# Applications of Multivalued Contractions on Graphs to Graph-Directed Iterated Function Systems

Authors: T. Dinevari; M. Frigon
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

We apply a fixed point result for multivalued contractions on complete metric spaces endowed with a graph to graph-directed iterated function systems. More precisely, we construct a suitable metric space endowed with a graphG and a suitable G-contraction such that its fixed points permit us to obtain more information on the attractor of a graph-directed iterated function system.


---

## Body

## 1. Introduction

Based on the work of Hutchinson [1] and being popularized by Barnsley [2], the method of iterated function systems (IFS) permits us to generate fractals by iterating a collection of transformations {
T
i
:
i
=
1
,
…
,
p
}. If each T
i is a contraction on a complete metric space M, it was shown in [1] that there exists a unique nonempty compact set K
⊂
M which is invariant with respect to {
T
i
:
i
=
1
,
…
,
p
}; that is,
(1)
K
=
⋃
i
=
1
p
‍
T
i
(
K
)
.
This attractor K is such that, for every compact A
⊂
M,
(2)
g
n
(
A
)
⟶
K
with
respect
to
the
Hausdorff
metric
,
where
(3)
g
(
A
)
=
⋃
i
=
1
p
‍
T
i
(
A
)
.
The existence of K can be deduced from the Banach fixed point theorem.A fixed point result which is, in some sense, a combination of the Banach contraction principle and the Knaster-Tarski fixed point theorem in a partially ordered set was obtained by Ran and Reurings [3] in 2004. They considered a monotone, order preserving single-valued map f defined on a complete metric space endowed with a partial ordering. They assumed that f satisfies a contraction condition not necessarily for all x and y, but for those such that x
≤
y. Subsequently, their result was generalized by many authors, in particular by Nieto, Rodríguez-López, Pouso, Petruşel, and Rus [4–7]. In 2008, Jachymski [8] presented a nice unification of most of the previous results by considering complete metric spaces endowed with a graph G. He introduced the notion of single-valued G-contraction for which he obtained fixed point results.Using those fixed point results, Gwóźdź-Łukawska and Jachymski [9] developed the Hutchinson-Barnsley theory on complete metric space endowed with a graph G for iterated function systems of single-valued G-contractions.Different extensions of the concept of single-valuedG contractions on complete metric spaces endowed with a graph to multivalued maps were presented by Dinevari and Frigon [10] and by Nicolae et al. [11]. Those extensions led to generalizations of Jachymski’s fixed point results and of the Nadler fixed point theorem for multivalued contractions.In 1988, Mauldin and Williams [12] introduced the notion of geometric graph-directed construction.Definition 1.
Ageometric graph-directed construction in R
m consists of (i)
a collection ofp nonoverlapping, compact, nonempty subsets of R
m, J
1
,
…
,
J
p with nonempty interior;
(ii)
a directed-graphH
=
(
V
(
H
)
,
E
(
H
)
) such that V
(
H
)
=
{
1
,
…
,
p
} is the set of its vertices, and, for each i
∈
V
(
H
), there exists some edge (
i
,
j
)
∈
E
(
H
);
(iii)
for each(
i
,
j
)
∈
E
(
H
), there is a similarity map T
i
,
j
:
R
m
→
R
m with similarity ratios r
i
,
j such that
(4)
⋃
(
i
,
j
)
∈
E
H
‍
T
i
,
j
J
j
⊂
J
i
;
(iv)
for eachi, {
T
i
,
j
(
J
j
)
:
(
i
,
j
)
∈
E
(
H
)
} is a nonoverlapping family of sets;
(v)
if[
i
1
,
…
,
i
q
-
1
,
i
q
=
i
1
] is a cycle in H, then
(5)
∏
k
=
1
q
‍
r
i
k
-
1
,
i
k
<
1
.They showed that a geometric graph-directed construction has an attractor.Theorem 2 (Mauldin and Williams [12]).
For a geometric graph-directed construction as above, there existsK
1
,
…
,
K
p a unique collection of nonempty compact sets such that
(6)
∀
i
∈
1
,
…
,
p
,
K
i
⊂
J
i
,
K
i
=
⋃
(
i
,
j
)
∈
E
(
H
)
‍
T
i
,
j
(
K
j
)
.
The set
(7)
K
=
⋃
i
=
1
p
‍
K
i
is called theattractor of this geometric graph-directed construction.Geometric graph-directed constructions have been studied and generalized by many authors; see [13–16]. In particular, it was shown in [13] that with an appropriate rescaling, condition (v) can be replaced by(v)′
for each(
i
,
j
)
∈
H, r
i
,
j
<
1.Also, in some of those generalizations, similarities on R
m were replaced by contractions on complete metric spaces and the terminology of graph-directed iterated function system was used. Again, the existence of an attractor K was established.In this paper, we take into account the graphH to obtain more information on the attractor K of a graph-directed iterated function system. To do so, we apply a fixed point result obtained by the authors [10] for multivalued contractions on complete metric spaces endowed with a graph.The paper is organized as follows. In Section2, we present some notations and we recall some results. In Section 3, we consider a space X such that K
∈
X and on which we define a suitable graph G and a suitable metric. In Section 4, we define an appropriate multivalued G-contraction F. In the last three sections, taking into account the maximal connected component of the graph H, we obtain more information on the attractor K from some fixed points of F.


## 2.H-Iterated Function System

First of all, we introduce the notion of MW-directed graph and we consider iterated function systems which takes into account the structure of anMW-directed graph.Definition 3.
A directed-graphH
=
(
V
(
H
)
,
E
(
H
)
) is called anMW-directed graph if V
(
H
)
=
{
1
,
…
,
p
}, H has no parallel edges, and for every i
∈
V
(
H
), there exists j
∈
V
(
H
) such that (
i
,
j
)
∈
E
(
H
).Definition 4.
LetH
=
(
V
(
H
)
,
E
(
H
)
) be an MW-directed graph. Agraph-directed iterated function system over the graph
H (H-IFS) is a collection of p nonempty, bounded, complete metric spaces, (
X
1
,
d
1
)
,
…
,
(
X
p
,
d
p
), and, for each (
i
,
j
)
∈
E
(
H
), a contraction T
i
,
j
:
X
j
→
X
i with constant of contraction λ
i
,
j. An H-IFS is denoted {
T
i
,
j
}
H.Definition 5.
Let{
T
i
,
j
}
H be an H-IFS. Anattractor
K of the H-IFS is a collection of nonempty compact sets K
=
{
K
i
}
H such that K
i
⊂
X
i and
(8)
K
i
=
⋃
(
i
,
j
)
∈
E
H
‍
T
i
,
j
(
K
j
)
∀
i
∈
1
,
…
,
p
.The Banach contraction principle insures the existence of an attractor of anH-IFS. We present the proof for sake of completeness. For more information on graph-directed iterated function systems, the reader is referred to [12, 15].Theorem 6.
AnH-IFS, {
T
i
,
j
}
H, has a unique attractor K.Proof.
Consider(9)
Y
=
S
1
,
…
,
S
p
⊂
∏
i
=
1
p
‍
X
i
:
S
i
is
a
compact
nonempty
subset
of
X
i
∏
i
=
1
p
‍
endowed with the metric
(10)
ρ
S
,
S
^
=
max
⁡
D
i
S
i
,
S
^
i
:
i
=
1
,
…
,
p
,
where D
i is the Hausdorff metric on X
i; that is,
(11)
D
i
(
S
i
,
S
^
i
)
=
inf
⁡
ε
>
0
:
S
i
⊂
B
S
^
i
,
ε
,
S
^
i
⊂
B
S
i
,
ε
,
where
(12)
B
(
S
i
,
ε
)
=
y
∈
X
i
:
∃
x
∈
S
i
such
that
d
i
x
,
y
<
ε
.
Let us define f
:
Y
→
Y by
(13)
f
i
(
S
)
=
⋃
(
i
,
j
)
∈
E
(
H
)
‍
T
i
,
j
(
S
j
)
.
Using the fact that every T
i
,
j is a contraction, one verifies that f is a contraction with constant of contraction
(14)
θ
=
max
⁡
λ
i
,
j
:
i
,
j
∈
E
H
.
The Banach contraction principle insures the existence of K
∈
Y a unique fixed point of f. Thus, K is the unique attractor of {
T
i
,
j
}
H.More information onK will be obtained by applying a fixed point result for multivalued contractions on complete metric spaces endowed with a graph. We recall the notion of G-contraction introduced in [10].For(
X
,
d
) a complete metric space, we consider G
=
(
V
(
G
)
,
E
(
G
)
) a directed graph such that X
=
V
(
G
), the diagonal in X
×
X is contained in E
(
G
), and G has no parallel edges.Definition 7.
LetF
:
X
→
X be a multivalued map with nonempty values. We say that F is a G-contraction if there exists α
∈
]
0,1
[ such that (
C
G
)
for all(
x
,
y
)
∈
E
(
G
) and all u
∈
F
(
x
), there exists v
∈
F
(
y
) such that (
u
,
v
)
∈
E
(
G
) and d
(
u
,
v
)
≤
α
d
(
x
,
y
).We consider suitable trajectories inX.Definition 8.
LetF
:
X
→
X be a multivalued mapping and x
0
∈
X. We say that a sequence {
x
n
} is a G
1-Picard trajectory from
x
0 if x
n
∈
F
(
x
n
-
1
) and (
x
n
-
1
,
x
n
)
∈
E
(
G
) for all n
∈
N. The set of all such G
1-Picard trajectories from x
0 is denoted by T
1
(
F
,
G
,
x
0
).The reader is referred to [10] for the proof of the following fixed point result for multivalued G-contractions.Theorem 9.
LetF
:
X
→
X be a multivalued G-contraction such that there exists (
x
0
,
x
1
)
∈
E
(
G
) such that x
1
∈
F
(
x
0
). In addition, assume that one of the following conditions holds. (i)
F is G
1
-Picard continuous from
x
0; that is, the limit of any convergent sequence {
x
n
}
∈
T
1
(
F
,
G
,
x
0
) is a fixed point of F.
(ii)
F has closed values and, for every {
x
n
} in T
1
(
F
,
G
,
x
0
) converging to some x
∈
X, there exists a subsequence {
n
k
} such that (
x
n
k
,
x
)
∈
E
(
G
) for all k
∈
N.Then, there exists a G
1-Picard trajectory from x
0, {
x
n
}, converging to x a fixed point of F. Moreover, every converging G
1-Picard trajectory from x
0 converges to a fixed point of F.In what follows, we considerH an MW-directed graph. We will use the following definitions and notations.Apath from
i
to
j in H is denoted by [
i
k
]
0
N
=
[
i
0
,
…
,
i
N
], where i
=
i
0, j
=
i
N, and (
i
k
-
1
,
i
k
)
∈
E
(
H
) for every k
=
1
,
…
,
N.We say that a subgraphC
=
(
V
(
C
)
,
E
(
C
)
) of H isconnected if for every i
,
j
∈
V
(
C
) there exists a path from i to j in C. Aconnected component of H is a maximal connected subgraph of H. We denote
(15)
C
(
H
)
=
C
:
C
is
a
connected
component
of
H
.
It follows from the definition of MW-directed graph that
(16)
∅
≠
C
(
H
)
=
C
α
:
α
∈
Λ
,
where
Λ
has
finite
cardinality
.
We can define a partial order on C
(
H
) as follows:
(17)
C
α
⪯
C
β
⟺
∃
i
k
0
N
a
path
in
H
such
that
i
0
∈
C
α
,
i
N
∈
C
β
.
We write C
α
≺
C
β to mean C
α
⪯
C
β and C
α
≠
C
β. We say that C
α and C
β areincomparable if C
α
⋠
C
β and C
β
⋠
C
α.We denote the set of vertices from which there is a path inH reaching i
∈
H by
(18)
i
←
=
j
∈
V
H
:
there
is
a
path
from
j
to
i
in
H
.
Similarly, for C
∈
C
(
H
), we denote the set of vertices from which there is a path in H reaching V
(
C
) by
(19)
C
←
=
⋃
i
∈
V
(
C
)
‍
i
←
.


