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Dendrite-type attractors of IFSs formed by two injective functions
Chaos, Solitons and Fractals 116 (2018) 433–438
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Dendrite-type attractors of IFSs formed by two injective functions Dan DUMITRU Faculty of Engineering, Computer Science and Geography, Spiru Haret University of Bucharest, 46G Fabricii Str, Bucharest, Romania
a r t i c l e
i n f o
Article history: Received 29 October 2017 Revised 11 September 2018 Accepted 17 September 2018
MSC: 28A80
a b s t r a c t The aim of this paper is to study the dendrite-type attractors of an iterated function system formed by two injective functions. We consider (X, d) a complete metric space and S = (X, {f0 , f1 }) an iterated function system (IFS), where f0 , f1 : X −→ X are injective functions and A is the attractor of S. Moreover, we suppose that f0 (A ) ∩ f1 (A ) = {a} and {a} = π (0m 1∞ ) = π (1n 0∞ ) with m, n ≥ 1, where π is the canonical projection on the attractor. We compute the connected components of the sets A\{π (0∞ )}, A\{π (1∞ )}, A\{π (0m 1∞ ) = π (1n 0∞ )} and deduce there are infinitely-many (countably) non-homeomorphic dendritetype attractors of iterated function systems formed by two injective functions. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Attractor Iterated function system Dendrite Connected component
1. Introduction Iterated function systems were conceived in the present form by J. Hutchinson in [9], popularized by M. Barnsley in [1] and are one of the most common and general ways to generate fractals. Many of the important examples of functions and sets with special and unusual properties turn out to be fractal sets or functions whose graphs are fractal sets and a great part of them are attractors of iterated function systems. There is a current effort to extend the classical Hutchinson’s framework to more general spaces and infinite iterated function systems or, more generally, to multifunction systems and to study them ([10,11,14,15,17]). Such example can be found in [12], where the Lipscomb’s space, which is an important example in dimension theory, can be obtained as an attractor of an infinite iterated function system defined in a very general setting. In those settings the attractor can be a closed bounded set, in contrast with the classical theory where only compact sets are considered. Although the fractal sets are defined with measure theory, being sets with non-integer Hausdorff dimension ([6–8,18,22]), it turns out that they have interesting topological properties ([3–5]). The topological properties of fractal sets have a great importance in analysis on fractals as we can see in ([5,10,11]). Generalized iterated function systems can be found in ([13,18–20]). Topological versions of an iterated function system have been studied in ([2,16,21]). In this article we intend to characterize the dendrites which are attractors of iterated function systems composed by two injective
functions. For a metric space (X, d), we denote by K∗ (X ) the set of nonempty compact subsets of X. Definition 1.1. Let (X, d) be a metric space. The function h : K∗ (X ) × K∗ (X ) −→ [0, +∞ ) defined by h(A, B ) = max (d(A, B), d(B, A)), where d(A, B) = sup d (x, B ) = sup infd (x, y ) x∈A
x∈A
y∈B
is called the
Hausdorff-Pompeiu metric. Remark 1.1 ([18]). The space (K∗ (X ), h ) is complete if (X, d) is complete, compact if (X, d) is compact and separable if (X, d) is separable. Definition 1.2. Let (X, d) be a metric space. For a function f : X −→ X let us denote by Lip( f ) ∈ [0, +∞] the Lipschitz constant associd ( f (x ), f (y )) ated to f, which is Lip( f ) = sup . We say that f is d (x, y ) x,y∈X ; x=y a Lipschitz function if Lip( f ) < +∞ and a contraction if Lip(f) < 1. Definition 1.3. An iterated function system (IFS) on a complete metric space (X, d) consists of a finite family of contractions { fk }k=1,n , fk : X −→ X for every k ∈ {1, 2, . . . , n} and it is denoted by S = (X, { fk }k=1,n ). Definition 1.4. For an iterated function system S = (X, { fk }k=1,n ), n the function FS : K∗ (X ) −→ K∗ (X ) defined by FS (B ) = fk (B ) is k=1
E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2018.09.031 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
called the fractal operator associated to the iterated function system S.
