CONNECTNESS OF ATTRACTORS OF CERTAIN FAMILY OF IFSS
FILIP STROBIN AND JAROSLAW SWACZYNA
Abstract. Let X be a Banach space and f, g : X → X be two contractions. We investigate the size
of the set
Wf,g = {w ∈ X : the attractor of IFS Sw = (f, g + w) is not connected}.
The motivation for our research comes from a recent paper of Mihail and Miculescu, where it was
shown that Wf,g is open and dense, provided f, g are linear bounded operators with k f k, k g k< 1
and such that f is finite–dimensional.
1. Introduction
Let X be a metric space, f1 , ..., fn : X → X be contiuous mappings. Then the system S =
(f1 , ..., fn ) generates a natural mapping FS : K(X) → K(X):
n
[
FS (D) :=
fk (D)
k=1
(K(X) is the space of all nonempty and compact subsets of X, considered as a metric space with
the Hausdorff metric H). It turns out that if all fi are Banach contractions, then so is FS . Hence it
satisfies the thesis of the Banach fixed point theorem, which means that there is a set AS ∈ K(X)
such that FS (AS ) = AS ), and for any H ∈ K(X), the sequence of iterates FSn (H) converges (in
Hausdorff metric) to AS . In fact, we have even more.
Let g : X → X be a mapping. If for some nondecreasing, upper semicontinuous function ϕ : [0, ∞) →
R with ϕ(t) < t for t > 0, we have
(1)
∀x,y∈X d(g(x), g(y)) ≤ ϕ(d(x, y)),
then we say that f is a ϕ-contraction.
The following facts are known. The proof of the first one can be found in [], and of the second one
in [].
Proposition 1.1. Let X be a complete metric space. If g : X → X is a ϕ-contraction, then it
satisfies the thesis of the Banach fixed point theorem.
1991 Mathematics Subject Classification. Primary: 28A80, Secondary: 54E52.
Key words and phrases. fractals, IFSs, fixed points, Baire category, porosity.
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FILIP STROBIN AND JAROSLAW SWACZYNA
Proposition 1.2. Let X be a metric space. If f1 , ..., fn are ϕ-contractions (with the same function
ϕ) and S = (f1 , ..., fn ). Then FS is also ϕ-contraction (with the function ϕ).
The above show that for a complete metric space X and a system of ϕ-contractions of X S =
(f1 , ..., fn ), the function FS satisfies the thesis of the Banach fixed point theorem. This leads to the
following definition:
Definition 1.3. Let X be a complete metric space and f1 , ..., fn be ϕ-contractions of X. Then we
call S = (f1 , ..., fn ) the iterated function system (IFS is short).
If S is an IFS, then the fixed point of FS is called attractor or Hutchinson–Barnsley fractal, and
denoted by AS .
Note that IFSs in such a general form were considered in literature (for exapmle in [Ma]), but
originally IFSs were considered as systems of Banach contractions ([H] and [B]).
Now we present some notions of porosity. Let X be a Banach space and M ⊂ X.
We say that M is c-porous, if its convex hull conv(M ) is nowhere dense.
We say that M is σ-c-porous, if M is a countable union of c-porous sets.
C-porosity was defined and investigated in [S]. It turns out that it is one of the most restrictive
notions of porosity – σ-c-porous sets can be considered as very small. In particular, they are meager,
but the converse is not true (for more information on porosity we refer the reader to survey papers
[Z1] and [Z2]). The following are trivial:
Proposition 1.4. Let X be infinite-dimensional Banach space and M ⊂ X. Then
(i) If M is compact, then it is c-porous;
(ii) If M is a finite dimensional subspace of X, then it is c-porous.
2. Results
Let X be a Banach space and f, g : X → X be ϕ-contractions. If w ∈ X, then we denote the IFS
(f, g + w) by Sw (clearly, g + w is a ϕ-contraction). Define
Wf,g := {w ∈ X : ASw is not connect}.
At first we show that Wf,g is open.
Theorem 2.1. Let X be a Banach space and f, g : X → X be ϕ-contractions. Then Wf,g is open.
We procede the proof with few lemmas. The first one is a mathematical folklore, so we skip the
proof.
CONNECTNESS OF ATTRACTORS OF CERTAIN FAMILY OF IFSS
3
Lemma 2.2. If X is a metric space, then the set U = {F ∈ K(X) : F is connect} is open.
Lemma 2.3. Let X be a metric space and let (Hn ), (An ) ⊂ K(X) be two sequences and H, A ∈ K(X).
If Hn → H, An → A, and for every n ∈ N, Hn ⊂ An , then A ⊂ H
Proof. For F ⊂ X and > 0, we set F () =
S
x∈F
B(x, ) (B(x, ) denotes an open ball).
For any > 0 there exists n ∈ N such that
H ⊂ Hn () ⊂ An () ⊂ A()() ⊂ A(2).
Hence H ⊂
T
>0 A()
= A.