## 3. A Suitable Metric Space Endowed with a Directed Graph

LetH be an MW-directed graph with V
(
H
)
=
{
1
,
…
,
p
}. For i
∈
V
(
H
), let (
X
i
,
d
i
) be a bounded complete metric space.In this section, usingH and the spaces X
i, we define a complete metric space endowed with a suitable directed graph. Let us recall that
(20)
C
(
H
)
=
C
:
C
is
a
connected
component
of
H
.We consider the spaceX of p-tuples A
=
(
A
1
,
…
,
A
p
) satisfying the following properties:(Xi)
A
i
⊂
X
i is compact for every i
=
1
,
…
,
p;
(Xii)
ifA
i
≠
∅ for some i
∈
V
(
C
) and C
∈
C
(
H
), then A
j
≠
∅ for all j
∈
V
(
C
);
(Xiii)
there existsC
∈
C
(
H
) and i
∈
V
(
C
) such that A
i
≠
∅.It is important to point out that, for A
=
(
A
1
,
…
,
A
p
)
∈
X, some A
i can be empty.We endowX with the metric
(21)
d
(
A
,
B
)
=
max
⁡
i
∈
{
1
,
…
,
p
}
D
¯
i
A
i
,
B
i
,
where
(22)
D
¯
i
(
A
i
,
B
i
)
=
D
i
A
i
,
B
i
,
if
A
i
≠
∅
,
B
i
≠
∅
,
0
,
if
A
i
=
∅
=
B
i
,
R
i
,
otherwise
,
where D
i is the Hausdorff metric in X
i and R
i
>
R is a constant which will be fixed later, with
(23)
R
=
max
⁡
diam
⁡
X
i
:
i
=
1
,
…
,
p
.
It is clear that (
X
,
d
) is a complete metric space.Taking into account the graphH, we want to endow X with a directed graph. To do so, we distinguish vertices of H which are in a connected component from the others. We set
(24)
V
c
=
⋃
C
∈
C
H
‍
V
C
,
(25)
V
e
=
V
H
∖
V
c
.We define the graphG as follows: V
(
G
)
=
X, and for A
,
B
∈
X, (
A
,
B
)
∈
E
(
G
) if and only if(G)
for everyi
∈
{
1
,
…
,
p
}, one of the following properties holds:
(i)
A
i
=
B
i
=
∅, or A
i
≠
∅ and B
i
≠
∅;
(ii)
A
i
=
∅, B
i
≠
∅, and one of the following statements is true:
(a)
i
∈
V
e and there exists j
∈
V
(
H
) such that (
i
,
j
)
∈
E
(
H
) and A
j
≠
∅;
(b)
i
∈
V
(
C
) for some C
∈
C
(
H
) and there exist k
∈
V
(
C
) and j
∈
V
(
H
) such that (
k
,
j
)
∈
E
(
H
) and A
j
≠
∅;
(iii)
A
i
≠
∅, B
i
=
∅, i
∈
V
e, and one of the following properties is satisfied:
(a)
there is noj
∈
V
(
H
) such that (
j
,
i
)
∈
E
(
H
);
(b)
for everyj
∈
V
(
H
) such that (
j
,
i
)
∈
E
(
H
), one has B
j
≠
∅.Example 10.
LetH be the MW-graph of Figure 1. We consider X the associated metric space satisfying (Xi)–(Xiii) endowed with the graph G satisfying the condition (G). Let A
i
k be nonempty compact subsets of X
i for all i
∈
{
1
,
…
,
9
} and k
∈
{
1
,
…
,
7
}. We consider the following elements of X:
(26)
A
1
=
(
∅
,
∅
,
A
3
1
,
A
4
1
,
A
5
1
,
∅
,
∅
,
∅
,
∅
)
,
A
2
=
(
∅
,
∅
,
A
3
2
,
A
4
2
,
A
5
2
,
A
6
2
,
∅
,
∅
,
∅
)
,
A
3
=
(
∅
,
∅
,
∅
,
∅
,
∅
,
∅
,
A
7
3
,
A
8
3
,
A
9
3
)
,
A
4
=
(
∅
,
∅
,
∅
,
∅
,
∅
,
A
6
4
,
A
7
4
,
A
8
4
,
A
9
4
)
,
A
5
=
(
∅
,
∅
,
A
3
5
,
A
4
5
,
A
5
5
,
∅
,
A
7
5
,
A
8
5
,
A
9
5
)
,
A
6
=
A
1
6
,
∅
,
A
3
6
,
A
4
6
,
A
5
6
,
∅
,
∅
,
∅
,
∅
,
A
7
=
(
∅
,
A
2
7
,
A
3
7
,
A
4
7
,
A
5
7
,
∅
,
∅
,
∅
,
∅
)
.
Here is the list of all edges of G between them:
(27)
(
A
1
,
A
7
)
,
(
A
2
,
A
1
)
,
(
A
2
,
A
7
)
,
(
A
3
,
A
4
)
,
(
A
3
,
A
4
)
,
(
A
4
,
A
5
)
,
(
A
6
,
A
1
)
,
(
A
6
,
A
7
)
,
(
A
7
,
A
6
)
⊂
E
(
G
)
.Figure 1
An MW-graphH.Now, we want to fixR
i in (22) in such a way that we will be able to define a suitable multivalued G-contraction on X in the next section. To this aim, we decompose V
(
H
) in appropriate subsets V
μ with μ
∈
I a totally ordered set.Lemma 11.
LetH be an MW-directed graph. Then there exist I a totally ordered set and {
V
μ
:
μ
∈
I
} a family of nonempty disjoint subsets, and, for every i
∈
{
1
,
…
,
p
}, there exists R
i
>
R such that (1)
V
(
H
)
=
∪
μ
∈
I
V
μ;
(2)
ifV
(
C
)
∩
V
μ
≠
∅ for some μ
∈
I and some C
∈
C
(
H
), then V
(
C
)
⊂
V
μ;
(3)
ifμ
<
ν in I, for all i
∈
V
μ, and j
∈
V
ν, then j
∉
[
i
]
←;
(4)
for everyμ
∈
I, one has R
i
=
R
j for every i
,
j
∈
V
μ;
(5)
for everyμ
<
ν
∈
I, one has R
i
<
R
j for every i
∈
V
μ, j
∈
V
ν.Proof.
We want to separate vertices ofH in suitable subsets. Let us recall that some vertices are in a connected component, and some others are not:
(28)
V
(
H
)
=
V
c
∪
V
e
,
where V
c and V
e are defined in (24) and (25), respectively.
First of all, we examine vertices inV
c. Let
(29)
L
=
max
⁡
C
α
n
n
∈
N
:
there
exists
a
chain
C
α
1
≺
⋯
≺
C
α
n
in
C
H
.
We denote
(30)
C
H
1
=
{
C
∈
C
(
H
)
:
∄
C
^
∈
C
(
H
)
such
that
C
^
≺
C
}
,
C
H
2
=
C
∈
C
H
∖
C
H
1
:
∄
C
^
∈
C
H
∖
C
H
1
such
that
C
^
≺
C
,
⋮
C
H
L
=
C
∈
C
(
H
)
∖
⋃
k
=
1
L
-
1
‍
C
H
k
:
∄
C
^
∈
C
(
H
)
∖
⋃
k
=
1
L
-
1
‍
C
H
k
such
that
C
^
≺
C
⋃
k
=
1
L
-
1
‍
.
We define
(31)
V
k
,
0
=
⋃
C
∈
C
H
k
‍
V
C
for
k
=
1
,
…
,
L
.
Observe that
(32)
V
c
=
⋃
k
=
1
L
‍
V
k
,
0
,
V
k
,
0
∩
V
j
,
0
=
∅
if
k
≠
j
.
Now, we separate vertices inV
e in suitable subsets. We first separate them in two sets: those which can be reached by a path starting from a vertex in a connected component, and those which cannot. This last set is denoted:
