 3 , 2 3 ) leads to the gaps ( 1 9 , 2 9 ) and ( 7 9 , 8 9 ). During the process, the gaps propagate and at the end sum up to a set of Lebesgue measure 1. To counteract, we modify the functions f 0 and g 0 . As ( 1 3 , 2 3 ) and its images generate gaps, we make the gaps smaller by mapping ( 1 3 , 2 3 ) to smaller sets than ( 1 9 , 2 9 ) and ( 7 9 , 8 9 ). We continue to modify the functions f 0 and g 0 such that the images of the smaller gaps are even smaller. This way, we obtain two sequences of strictly increasing contractions that converge uniformly to strictly increasing contractions that form our IFS.
4.1. Key sequences. First, we need two sequences (ε k ) k∈N and (w k ) k∈N of parameters that will determine how we modify the functions f 0 and g 0 .
We let Having defined ε l > 0 and w l ∈ R for all 1 ≤ l ≤ k, we let and choose ε k+1 > 0 such that the following conditions are satisfied To see that the sequences (ε k ) k∈N and (w k ) k∈N are well-defined, we only have to show that we really can satisfy (4.6). First, we observe that w1 4 − ε1 2 = 1 4 − 1 12 > 0. Thus we can choose ε 2 . Fix k ∈ N and assume that we have already chosen ε k+1 and w k+1 . Then, using (4.3) and (4.6), we have w k+1 2 > 0, which shows that we can choose ε k+2 .
Condition (4.4) will be used to show that the attractor of the constructed IFS has positive Lebesgue measure. To guarantee that our functions are contractions, we will need condition (4.5). Finally, conditions (4.6) and (4.3) will guarantee that all the intervals where the modifications will take place are non-degenerated but small enough.  0 < w k ≤ 1 2 k−1 and Proof. Conditions (4.7) and (4.8) are clearly true for k = 1.
If k ≥ 2, then using (4.3) and (4.6), we get w k+1 = w k 2 − ε k > 2ε k+1 > 0. Thus the first inequality in (4.7) is proved. To prove the second one and (4.8), it is enough to observe that applying (4.3) we have w k+1 = w k 2 − ε k < w k 2 and proceed by induction on k.

4.2.
Intervals where the modifications take place. We inductively define a sequence of collections of intervals as follows. Put and observe that the only interval in I 1 has length w 1 .
Fix k ∈ N and assume that the collection I k has been defined in such a way that b − a = w k for each [a, b] ∈ I k ; note that w k > 0 by (4.7). Next observe that if [a, b] ∈ I k , then according to (4

Now we put
In this way we have constructed a sequence (I k ) k∈N of collections of intervals. Let us write down some of the sequence's properties in the next lemma.
as its first step.

Attractor.
We now define a set, which turns out to be the attractor of our IFS. For every k ∈ N we let A k = I k and observe that by assertion (iii) of Lemma 4.2 we have (4.9) A k+1 A k . Proof. It is easy to see that 0 ∈ A * , so A * = ∅. From assertion (i) of Lemma 4.2 we conclude that each A k is closed and bounded. Hence A * is compact.

Moreover, assertion (iv) of Lemma 4.2 yields
For showing that A * is nowhere dense, suppose the contrary and choose a point  Proof. By Lemma 4.3 the set A * is Lebesgue measurable. We calculate the measure of the complement of A * . In the course of the computation, we need (4.9), assertion (i) of Lemma 4.2 and (4.4), as well as Consequently, .  1]. Note that f 1 is a strictly increasing contraction with the minimal Lipschitz constant strictly smaller than 1 2 ; here we use that ε 2 ∈ (0, 1 16 ) by (4.4) and apply Lemma 2.1. Moreover, simple calculations (some of them with the use of (4.3)) give  Figure 1.
see Figure 1. Note that the above formula is consistent with the definition of f 1 by (4.3) and simple calculations. In this way we have defined f k on I k . Now we put The next lemma collects essential properties of the just defined sequence (f k ) k∈N .
(iv) The function f k is strictly increasing.
(v) The function f k is a contraction with Lipschitz constant strictly smaller than 1 2 .  Moreover, from Lemma 4.7 and Lemma 4.8 we see that our IFS consists of strictly increasing contractions. If we show that its attractor is the set A * , then the example will be complete. Proof. We first prove that (4.11) for every k ∈ N. Fix k ∈ N. From assertion (viii) of Lemma 4.5 we conclude that A k+1 ⊂ f (A k ) ∪ g(A k ). If we prove that f (A k ) ∪ g(A k ) ⊂ A k+1 , the assertion follows. Making also use of assertion (vi) of Lemma 4.5 and assertion (iv) of Lemma 4.2 we get which proves (4.11). In Lemma 4.3 we have recorded already that A * is nonempty and compact. According to Theorem 2.2 it remains to prove that A * = f (A * ) ∪ g(A * ).
As f and g are strictly increasing as verified in the proof of Lemma 4.7, we have, using (4.9), The proof is complete.