Lemma 2.4. Let (X, d) be a complete metric space and (fn ) be a sequence of ϕ-contractions of X
(with the same function ϕ), pointwise convergent to a ϕ-contraction f . Let xn be the fixed point of
fn , n ∈ N, and x be the fixed point of f . If {xn : n ∈ N} is bounded, then xn → x.
Proof. For any n ∈ N, we have:
d(xn , x) ≤ d(fn (xn ), fn (x)) + d(fn (x), f (x)) ≤ ϕ(d(xn , x)) + d(fn (x), f (x)).
Now let (d(xkn , x)) be any subsequence of (d(xn , x)). By Weierstrass theorem, it has a subsequence
(d(xln , x)), which has a limit t ∈ [0, ∞]. By our assumptions, t < ∞. Since ϕ is upper semicontinuous,
we get
t = lim sup d(xln , x) ≤ lim sup ϕ(d(xn , x)) + 0 ≤ ϕ(t).
n→∞
n→∞
Hence t = 0 and the result follows.
We are ready to give the proof of Theorem 2.1.
Proof. (of Theorem 2.1)
We will show that X \ Wf,g is closed. Let (wk ) ⊂ X \ Wf,g and assume that wk → w. By Lemma
2.2, it is enough to prove that ASwk → ASw . Hence, by Lemma 2.4 and an easy fact that FSwk is
pointwise convergent to FSw , we only have to show that {ASwk : n ∈ N} is bounded, in other words,
[
(2)
ASwk is bounded subset of X.
k∈N
Consider the following mapping G : K(X) → K(X) given by
[
G(F ) = f (F ) ∪ (g(F ) + w) ∪
(g(F ) + wk ).
k∈N
The mapping G is well defined since (g(F ) + w) ∪
S
k∈N (g(F ) + wk )
is compact as {wk : k ∈ N} ∪ {w}
and g(F ) are compact. Moreover, G is ϕ-contraction since for all F, D ∈ K(X), we have
H(G(F ), G(D)) ≤
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FILIP STROBIN AND JAROSLAW SWACZYNA
≤ sup{H(f (F ), f (D)), H(g(F )+w, g(D)+w), H(g(F )+w1 , g(D)+w1 ), H(g(F )+w2 , g(D)+w2 ), ...} ≤
≤ ϕ(H(F, D)).
Hence there is a compact (in particular, bounded) set FG ⊂ X with G(FG ) = FG and for each
D ∈ K(X), the sequence of itrates (Gn (D)) converges to FG . Now let k ∈ N and fix any D ∈ K(X).
Then FSwk (D) ⊂ G(D), and as a consequence,
∀n∈N FSnw (D) ⊂ Gn (D).
k
Since FSnw (D) →n→∞ ASwk and Gn (D) →n→∞ FG , by Lemma 2.3 we have ASwk ⊂ FG . Since k was
k
arbitrary and FG is bounded, we get (2). The result follows.
Let us point out that Wf,g can be very small:
Example 2.5. Let X = R and f (x) = 34 x for x ∈ R. Then for every w ≥ 0 we have
f ([0, 4w]) ∪ (f ([0, 4w] + w) = [0, 3w] ∪ [w, 4w] = [0, 4w],
and for every w < 0,
f ([4w, 0]) ∪ (f ([4w, 0] + w) = [3w, 0] ∪ [4w, w] = [4w, 0].
Hence in this case Wf,f = ∅.
Now we formulate the main result of the paper.
Theorem 2.6. Let X be infinite-dimensional Banach space and f, g be ϕ-contractions of X. Assume
that f is a compact operator (i.e., f (D) is compact for all bounded D; f need not be linear), and g
satisfies one of the following conditions:
(i) g is linear and k g k< 1;
(ii) Lip(g) ≤ 12 ;
(iii) g is compact operator.
Then the set X \ Wf,g is a countable union of relatively compact sets. In particula, it is σ-c-porous.
Again, we have to start with lemmas.
Lemma 2.7. Let X be a complete metric space and f, g be ϕ-contractions. For S = (f, g), we have
[
g(As ) =
g n (f (As )) ∪ {e},
n∈N
where e is a fixed point of g.
Proof.
CONNECTNESS OF ATTRACTORS OF CERTAIN FAMILY OF IFSS
5
Lemma 2.8. Let X be a Banach space and g be a ϕ-contraction of X. Assume that g satisfies one
of the following conditions:
(i) g is linear and k g k< 1;
(ii) Lip(g) ≤ 12 ;
(iii) g is compact operator.
Then for any n ∈ N and D ∈ K(X), the set
n
Mg,n,D = {w ∈ X : D ∩ gw
(D) 6= ∅}
is a countable union of relatively compact sets.
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FILIP STROBIN AND JAROSLAW SWACZYNA
E-mail address: [email protected]
E-mail address: [email protected]
Institute of Mathematics, Technical University of Lódź, Wólczańska 215, 93-005 Lódź, Poland
Institute of Mathematics, Technical University of Lódź, Wólczańska 215, 93-005 Lódź, Poland