(33)
V
0
=
j
∈
V
e
:
V
c
∩
j
←
=
∅
.
If V
0
≠
∅, let
(34)
N
0
=
max
⁡
n
:
there
is
a
path
i
k
1
n
such
that
i
k
∈
V
0
for
every
k
=
1
,
…
,
N
0
i
k
1
n
.
We define
(35)
V
0,1
=
i
∈
V
0
:
∃
j
∈
V
0
such
that
j
,
i
∈
E
H
,
V
0,2
=
i
∈
V
0
∖
V
0,1
:
∃
j
∈
V
0
∖
V
0,1
such
that
j
,
i
∈
E
H
,
⋮
V
0
,
N
0
=
i
∈
V
0
∖
⋃
k
=
1
N
0
-
1
‍
V
0
,
k
:
∃
j
∈
V
0
∖
⋃
k
=
1
N
0
-
1
‍
V
0
,
k
such
that
(
j
,
i
)
∈
E
(
H
)
⋃
k
=
1
N
0
-
1
‍
.
Observe that
(36)
V
0
=
⋃
k
=
1
N
0
‍
V
0
,
k
,
V
0
,
k
∩
V
0
,
j
=
∅
if
k
≠
j
.
IfV
e
∖
V
0
≠
∅, it follows from Definition 3 that, for every j
∈
V
e
∖
V
0, there exist C
α
,
C
β
∈
C
(
H
) such that
(37)
C
α
≺
C
β
,
V
C
α
⊂
j
←
,
j
∈
C
β
←
.
In other words, j is on a path from C
α to C
β. Hence, L
>
1, where L is defined in (29).
IfL
≥
2, we first examine vertices on a path from some i
∈
V
1,0 to some j
∈
V
2,0. Let
(38)
N
1
=
max
⁡
n
:
there
is
a
path
i
k
0
1
+
n
such
that
i
0
∈
V
1,0
,
i
1
+
n
∈
V
2,0
and
i
k
∈
V
e
∀
k
=
1
,
…
,
n
i
k
0
1
+
n
.
If N
1
≥
1, we define
(39)
V
1,1
=
i
∈
V
e
:
∃
i
k
0
1
+
N
1
with
i
=
i
1
,
i
0
∈
V
1,0
,
i
1
+
N
1
∈
V
2,0
,
i
k
∈
V
e
for
k
=
1
,
…
,
N
1
.
If N
1
≥
2, we define
(40)
V
1,2
=
i
∈
V
e
∖
V
1,1
:
∃
i
k
1
1
+
N
1
with
i
=
i
2
,
i
1
∈
V
1,0
∪
V
1,1
,
i
1
+
N
1
∈
V
2,0
,
i
k
∈
V
e
∖
V
1,1
for
k
=
2
,
…
,
N
1
.
We define inductively V
1,1
,
…
,
V
1
,
N
1.
We denote the set of vertices on a path fromV
1,0 to V
2,0 by
(41)
V
1
=
V
1,0
∪
V
2,0
∪
⋃
k
=
1
N
1
‍
V
1
,
k
.
IfL
≥
3, we examine vertices on a path from some i
∈
V
1 to some j
∈
V
3,0. Let
(42)
N
2
=
max
⁡
n
:
there
is
a
path
i
k
0
1
+
n
such
that
i
0
∈
V
1
,
i
1
+
n
∈
V
3,0
and
i
k
∈
V
e
∖
V
1
∀
k
=
1
,
…
,
n
.
If N
2
≥
1, we define
(43)
V
2,1
=
i
∈
V
e
∖
V
1
:
∃
i
k
0
1
+
N
2
with
i
=
i
1
,
i
0
∈
V
1
,
i
1
+
N
2
∈
V
3,0
,
i
k
∈
V
e
∖
V
1
for
k
=
1
,
…
,
N
2
.
If N
2
≥
2, we define
(44)
V
2,2
=
i
∈
V
e
∖
V
1
∪
V
2,1
:
∃
i
k
1
1
+
N
2
with
i
=
i
2
,
i
1
∈
V
1
∪
V
2,1
,
i
1
+
N
2
∈
V
3,0
,
and
i
k
∈
V
e
∖
V
1
∪
V
2,1
for
k
=
2
,
…
,
N
2
.
Similarly, we define V
2
,
j for j
≤
N
2.
So, inductively, we define the following subsets ofV
e
∖
V
0:
(45)
V
1,1
,
…
,
V
1
,
N
1
,
V
2,1
,
…
,
V
2
,
N
2
,
…
,
V
L
-
1,1
,
…
,
V
L
-
1
,
N
L
-
1
.
Each vertex in one of those sets is on a path from one connected component to another.
We have decomposedV
(
H
) in a collection of disjoint sets:
(46)
V
0,1
,
…
,
V
0
,
N
0
,
V
1,0
,
V
1,1
,
…
,
V
1
,
N
1
,
V
2,0
,
…
,
V
L
-
1
,
N
L
-
1
,
V
L
,
0
.
We denote
(47)
I
=
k
,
0
:
1
≤
k
≤
L
∪
k
,
l
:
0
≤
k
≤
L
-
1,1
≤
l
≤
N
k
.
We endow I with the order
(48)
k
1
,
l
1
≤
k
2
,
l
2
⟺
k
1
<
k
2
,
or
k
1
=
k
2
,
l
1
≤
l
2
.
By construction,
(49)
V
H
=
⋃
μ
∈
I
‍
V
μ
,
V
μ
∩
V
ν
=
∅
if
μ
≠
ν
.
Also, for every C
∈
C
(
H
), there exists μ
∈
I such that V
(
C
)
⊂
V
μ. Moreover, for μ
,
ν
∈
I such that μ
<
ν, one has j
∉
[
i
]
← for every i
∈
V
μ, and j
∈
V
ν.
Finally, we chooseσ
:
I
→
]
1
,
∞
[ a strictly increasing map. We define
(50)
R
i
=
σ
μ
R
for
i
∈
1
,
…
,
p
∩
V
μ
,
μ
∈
I
.
By construction, statements (4) and (5) are satisfied.Remark 12.
Let(
A
,
B
)
∈
E
(
G
). From the definition of the graph G and Lemma 11, we can make the following observations. (1)
If for somei
∈
V
(
H
), (G)(ii) holds with some j
∈
V
(
H
) such that A
j
≠
∅; let μ
i
,
μ
j
∈
I be such that i
∈
V
μ
i and j
∈
V
μ
j. Then, μ
i
<
μ
j.
(2)
If for somei
∈
V
(
H
), (G)(iii)(b) holds, let μ
i
∈
I be such that i
∈
V
μ
i. Then, for all j
∈
V
(
H
) such that (
j
,
i
)
∈
E
(
H
), there is μ
j
∈
I such that j
∈
V
μ
j and one has μ
j
<
μ
i.Example 13.
We considerH the MW-graph of Figure 2 for which we describe the collection of subsets V
μ constructed as in the proof of Lemma 11. In this graph,
(51)
C
(
H
)
=
{
C
1
,
…
,
C
6
}
,
V
c
=
2,3
,
4,5
,
6,7
,
10,11,12,13,14,15
,
18,19,20,26,27,28
,
V
e
=
1,8
,
9,16,17,21,22,23,24,25,29
.
Since C
2
≺
C
4
≺
C
6
≺
C
5, one has L
=
4, and
(52)
V
1,0
=
2,3
,
4,5
,
6,7
,
V
2,0
=
10,11,12,13,14,15
,
V
3,0
=
18,19,20
,
V
4,0
=
26,27,28
.
By considering the paths from V
1,0 to V
2,0, one sees that N
1
=
2, and
(53)
V
1,1
=
8
,
V
1,2
=
9
.
By considering the paths from V
1,0
∪
V
1,1
∪
V
1,2
∪
V
2,0 to V
3,0, one sees that N
2
=
2, and
(54)
V
2,1
=
{
16
}
,
V
2,2
=
{
17
}
.
Similarly, one has N
3
=
3, and
(55)
V
3,1
=
{
23
}
,
V
3,2
=
{
24
}
,
V
3,3
=
{
25,29
}
.
So, the vertices which are not in one of the previous sets are in V
0
=
{
1,21,22
}. Similarly, N
0
=
2, and
(56)
V
0,1
=
{
1,21
}
,
V
0,2
=
{
22
}
.
So, I is the totally ordered set:
(57)
0,1
<
(
0,2
)
<
(
1,0
)
<
(
1,1
)
<
(
1,2
)
<
(
2,0
)
<
(
2,1
)
<
(
2,2
)
<
(
3,0
)
<
(
3,1
)
<
(
3,2
)
<
(
3,3
)
<
(
4,0
)
,
V
(
H
)
=
{
1
,
…
,
29
}
=
⋃
μ
∈
I
‍
V
μ
.Figure 2
An MW-graphH with C
(
H
)
=
{
C
1
,
…
,
C
6
}.


## 4. AG-Contraction

In this section, we consider a graph-directed iterated function system over the graphH, {
T
i
,
j
}
H. We will define an appropriate multivalued G-contraction on X, where G and X are, respectively, the graph and the metric space endowed with this graph and defined in the previous section. This G-contraction will be used to get more information on the attractor of this H-IFS.LetA
∈
X. For each j such that A
j
≠
∅, T
i
,
j
(
A
j
)
≠
∅ for all i such that (
i
,
j
)
∈
E
(
H
). So, it is important to distinguish all those edges. To this aim, we introduce the following notations.LetV
e be the subset of vertices in V
(
H
) which are not in connected components of H and defined in (25). So, for i
∈
V
e, we denote
(58)
E
i
(
A
)
=
i
,
j
∈
E
H
:
A
j
≠
∅
.
For ∅
≠
P
⊂
E
i
(
A
), we define
(59)
U
i
e
(
A
,
P
)
=
⋃
(
i
,
j
)
∈
P
‍
T
i
,
j
(
A
j
)
.LetV
c be the subset of vertices in V
(
H
) which are in connected components of H and defined in (24). So, for i
∈
V
c, there exists C
∈
C
(
H
) such that i
∈
V
(
C
). We consider the set of edges from a vertex of C to a vertex outside of C for which the component of A is nonempty:
(60)
E
C
(
A
)
=
k
,
j
∈
E
H
:
k
∈
V
C
,
j
∉
V
C
,
A
j
≠
∅
.
For k
∈
V
(
C
), we denote
(61)
i
→
C
k
=
i
k
0
N
which
is
a
path
in
C
from
i
=
i
0
to
k
=
i
N
and
containing
no
cycle
⁡
i
k
0
N

,
and we define T
i
→
k
:
X
k
→
X
i by
(62)
T
i
→
k
(
x
)
=
⋃
[
i
k
]
0
N
∈
i
→
C
k
‍
T
i
0
,
i
1
∘
⋯
∘
T
i
N
-
1
,
i
N
(
x
)
.
We define
(63)
U
i
c
(
A
,
P
)
=
∅
,
if
P
=
∅
,
⋃
(
k
,
j
)
∈
P
‍
T
i
→
k
∘
T
k
,
j
A
j
,
if
∅
≠
P
⊂
E
C
A
.
We also define
(64)
W
i
(
A
)
=
∅
,
if
A
i
=
∅
,
⋃
(
i
,
j
)
∈
E
C
‍
T
i
,
j
A
j
,
if
A
i
≠
∅
,
where E
(
C
)
=
{
(
k
,
j
)
∈
E
(
H
)
:
k
,
j
∈
V
(
C
)
}.We have all the ingredients to define the multivalued mapF
:
X
→
X. For A
∈
X,
(65)
U
=
(
U
1
,
…
,
U
p
)
∈
F
(
A
)
⟺
U
i
∈
F
i
(
A
)
,
where F
i
(
A
) is defined as follows.Fori
∈
V
e,
(66)
F
i
(
A
)
=
∅
,
if
E
i
A
=
∅
,
U
i
e
A
,
P
:
∅
≠
P
⊂
E
i
A
,
if
E
i
A
≠
∅
.
For i
∈
V
(
C
) for some C
∈
C
(
H
),
(67)
F
i
(
A
)
=
∅
,
if
A
i
=
∅
,
E
C
(
A
)
=
∅
,
U
i
c
A
,
P
:
∅
≠
P
⊂
E
C
A
,
if
A
i
=
∅
,
E
C
(
A
)
≠
∅
,
W
i
A
∪
U
i
c
A
,
P
:
P
⊂
E
C
A
,
if
A
i
≠
∅
.Observe thatF is well defined. Indeed, if U
∈
F
(
A
) is such that U
i
≠
∅ for i in some V
(
C
), then U
j
≠
∅ for all j
∈
V
(
C
). Also, there exists C
∈
C
(
H
) such that U
i
≠
∅ for all i
∈
V
(
C
). Moreover, the values of F are finite and hence closed.We show thatF is a multivalued G-contraction.Proposition 14.
LetF
:
X
→
X be the multivalued map defined above. Then F is a G-contraction.Proof.
We want to show thatF is a G-contraction with constant of contraction:
(68)
λ
=
max
⁡
R
R
i
:
i
∈
1
,
…
,
p
max
⁡
λ
i
,
j
:
i
,
j
∈
E
H
,
max
⁡
R
R
i
:
i
∈
1
,
…
,
p
,
max
⁡
R
i
R
j
:
i
∈
V
μ
i
,
j
∈
V
μ
j
for
μ
i
,
μ
j
∈
I
such
that
μ
i
<
μ
j
R
i
R
j
R
R
i
:
i
∈
1
,
…
,
p

,
where R
i, I, and V
μ for μ
∈
I are given in Lemma 11. For i
,
k
∈
V
(
C
) for some C
∈
C
(
H
), we denote
(69)
λ
i
→
k
=
max
⁡
λ
i
0
,
i
1
⋯
λ
i
N
-
1
,
i
N
:
i
k
0
N
∈
i
→
C
k
,
where {
i
→
C
k
} is given in (61). Observe that λ
i
→
k
≤
λ.
LetA
,
B
∈
X be such that (
A
,
B
)
∈
E
(
G
) and U
∈
F
(
A
). We look for U
^
∈
F
(
B
) such that (
U
,
U
^
)
∈
E
(
G
) and d
(
U
,
U
^
)
≤
λ
d
(
A
,
B
).
Step  1 (
i
∈
V
e
). Let μ
∈
I be such that i
∈
V
μ.
Case  1 (
U
i
=
∅
and
U
~
i
≠
∅
for every
U
~
∈
F
(
B
)
). In this case, E
i
(
A
)
=
∅ and E
i
(
B
)
≠
∅ by (66). Choose some (
i
,
j
)
∈
E
i
(
B
). Therefore, A
j
=
∅, B
j
≠
∅, and for ν
∈
I such that j
∈
V
ν, one has μ
<
ν.
By condition (G)(ii)(a), ifj
∈
V
e, there exists l
∈
V
(
H
) such that (
j
,
l
)
∈
E
(
H
) and A
l
≠
∅. So, (
j
,
l
)
∈
E
j
(
A
).
On the other hand, ifj
∈
V
(
C
) for some C
∈
C
(
H
), by condition (G)(ii)(b), there exist k
∈
V
(
C
) and l
∈
V
(
H
) such that (
k
,
l
)
∈
E
(
H
) and A
l
≠
∅. So, (
k
,
l
)
∈
E
C
(
A
) and j
,
k
∈
V
(
C
).
So, for the casej
∈
V
e and the case j
∈
V
c, we obtain by (66) and (67),
(70)
U
i
=
∅
,
U
~
i
≠
∅
,
U
j
≠
∅
for
some
i
,
j
∈
V
H
.
Moreover, by (21), (22), and (68),
(71)
D
¯
i
U
i
,
U
~
i
=
R
i
=
R
i
R
j
D
¯
j
(
A
j
,
B
j
)
≤
λ
d
(
A
,
B
)
∀
U
~
∈
F
(
B
)
.
Case 2 (
U
i
≠
∅
and
U
~
i
=
∅
for every
U
~
∈
F
(
B
)
). In this case, E
i
(
A
)
≠
∅ and E
i
(
B
)
=
∅ by (66). Choose some (
i
,
j
)
∈
E
i
(
A
). Therefore, A
j
≠
∅, B
j
=
∅, and for ν
∈
I such that j
∈
V
ν, one has μ
<
ν. By conditions (G)(i) and (G)(iii), one has j
∈
V
e and B
i
≠
∅. By (66), (67) and since B
i
≠
∅, one has
(72)
U
i
≠
∅
,
U
~
i
=
∅
and
one
of
the
following
situations
hold:
(
i
)
there
is
no
k
∈
V
(
H
)
such
that
k
,
i
∈
E
(
H
)
;
(
ii
)
∀
k
∈
V
(
H
)
such
that
(
k
,
i
)
∈
E
(
H
)
,
U
~
k
≠
∅
∀
U
~
∈
F
(
B
)
.
Also, by (21), (22), and (68),
(73)
D
¯
i
(
U
i
,
U
~
i
)
=
R
i
=
R
i
R
j
D
¯
j
(
A
j
,
B
j
)
≤
λ
d
(
A
,
B
)
∀
U
~
∈
F
(
B
)
.
Case 3 (
U
i
≠
∅
and
U
~
i
≠
∅
for every
U
~
∈
F
(
B
)
). In this case, U
i
=
U
i
e
(
A
,
P
) for some ∅
≠
P
⊂
E
i
(
A
), and E
i
(
B
)
≠
∅ by (66).
IfP
⊂
E
i
(
B
), one has by (21), (59), and (68),
(74)
D
i
U
i
e
A
,
P
,
U
i
e
B
,
P
=
D
i
⋃
(
i
,
j
)
∈
P
‍
T
i
,
j
(
A
j
)
,
⋃
(
i
,
j
)
∈
P
‍
T
i
,
j
(
B
j
)
≤
max
⁡
(
i
,
j
)
∈
P
D
i
T
i
,
j
A
j
,
T
i
,
j
B
j
≤
max
⁡
(
i
,
j
)
∈
P
λ
i
,
j
D
j
A
j
,
B
j
≤
λ
d
A
,
B
.
IfP
⊄
E
i
(
B
), choose some i
,
j
∈
P
∖
E
i
(
B
). So, A
j
≠
∅, B
j
=
∅, and, for ν
∈
I such that j
∈
V
ν, one has μ
<
ν. Thus, by (21), (22), and (68),
(75)
D
¯
i
(
U
i
,
U
~
i
)
≤
R
=
R
R
j
D
¯
j
(
A
j
,
B
j
)
≤
λ
d
(
A
,
B
)
∀
U
~
∈
F
(
B
)
.
Combining (74) and (75), for U
i
=
U
i
e
(
A
,
P
) for some P
⊂
E
i
(
A
), we choose U
~
i
∈
F
i
(
B
) such that
(76)
U
~
i
=
U
i
e
B
,
P
,
if
P
⊂
E
C
(
A
)
∩
E
C
(
B
)
,
U
~
i
,
otherwise
,
with
some
U
~
i
∈
F
i
(
B
)
;
and we get
(77)
D
¯
i
(
U
i
,
U
~
i
)
≤
λ
d
(
A
,
B
)
.
Step 2 (
i
∈
V
(
C
)
for some
C
∈
C
(
H
)
). Let μ
∈
I be such that i
∈
V
μ.
Case 4 (
U
i
=
∅
and
U
~
i
≠
∅
for every
U
~
∈
F
(
B
)
). In this case, A
i
=
E
C
(
A
)
=
∅ and B
i
∪
E
C
(
B
)
≠
∅ by (67).
IfB
i
≠
∅, by condition (G)(ii)(b), there exist k
∈
V
(
C
) and j
∈
V
(
H
) such that (
k
,
j
)
∈
E
(
H
) and A
j
≠
∅. So, (
k
,
j
)
∈
E
C
(
A
). This contradicts the fact that E
C
(
A
)
=
∅.
IfE
C
(
B
)
≠
∅, by (60), there exist k
∈
V
(
C
) and j
∈
V
H
∖
V
(
C
) such that (
k
,
j
)
∈
E
(
H
) and B
j
≠
∅ and, for ν
∈
I such that j
∈
V
ν, one has μ
<
ν. Since E
C
(
A
)
=
∅, A
j
=
∅. If j
∈
V
e, by condition (G)(ii)(a), there exists l
∈
V
(
H
) such that (
j
,
l
)
∈
E
(
H
) and A
l
≠
∅. So, E
j
(
A
)
≠
∅, and U
j
≠
∅ by (66). On the other hand, if j
∈
V
(
C
^
) for some C
^
∈
C
(
H
), by condition (G)(ii)(b), there exist m
∈
V
(
C
^
), l
∈
V
(
H
) such that (
m
,
l
)
∈
E
(
H
) and A
l
≠
∅. So, E
C
^
(
A
)
≠
∅ and U
j
≠
∅ by (67). Thus, for the case j
∈
V
e and the case j
∈
V
c, we obtain
(78)
U
i
=
∅
,
U
~
i
≠
∅
,
U
j
≠
∅
for
some
k
,
j
∈
E
C
(
B
)
.
Moreover, by (21), (22), and (68),
(79)
D
¯
i
(
U
i
,
U
~
i
)
=
R
i
=
R
i
R
j
D
¯
j
(
A
j
,
B
j
)
≤
λ
d
(
A
,
B
)
∀
U
~
∈
F
(
B
)
.
Case 5 (
U
i
≠
∅
and
U
~
i
=
∅
for every
U
~
∈
F
(
B
)
). In this case, A
i
∪
E
C
(
A
)
≠
∅ and B
i
∪
E
C
(
B
)
=
∅ by (67). From condition (G)(iii), we deduce that A
i
=
B
i
=
∅. Let (
k
,
j
)
∈
E
C
(
A
). One has A
j
≠
∅ and B
j
=
∅ since (
k
,
j
)
∉
E
C
(
B
). By condition (G)(iii), j
∈
V
e and B
k
≠
∅ since (
k
,
j
)
∈
E
(
H
). This implies that B
i
≠
∅ by condition (Xii) since i
,
k
∈
V
(
C
). This is a contradiction. Thus,
(80)
U
i
≠
∅
,
U
~
i
=
∅
∀
U
~
∈
F
(
B
)
is
impossible
.
Case 6 (
U
i
≠
∅
and
U
~
i
≠
∅
for every
U
~
∈
F
(
B
)
). In this case, A
i
∪
E
C
(
A
)
≠
∅ and B
i
∪
E
C
(
B
)
≠
∅ by (67).
IfA
i
≠
∅, by condition (G)(iii), B
i
≠
∅. So W
i
(
A
)
≠
∅, W
i
(
B
)
≠
∅, and, by (21), (64), and (68),
(81)
D
i
W
i
A
,
W
i
B
=
D
i
⋃
(
i
,
j
)
∈
E
(
C
)
‍
T
i
,
j
(
A
j
)
,
⋃
(
i
,
j
)
∈
E
(
C
)
‍
T
i
,
j
(
B
j
)
≤
max
⁡
(
i
,
j
)
∈
E
C
D
i
T
i
,
j
A
j
,
T
i
,
j
B
j
≤
max
⁡
(
i
,
j
)
∈
E
C
λ
i
,
j
D
j
A
j
,
B
j
≤
λ
max
⁡
(
i
,
j
)
∈
E
(
C
)
D
j
(
A
j
,
B
j
)
≤
λ
d
(
A
,
B
)
.
IfE
C
(
A
)
≠
∅, for ∅
≠
P
⊂
E
C
(
A
) such that P
⊂
E
C
(
B
), one has by (21), (62), (63), (68), and (69),
(82)
D
i
U
i
c
A
,
P
,
U
i
c
B
,
P
=
D
i
⋃
(
k
,
j
)
∈
P
‍
T
i
→
k
∘
T
k
,
j
(
A
j
)
,
⋃
(
k
,
j
)
∈
P
‍
T
i
→
k
∘
T
k
,
j
(
B
j
)
≤
max
⁡
k
,
j
∈
P
λ
i
→
k
D
k
T
k
,
j
A
j
,
T
k
,
j
B
j
≤
max
⁡
k
,
j
∈
P
λ
i
→
k
λ
k
,
j
D
j
A
j
,
B
j
≤
λ
max
⁡
k
,
j
∈
P
D
j
A
j
,
B
j
≤
λ
d
A
,
B
.
If P
⊂
E
C
(
A
) and P
⊄
E
C
(
B
), there exists (
k
,
j
)
∈
P such that A
j
≠
∅, B
j
=
∅ and, for ν
∈
I such that j
∈
V
ν, one has μ
<
ν. Hence, by (21), (22), and (68),
(83)
D
¯
i
(
U
i
,
U
~
i
)
≤
R
=
R
R
j
D
¯
j
(
A
j
,
B
j
)
≤
λ
d
(
A
,
B
)
∀
U
~
∈
F
(
B
)
.
Combining (67), (81), (82), and (83), we choose U
~
i
∈
F
i
(
B
) such that
(84)
U
~
i
=
W
i
(
B
)
,
if
U
i
=
W
i
(
A
)
,
U
i
c
(
B
,
P
)
,
if
U
i
=
U
i
c
(
A
,
P
)
for
∅
≠
P
⊂
E
C
(
A
)
∩
E
C
(
B
)
,
W
i
(
B
)
∪
U
i
c
(
B
,
P
)
,
if
U
i
=
W
i
(
A
)
∪
U
i
c
(
A
,
P
)
for
∅
≠
P
⊂
E
C
(
A
)
∩
E
C
(
B
)
,
U
~
i
,
otherwise
,
with
some
U
~
i
∈
F
i
(
B
)
;
and we get
(85)
D
¯
i
(
U
i
,
U
~
i
)
≤
λ
d
(
A
,
B
)
.
Step  3 (choice of an appropriate
U
~
∈
F
(
B
)
). Finally, we choose U
~
∈
F
(
B
) as follows:
(86)
U
~
i
=
∅
,
if
i
∈
V
e
,
E
i
(
B
)
=
∅
,
some
U
~
i
∈
F
i
(
B
)
,
if
i
∈
V
e
,
U
i
=
∅
,
E
i
(
B
)
≠
∅
,
U
~
i
given
by
7
6
,
if
i
∈
V
e
,
U
i
≠
∅
,
E
i
(
B
)
≠
∅
,
∅
,
if
i
∈
V
(
C
)
,
B
i
∪
E
C
(
B
)
=
∅
,
some
U
~
i
∈
F
i
(
B
)
,
if
i
∈
V
(
C
)
,
U
i
=
∅
,
U
~
i
given
by
8
4
,
if
i
∈
V
(
C
)
,
U
i
≠
∅
,
B
i
∪
E
C
(
B
)
≠
∅
.
It follows from (70), (72), (78), and (80) that
(87)
(
U
,
U
~
)
∈
E
(
G
)
.
Finally, from (71), (73), (77),   (79), and (85), we deduce that
(88)
d
(
U
,
U
~
)
≤
λ
d
(
A
,
B
)
.
Therefore, F is a G-contraction.Here is another property satisfied by the multivalued mapF.Lemma 15.
LetF
:
X
→
X be the multivalued map defined above. Then, for every A
0
∈
X and every {
A
n
}
G
1-Picard trajectory from A
0 converging to some A
∈
X, there exists N
∈
N such that (
A
n
,
A
)
∈
E
(
G
) for all n
>
N.Proof.
LetA
0
∈
X and {
A
n
} a G
1-Picard trajectory from A
0 such that A
n
→
A. Thus, there exists N
∈
N such that d
(
A
n
,
A
)
<
R for all n
>
N. So, by (21) and (22), A
n
=
(
A
1
n
,
…
,
A
p
n
) and A
=
(
A
1
,
…
,
A
p
) are such that, for all n
>
N and all i
∈
V
(
H
), A
i
n
=
∅ if and only if A
i
=
∅. Thus, (G)(i) is satisfied and (
A
n
,
A
)
∈
E
(
G
) for all n
>
N.


## 5. Attractor of anH-IFS and Elements of C
(
H
)

ForH
=
(
V
(
H
)
,
E
(
H
)
) an MW-directed graph, and {
T
i
,
j
}
H a graph-directed iterated function system over the graph H, we consider K the attractor of this H-IFS insured by Theorem 6. We want to get more information on K by taking into account the connected components of H.Theorem 16.
LetH
=
(
V
(
H
)
,
E
(
H
)
) be an MW-directed graph. Let {
T
i
,
j
}
H be an H-IFS and K its attractor. Then the following statements hold. (1)
For everyC
∈
C
(
H
), there exists K
+
(
C
)
⊂
K such that
(a)
K
i
+
(
C
)
≠
∅ for every i
∈
V
(
C
);
(b)
K
i
+
(
C
)
≠
∅ for every i
∈
[
C
]
←, where [
C
]
← is defined in (19).
(c)
K
i
+
(
C
)
=
∅ for every i
∉
[
C
]
←.
(2)
IfC
1
,
C
2
∈
C
(
H
) are such that C
1
⪯
C
2, then K
+
(
C
1
)
⊂
K
+
(
C
2
).
(3)
IfC
1
,
C
2
∈
C
(
H
) are incomparable, then
(89)
K
i
+
(
C
1
)
∩
K
i
+
(
C
2
)
=
∅
∀
i
∉
C
1
←
∩
(
C
2
←
)
.
(4)
There existsK
-
∈
X such that K
-
⊂
K and
(a)
for everyC
∈
C
(
H
), K
i
-
=
K
i
+
(
C
) for every i
∈
V
(
C
) and K
i
-
⊂
K
i
+
(
C
) for every i
∈
[
C
]
←;
(b)
ifC
1
,
C
2
∈
C
(
H
) are such that C
1
⪯
C
2, then
(90)
K
i
-
⊂
K
i
+
(
C
1
)
⊂
K
i
+
(
C
2
)
∀
i
∈
C
1
←
;
(c)
ifC
1
,
C
2
∈
C
(
H
) are incomparable, then,
(91)
K
i
-
⊂
K
i
+
(
C
1
)
∩
K
i
+
(
C
2
)
∀
i
∈
C
1
←
∩
C
2
←
.Proof.
(
1
) Let F
:
X
→
X be the multivalued map defined in (65), (66), and (67). We know that F is a G-contraction by Proposition 14. Also, it follows from Lemma 15 that F satisfies condition (ii) of Theorem 9.
Theorem6 and the definition of F imply that fixed points of F are included in K.
LetC
∈
C
(
H
). We want to show that there exists K
+
(
C
) a fixed point of F satisfying the required properties. Fix
(92)
A
0
=
A
1
0
,
…
,
A
p
0
∈
X
such
that
A
i
0
≠
∅
⟺
i
∈
V
(
C
)
.
Forn
∈
N
∪
{
0
}, we choose inductively
(93)
A
n
+
1
∈
F
A
n
the
biggest
element
of
F
A
n
.
That is, by (66) and (67), A
n
+
1
=
(
A
1
n
+
1
,
…
,
A
p
n
+
1
)
∈
F
(
A
n
) is chosen as follows.
Fori
∈
V
e,
(94)
A
i
n
+
1
=
∅
,
if
E
i
(
A
n
)
=
∅
;
U
i
e
A
n
,
E
i
A
n
,
if
E
i
(
A
n
)
≠
∅
,
where E
i
e and U
i
e are defined in (58) and (59), respectively.
Fori
∈
V
(
C
^
) for some C
^
∈
C
(
H
),
(95)
A
i
n
+
1
=
∅
,
if
A
i
n
=
E
C
^
(
A
n
)
=
∅
;
U
i
c
A
n
,
E
C
^
A
n
,
if
A
i
n
=
∅
,
E
C
^
(
A
n
)
≠
∅
;
W
i
(
A
n
)
∪
U
i
c
A
n
,
E
C
^
A
n
,
if
A
i
n
≠
∅
,
where E
C
^, U
i
c, and W
i are defined in (60), (63), and (64), respectively.
Arguing as in the proof of Proposition14, one has that (
A
n
-
1
,
A
n
)
∈
E
(
G
) and
(96)
d
(
A
n
,
A
n
+
1
)
≤
λ
d
(
A
n
-
1
,
A
n
)
∀
n
∈
N
.
By Theorem 9, {
A
n
} is a G
1-Picard trajectory converging to some K
+
(
C
)
∈
X a fixed point of F.
Observe that, for everyn
∈
N and every i
∈
V
(
C
), A
i
n
≠
∅. Therefore,
(97)
K
i
+
C
≠
∅
∀
i
∈
V
(
C
)
.
Similarly, observe that, by construction, A
i
n
=
∅ for every i
∉
[
C
]
←. Indeed, for such i, E
i
(
A
n
-
1
)
=
∅ if i
∈
V
e, and A
i
n
-
1
=
E
C
^
(
A
n
-
1
)
=
∅ if i
∈
V
(
C
^
) for some C
^
∈
V
(
C
). Thus,
(98)
K
i
+
(
C
)
=
∅
∀
i
∉
C
←
.
On the other hand, let
(99)
N
C
=
max
⁡
i
∈
C
←
min
⁡
i
k
0
N
N
:
i
=
i
0
,
i
N
∈
V
(
C
)
,
i
k
0
N
is
a
path
in
H
from
i
to
i
N
.
Again by construction, A
i
n
≠
∅ for all n
>
N
C, for all i
∈
[
C
]
←. So,
(100)
K
i
+
(
C
)
≠
∅
∀
i
∈
C
←
.
Finally, observe thatK
+
(
C
) is independent of A
0
⊂
X chosen as in (92). Indeed, for
(101)
A
~
0
=
(
A
~
1
0
,
…
,
A
~
p
0
)
∈
X
such
that
A
~
i
0
≠
∅
⟺
i
∈
V
(
C
)
,
we define inductively A
~
n
+
1
∈
F
(
A
~
n
) as in (93). Observe that (
A
n
,
A
~
n
)
∈
E
(
G
) for all n
∈
N
∪
{
0
}. Arguing as in Proposition 14, one has
(102)
d
(
A
n
+
1
,
A
~
n
+
1
)
≤
λ
d
(
A
n
,
A
~
n
)
∀
n
∈
N
.
This inequality combined with the fact that A
n
→
K
+
(
C
) implies that A
~
n
→
K
+
(
C
).
(
2
) Let C
1
,
C
2
∈
C
(
H
) be such that C
1
⪯
C
2. One has
(103)
i
∈
C
1
←
⊂
i
∈
C
2
←
.
Let B
0
=
(
B
1
0
,
…
,
B
p
0
)
∈
X be such that
(104)
B
j
0
=
K
j
+
(
C
2
)
,
if
j
∈
C
1
←
,
∅
,
if
j
∉
C
1
←
.
By (
1
) and (G)(i), one has (
K
+
(
C
1
)
,
B
0
)
∈
E
(
G
) and K
+
(
C
1
)
∈
F
(
K
+
(
C
1
)
). Let B
1 be the biggest element in F
(
B
0
); that is, B
1 is chosen similarly to (94) and (95). Observe that B
1
⊂
K
+
(
C
2
), since B
0
⊂
K
+
(
C
2
), K
+
(
C
2
)
∈
F
(
K
+
(
C
2
)
), and by the definitions of F and K
+
(
C
2
). Arguing as in the proof of Proposition 14, one has (
K
+
(
C
1
)
,
B
1
)
∈
E
(
G
) and
(105)
d
(
K
+
(
C
1
)
,
B
1
)
≤
λ
d
(
K
+
(
C
1
)
,
B
0
)
.
Repeating this argument, we obtain {
B
n
} a G
1-Picard trajectory from B
0 such that
(106)
B
n
⊂
K
+
C
2
,
d
K
+
C
1
,
B
n
≤
λ
n
d
K
+
C
1
,
B
0
∀
n
∈
N
.
Therefore, B
n
→
K
+
(
C
1
) and
(107)
K
+
(
C
1
)
⊂
K
+
(
C
2
)
.
(
3
) If C
1
,
C
2
∈
C
(
H
) are incomparable, it follows directly from (1)(c) that
(108)
K
i
+
(
C
1
)
∩
K
i
+
(
C
2
)
=
∅
∀
i
∉
C
1
←
∩
C
2
←
.
(4) For everyC
∈
C
(
H
), C
=
(
V
(
C
)
,
E
(
C
)
) is an MW-directed graph and
(109)
T
i
,
j
:
i
,
j
∈
E
C
is a graph-directed iterated function system over the graph C. Let
(110)
K
-
(
C
)
=
K
i
-
i
∈
V
(
C
)
be the attractor of this graph-directed iterated system insured by Theorem 6.
We defineK
-
∈
X by
(111)
K
-
=
(
K
1
-
,
…
,
K
p
-
)
,
where
K
i
-
=
K
i
-
(
C
)
,
if
i
∈
V
(
C
)
for
some
C
∈
C
(
H
)
,
∅
,
if
i
∈
V
e
.
LetC
∈
C
(
H
) and {
A
n
} the G
1-Picard trajectory from A
0 defined in (92) and (93). By (95), for all n
∈
N, E
C
(
A
n
-
1
)
=
∅ and A
i
n
=
W
i
(
A
n
-
1
) for all i
∈
V
(
C
). So, using (64) and (67) and the fact that A
i
n
→
K
i
+
(
C
)
∈
F
i
(
K
+
(
C
)
) for every i
∈
V
(
C
), we deduce that
(112)
K
i
+
(
C
)
=
⋃
(
i
,
j
)
∈
E
C
‍
T
i
,
j
K
j
+
C
∀
i
∈
V
(
C
)
.
By definition of K
-,
(113)
K
i
-
=
⋃
(
i
,
j
)
∈
E
(
C
)
‍
T
i
,
j
(
K
j
-
)
∀
i
∈
V
(
C
)
.
The uniqueness of the fixed point of this operator implies that
(114)
K
i
+
(
C
)
=
K
i
-
∀
i
∈
V
(
C
)
.
On the other hand, ifi
∈
V
e
∩
[
C
]
←, one has ∅
=
K
i
-
⊂
K
i
+
(
C
). If i
∈
V
(
C
^
)
∩
[
C
]
← for some C
≠
C
^
∈
C
(
H
), then C
^
⪯
C. It follows from (114) and (2) that K
i
-
=
K
i
+
(
C
^
)
⊂
K
i
+
(
C
).
The properties(
4
)(b) and (
4
)(c) follow directly from (
2
) and (
4
)(a).Example 17.
LetH be the MW-graph of Figure 3.
We consider theH-IFS, {
T
i
,
j
}
H, with the metric spaces:
(115)
X
1
=
[
1,2
]
×
[
0,1
]
,
X
2
=
[
2,3
]
×
[
0,1
]
,
X
3
=
[
1,2
]
×
[
1,2
]
,
X
4
=
[
2,3
]
×
[
1,2
]
,
X
5
=
[
0,1
]
×
[
0,1
]
,
X
6
=
[
-
1,0
]
×
[
0,1
]
,
X
7
=
[
0,1
]
×
[
1,2
]
,
X
8
=
[
-
1,0
]
×
[
1,2
]
,
and the contractions:
(116)
T
1,2
(
x
)
=
M
1
x
+
-
2
5
,
1
5
,
T
1,3
(
x
)
=
M
1
x
+
1
5
,
-
4
5
,
T
1,4
(
x
)
=
M
3
x
+
-
1
3
,
-
1
3
,
T
2,1
(
x
)
=
M
2
x
+
14
8
,
3
8
,
T
3,1
(
x
)
=
M
2
x
+
3
8
,
1
,
T
4,1
(
x
)
=
M
4
x
+
5
4
,
5
4
,
T
5,1
(
x
)
=
M
4
x
+
-
2
4
,
1
4
,
T
5,2
(
x
)
=
M
3
x
+
-
1,0
,
T
5,6
(
x
)
=
M
1
x
+
(
1,0
)
,
T
5,7
(
x
)
=
M
1
x
+
0
,
-
3
5
,
T
5,8
(
x
)
=
M
3
x
+
2
3
,
-
2
3
,
T
6,5
(
x
)
=
M
2
x
+
-
5
8
,
0
,
T
7,5
(
x
)
=
M
2
x
+
0
,
11
8
,
T
8,5
(
x
)
=
M
4
x
+
(
-
1,1
)
,
where
(117)
M
1
=
4
5
0
0
4
5
,
M
2
=
5
8
0
0
5
8
,
M
3
=
2
3
0
0
2
3
,
M
4
=
3
4
0
0
3
4
.
Figures 4 and 5 present K
+
(
C
2
) and K
-, respectively.Figure 3
An MW-graphH with C
(
H
)
=
{
C
1
,
C
2
}.Figure 4
The setK
+
(
C
2
).Figure 5
The setK
-.


## 6. Attractor of anH-IFS and Subsets of C
(
H
)

We obtain other pieces of information on the attractor of the graph-directed iterated function system by considering subsets ofC
(
H
).Theorem 18.
LetH
=
(
V
(
H
)
,
E
(
H
)
) be an MW-directed graph. Let {
T
i
,
j
}
H be an H-IFS and K its attractor. Then the following statements hold: (1)
for everyS
⊂
C
(
H
), there exists K
+
(
S
)
⊂
K such that
(a)
K
+
(
C
)
⊂
K
+
(
S
) for every C
∈
S;
(b)
K
i
+
(
C
)
=
K
i
+
(
S
) for every i
∈
V
(
C
) and every maximal element C
∈
S;
(c)
K
i
+
(
S
)
≠
∅ if and only if i
∈
⋃
C
∈
S
‍
[
C
]
←
(2)
ifS
1
,
S
2
⊂
C
(
H
) are such that, for every C
1
∈
S
1, there exists C
2
∈
S
2 such that C
1
⪯
C
2, then K
+
(
S
1
)
⊂
K
+
(
S
2
),
(3)
forS
1
,
S
2
⊂
C
(
H
), one has
(118)
K
+
(
S
1
)
∩
K
+
(
S
2
)
=
∅
i
f
⋃
C
∈
S
1
‍
C
←
∩
⋃
C
∈
S
2
‍
C
←
=
∅
(4)
the attractorK is such that K
=
K
+
(
C
(
H
)
).Proof.
(
1
) By Proposition 14 and Lemma 15, the map F
:
X
→
X defined in (65), (66), and (67) is a G-contraction satisfying condition (ii) of Theorem 9. Also, from the proof of Theorem 16, we know that fixed points of F are included in K.
LetS
⊂
C
(
H
). We want to show that there exists K
+
(
S
) a fixed point of F satisfying the required properties. Fix
(119)
A
^
0
=
A
^
1
0
,
…
,
A
^
p
0
∈
X
such
that
A
^
i
0
≠
∅
⟺
i
∈
⋃
C
∈
S
‍
V
C
,
A
^
i
0
=
A
i
0
if
i
∈
V
C
for
C
∈
S
,
where
A
0
is
defined
in
(
9
2
)
.
For n
∈
N
∪
{
0
}, we choose inductively
(120)
A
^
n
+
1
∈
F
A
^
n
the
biggest
element
of
F
A
^
n
.
Arguing as in the proof of Theorem 16, one deduces that {
A
^
n
} is a G
1-Picard trajectory converging to some K
+
(
S
)
∈
X a fixed point of F. Also, K
+
(
S
) is independent of A
^
0 chosen as in (119).
ForC
∈
S, observe that A
n
⊂
A
^
n for all n
∈
N
∪
{
0
}, where A
n is defined in (92) and (93). Since A
^
n
→
K
+
(
S
) and A
n
→
K
+
(
C
), we deduce that
(121)
K
+
(
C
)
⊂
K
+
(
S
)
.
It follows from this inclusion and Theorem16(1)(b) that
(122)
K
i
+
S
≠
∅
∀
i
∈
⋃
C
∈
S
‍
C
←
.
On the other hand,
(123)
A
^
i
n
=
∅
∀
i
∉
⋃
C
∈
S
‍
C
←
,
∀
n
∈
N
.
Thus, (1)(c) holds.
In the particular case whereC
∈
S is maximal, one has
(124)
A
i
n
=
A
^
i
n
∀
i
∈
V
(
C
)
,
∀
n
∈
N
∪
{
0
}
,
where A
n is defined in (93). Since
(125)
A
i
n
⟶
K
i
+
C
,
A
^
i
n
⟶
K
i
+
(
S
)
,
one has
(126)
K
i
+
(
C
)
=
K
i
+
(
S
)
∀
i
∈
V
(
C
)
.
((2) and (3)) The proofs are, respectively, analogous to those of (2) and (3) in Theorem16.
(4) LetS
=
C
(
H
). Since K
+
(
C
(
H
)
) is independent of the choice of A
^
0 in (119), we can fix
(127)
A
^
0
=
(
A
^
1
0
,
…
,
A
^
p
0
)
∈
X
such
that
A
^
i
0
=
K
i
,
if
i
∈
V
c
,
∅
,
if
i
∈
V
e
,
where V
c and V
e are defined in (24) and (25), respectively. Let A
^
n be defined as in (120). We know that A
^
n
→
K
+
(
C
(
H
)
). On the other hand, since K is the unique attractor of this H-IFS obtained in Theorem 6, we deduce that K
=
K
+
(
C
(
H
)
).In the following result, we see that the maximal elements ofC
(
H
) play a key role.Corollary 19.
LetH
=
(
V
(
H
)
,
E
(
H
)
) be an MW-directed graph and {
T
i
,
j
}
H an H-IFS. Then, for every S
1
,
S
2
⊂
C
(
H
) such that
(128)
C
∈
S
1
:
C
i
s
a
m
a
x
i
m
a
l
e
l
e
m
e
n
t
o
f
S
1
=
C
∈
S
2
:
C
i
s
a
m
a
x
i
m
a
l
e
l
e
m
e
n
t
o
f
S
2
,
one has
(129)
K
+
(
S
1
)
=
K
+
(
S
2
)
.Proof.
LetS
⊂
C
(
H
) and let
(130)
S
m
=
C
∈
S
:
C
is
a
maximal
element
of
S
.
To conclude, it is sufficient to show that
(131)
K
+
(
S
)
=
K
+
(
S
m
)
.
It follows from Theorem 18(2) that
(132)
K
+
S
⊂
K
+
S
m
,
K
+
(
S
m
)
⊂
K
+
(
S
)
.


## 7. Other Fixed Points of OurG-Contraction

In the proofs of Theorems16 and 18, K
+
(
C
) and K
+
(
S
) were obtained as fixed points of the multivalued G-contraction F. In fact, much more fixed points of F can be obtained in order to get more information on the attractor K.LetS
⊂
C
(
H
). For a vertex i
∈
V
e, we consider the set of edges from i on a path to some vertex in S
:
(133)
E
i
(
S
)
=
∅
,
if
i
∉
⋃
C
∈
S
‍
C
←
,
i
,
j
∈
E
H
:
i
,
j
∈
⋃
C
∈
S
‍
C
←
,
otherwise
.
Similarly, for C
^
∈
C
(
H
), we consider
(134)
E
C
^
(
S
)
=
∅
,
if
V
(
C
^
)
⊄
⋃
C
∈
S
‍
C
←
,
⋃
C
∈
S
‍
(
i
,
j
)
∈
E
(
H
)
:
i
∈
V
(
C
^
)
,
j
∉
V
(
C
^
)
,
j
∈
⋃
C
∈
S
‍
C
←
,
otherwise
.
Finally, we consider suitable subsets of edges on paths in H reaching S, that is, subsets of E
i
(
S
) and E
C
^
(
S
),
(135)
Q
S
=
Q
=
Q
i
i
∈
V
e
×
Q
C
^
C
^
∈
C
(
H
)
:
Q
C
^
⊂
E
C
^
(
S
)
∀
C
^
∈
C
H
,
∀
i
∈
V
e
,
Q
i
⊂
E
i
S
,
Q
i
≠
∅
if
E
i
(
S
)
≠
∅
Q
C
^
C
^
∈
C
(
H
)
.UsingQ
(
S
), we can obtain more information on K
+
(
S
).Theorem 20.
LetH
=
(
V
(
H
)
,
E
(
H
)
) be an MW-directed graph and {
T
i
,
j
}
H an H-IFS. Then, the following statements hold. (1)
For everyS
⊂
C
(
H
) and every Q
∈
Q
(
S
), there exists K
(
S
,
Q
)
∈
X such that
(a)
K
(
S
,
Q
)
⊂
K
+
(
S
);
(b)
K
i
(
S
,
Q
)
≠
∅ if and only if i
∈
⋃
C
∈
S
‍
[
C
]
←;
(c)
K
i
(
S
,
Q
)
=
K
i
+
(
S
) for every i
∈
V
(
C
) and every C
∈
S maximal element in S.
(2)
For everyS
⊂
C
(
H
), if Q
,
Q
^
∈
Q
(
S
) are such that Q
⊂
Q
^, then K
(
S
,
Q
)
⊂
K
(
S
,
Q
^
).
(3)
LetS
1
,
S
2
⊂
C
(
H
) be such that S
1
⊂
S
2. If Q
∈
Q
(
S
1
)
∩
Q
(
S
2
), then K
(
S
1
,
Q
)
⊂
K
(
S
2
,
Q
).
(4)
LetS
1
,
S
2
⊂
C
(
H
) be such that, for every C
1
∈
S
1, there exists C
2
∈
S
2 such that C
1
⪯
C
2. If Q
1
∈
Q
(
S
1
) and Q
2
∈
Q
(
S
2
) are such that Q
1
⊂
Q
2, then K
(
S
1
,
Q
1
)
⊂
K
(
S
2
,
Q
2
).
(5)
For everyS
⊂
C
(
H
) and every Q
∈
Q
(
S
), K
i
-
⊂
K
i
(
S
,
Q
) for every i
∈
V
(
C
^
) and every C
^
∈
C
(
H
) such that V
(
C
^
)
⊂
⋃
C
∈
S
‍
[
C
]
←.Proof.
(
1
) Let Q
∈
Q
(
S
). From Proposition 14 and Lemma 15, F
:
X
→
X the multivalued map defined in (65), (66), and (67) is a G-contraction satisfying condition (ii) of Theorem 9. We want to show that there exists K
(
S
,
Q
) a fixed point of F satisfying the required properties.
Fix(136)
A
n
(
S
,
Q
)
=
A
^
n
∈
X
∀
n
=
0
,
…
,
p
,
where A
^
n is defined in (119) and (120). From the definition of F, we can observe that
(137)
A
p
(
S
,
Q
)
=
A
1
p
S
,
Q
,
…
,
A
p
p
S
,
Q
∈
X
is such that
(138)
A
i
p
(
S
,
Q
)
≠
∅
⟺
i
∈
⋃
C
∈
S
‍
C
←
.
Moreover, for every i
∈
V
e,
(139)
Q
i
⊂
E
i
A
p
S
,
Q
∀
i
∈
⋃
C
∈
S
‍
C
←
,
Q
i
=
E
i
A
p
S
,
Q
=
∅
∀
i
∉
⋃
C
∈
S
‍
C
←
,
where E
i
(
A
p
(
S
,
Q
)
) is defined in (58). Similarly, for every C
^
∈
C
(
H
),
(140)
Q
C
^
⊂
E
C
^
A
p
S
,
Q
if
V
(
C
^
)
⊂
⋃
C
∈
S
‍
C
←
,
Q
C
^
=
E
C
^
A
p
S
,
Q
=
∅
if
V
(
C
^
)
⊄
⋃
C
∈
S
‍
C
←
,
where E
C
^
(
A
p
(
S
,
Q
)
) is defined in (60).
Forn
>
p, we choose inductively
(141)
A
n
(
S
,
Q
)
=
A
1
n
S
,
Q
,
…
,
A
p
n
S
,
Q
∈
F
A
n
-
1
S
,
Q
with
(142)
A
i
n
(
S
,
Q
)
=
∅
,
if
i
∉
⋃
C
∈
S
‍
C
←
,
U
i
e
A
n
-
1
S
,
Q
,
Q
i
,
if
i
∈
V
e
∩
⋃
C
∈
S
‍
C
←
,
W
i
A
n
-
1
S
,
Q
∪
U
i
c
A
n
-
1
S
,
Q
,
Q
C
^
,
if
C
^
∈
C
H
,
i
∈
V
(
C
^
)
∩
⋃
C
∈
S
‍
C
←
,
where U
i
e, U
i
c, and W
i are defined in (59), (63), and (64), respectively.
Arguing as in the proof of Theorem16, one deduces that {
A
n
(
S
,
Q
)
} is a G
1-Picard trajectory converging to some K
(
S
,
Q
)
∈
X a fixed point of F. So, K
(
S
,
Q
) satisfies (1)(b). Again, it can be shown that K
(
S
,
Q
) is independent of A
0
(
S
,
Q
) chosen as in (136).
Observe that(143)
A
n
(
S
,
Q
)
=
A
1
n
S
,
Q
,
…
,
A
p
n
S
,
Q
⊂
A
^
n
=
A
^
1
n
,
…
,
A
^
p
n
∀
n
,
where A
^
n is defined in (120) and A
^
n
→
K
+
(
S
). Moreover, for every C maximal element in S, E
C
(
S
)
=
∅ and
(144)
A
i
n
(
S
,
Q
)
=
A
^
i
n
∀
i
∈
V
(
C
)
.
Therefore, K
(
S
,
Q
) satisfies (1)(a),(c).
(2) LetQ
,
Q
^
∈
Q
(
S
) be such that Q
⊂
Q
^. From (141) and (142), one sees that
(145)
A
n
(
S
,
Q
)
⊂
A
n
(
S
,
Q
^
)
∀
n
∈
N
.
Since A
n
(
S
,
Q
)
→
K
(
S
,
Q
) and A
n
(
S
,
Q
^
)
→
K
(
S
,
Q
^
), one has that
(146)
K
(
S
,
Q
)
⊂
K
(
S
,
Q
^
)
.
(3) LetS
1
,
S
2
⊂
C
(
H
) be such that S
1
⊂
S
2 and let Q
∈
Q
(
S
1
)
∩
Q
(
S
2
). From (141) and (142), one sees that
(147)
A
n
(
S
1
,
Q
)
⊂
A
n
(
S
2
,
Q
)
∀
n
∈
N
.
Since A
n
(
S
1
,
Q
)
→
K
(
S
1
,
Q
) and A
n
(
S
2
,
Q
)
→
K
(
S
2
,
Q
), one has that
(148)
K
(
S
1
,
Q
)
⊂
K
(
S
2
,
Q
)
.
(4) LetS
1
,
S
2
⊂
C
(
H
) be such that, for every C
1
∈
S
1, there exists C
2
∈
S
2 such that C
1
⪯
C
2. One has
(149)
i
∈
⋃
C
1
∈
S
1
‍
C
1
←
⊂
i
∈
⋃
C
2
∈
S
2
‍
C
2
←
.
Let Q
1
∈
Q
(
S
1
) and Q
2
∈
Q
(
S
2
) be such that Q
1
⊂
Q
2. Fix
(150)
B
p
S
1
,
Q
1
=
B
1
p
S
1
,
Q
1
,
…
,
B
p
p
S
1
,
Q
1
∈
X
to be such that
(151)
B
j
p
S
1
,
Q
1
=
K
j
S
2
,
Q
2
,
if
j
∈
⋃
C
1
∈
S
1
‍
C
1
←
,
∅
,
if
j
∉
⋃
C
1
∈
S
1
‍
C
1
←
.
One has (
K
(
S
1
,
Q
1
)
,
B
p
(
S
1
,
Q
1
)
)
∈
E
(
G
) and K
(
S
1
,
Q
1
)
∈
F
(
K
(
S
1
,
Q
1
)
). For n
=
p
+
1, we define
(152)
B
n
(
S
1
,
Q
1
)
=
B
1
n
S
1
,
Q
1
,
…
,
B
p
n
S
1
,
Q
1
∈
F
B
p
S
1
,
Q
1
by
(153)
B
i
n
(
S
1
,
Q
1
)
=
∅
,
if
i
∉
⋃
C
∈
S
1
‍
C
←
,
U
i
e
B
p
S
1
,
Q
1
,
Q
i
1
,
if
i
∈
V
e
∩
⋃
C
∈
S
1
‍
C
←
,
W
i
B
p
S
1
,
Q
1
∪
U
i
c
B
p
S
1
,
Q
1
,
Q
C
^
1
,
if
C
^
∈
C
H
,
i
∈
V
C
^
∩
⋃
C
∈
S
1
‍
C
←
.
Since B
p
(
S
1
,
Q
1
)
⊂
K
(
S
2
,
Q
2
), K
(
S
2
,
Q
2
)
∈
F
(
K
(
S
2
,
Q
2
)
), Q
1
⊂
Q
2 and using the definitions of F and K
(
S
2
,
Q
2
), we deduce that B
p
+
1
(
S
1
,
Q
1
)
⊂
K
(
S
2
,
Q
2
). Also, (
K
(
S
1
,
Q
1
)
,
B
p
+
1
(
S
1
,
Q
1
)
)
∈
E
(
G
). Arguing as in the proof of Proposition 14, one has
(154)
d
K
S
1
,
Q
1
,
B
p
+
1
S
1
,
Q
1
≤
λ
d
K
S
1
,
Q
1
,
B
p
S
1
,
Q
1
.
Repeating this argument, we obtain for every n
≥
p, B
n
(
S
1
,
Q
1
)
∈
K
(
S
2
,
Q
2
) such that B
n
(
S
1
,
Q
1
)
→
K
(
S
1
,
Q
1
). Therefore,
(155)
K
(
S
1
,
Q
1
)
⊂
K
(
S
2
,
Q
2
)
.
(5) LetS
⊂
C
(
H
) and C
^
∈
C
(
H
) be such that V
(
C
^
)
⊂
⋃
C
∈
S
‍
[
C
]
←. Let
(156)
Q
=
Q
i
i
∈
V
e
×
Q
C
C
∈
C
(
H
)
∈
Q
(
S
)
.
We define
(157)
Q
^
=
Q
^
i
i
∈
V
e
×
Q
^
C
C
∈
C
(
H
)
by
(158)
Q
^
i
=
Q
i
,
if
i
∈
V
e
,
E
i
(
C
^
)
≠
∅
,
∅
,
if
i
∈
V
e
,
E
i
(
C
^
)
=
∅
;
Q
^
C
=
∅
,
for
C
∈
C
H
.
Clearly, Q
^
∈
Q
(
C
^
) and Q
^
⊂
Q. It follows from (2), (4), and Theorem 16(4) that
(159)
K
C
^
,
Q
^
⊂
K
S
,
Q
,
K
i
(
C
^
,
Q
^
)
=
K
i
+
(
C
^
)
=
K
i
-
∀
i
∈
V
(
C
^
)
.Example 21.
Let{
T
i
,
j
}
H be the H-IFS considered in Example 17. One has C
(
H
)
=
{
C
1
,
C
2
}, V
e
=
∅, E
C
2
(
C
2
)
=
∅, and E
C
1
(
C
2
)
=
{
(
5,1
)
,
(
5,2
)
}. For k
=
1,2 let Q
k
=
Q
C
1
k
×
Q
C
2
k
∈
Q
(
C
2
) be given by
(160)
Q
C
1
1
=
5,1
,
Q
C
1
2
=
5,2
,
Q
C
2
1
=
Q
C
2
2
=
∅
.
Figures 6 and 7 present K
(
C
2
,
Q
1
) and K
(
C
2
,
Q
2
), respectively. Observe that
(161)
K
C
2
,
Q
1
≠
K
C
2
,
Q
2
,
K
C
2
,
Q
1
⊊
K
+
C
2
,
K
C
2
,
Q
2
⊊
K
+
(
C
2
)
,
where K
+
(
C
2
) is presented in Figure 4.Figure 6
The setK
(
C
2
,
Q
1
).Figure 7
The setK
(
C
2
,
Q
2
).



---

*Source: 345856-2015-01-28.xml*