434
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
Remark 1.2 ([1]). The function FS is a contraction satisfying Lip(FS ) ≤ max Lip( fk ). k=1,n
Using Banach’s contraction theorem there exists, for an iterated function system S = (X, { fk }k=1,n ), a unique set A ∈ K∗ (X ) such that FS (A ) = A, which is called the attractor of the iterated function system S. More precisely we have the following well-known result.
(3) A =
{aω }, Aα =
ω ∈
{aαω } for every α ∈ ∗ , A =
ω ∈
for every m ∈ N∗ and, more general, Aα =
ω ∈ m
ω ∈ m
Aω
Aαω for every α ∈ ∗
and every m ∈ N∗ . (4) The set {e[ω]m | ω ∈ and m ∈ N∗ } is dense in A. (5) The function π : −→ A defined by π (ω ) = aω is continuous and surjective. 2. Main results
Theorem 1.1 ([1,8,18]). Let (X, d) be a complete metric space and S = (X, { fk }k=1,n ) an iterated function system with c = max Lip( fk ) < 1. k=1,n
Then there exists a unique set A = A(S ) ∈ K∗ (X ) such that FS (A ) = A. Moreover, for any H0 ∈ K∗ (X ) the sequence (Hn )n ≥ 1 defined by Hn+1 = FS (Hn ) is convergent to A. For the speed of the convergence cn we have the following estimate: h(Hn , A ) ≤ h(H0 , H1 ) for every 1−c n ≥ 1. Now we briefly present the shift space of an iterated function system. For more details one can see [11]. We start with some set notations: N denotes the set of natural numbers, N∗ = N \ {0}, N∗n = {1, 2, . . . , n}. For two nonempty sets A and B, BA denotes the set of functions from A to B. By = (B ) we will under∗ stand the set BN and by n = n (B ) we will understand the set ∗ ∗ BNn . The elements of = (B ) = BN will be written as infinite words ω = ω1 ω2 . . . ωm ωm+1 . . . , where ωm ∈ B and the elements of ∗ n = n (B ) = BNn will be written as finite words ω = ω1 ω2 . . . ωn . By λ we will understand the empty word. Let us remark that ∗ (B ) we will understand the set of all finite 0 (B) = {λ}. By ∗ = words ∗ = ∗ (B ) = n ( B ) . n≥0
We denote by |ω| the length of the word ω. An element of = (B ) is said to have length +∞. If ω = ω1 ω2 . . . ωm ωm+1 . . . or if ω = ω1 ω2 . . . ωn and n ≥ m, then [ω]m : = ω1 ω2 . . . ωm . More generally, if l < m then [ω]lm = ωl+1 ωl+2 . . . ωm and we have [ω]m = [ω]l [ω]lm for every ω ∈ n (B), where n ≥ m > l ≥ 1. For two words α , β ∈ ∗ (B) ∪ (B), α ≺β means that |α | ≤ |β | and [β ]|α| = α . For α ∈ n (B) and β ∈ m (B) or β ∈ (B) by αβ we will understand the joining of the words α and β , namely αβ = α1 α2 . . . αn β1 β2 . . . βm and respectively αβ = α1 α2 . . . αn β1 β2 . . . βm βm+1 . . .. ∗ On = (N∗n ) = (N∗n )N , we can consider the metric ds (α , β ) = ∞
β 1−δα k
1, if x = y y k , where δx = and α = α1 α2 . . . , β = β1 β2 . . .. 3k 0, if x = y k=1 Let (X, d) be a complete metric space, S = (X, { fk }k=1,n ) an iterated function system on X and A = A(S ) the attractor of the iterated function system S. For ω = ω1 ω2 . . . ωm ∈ m (N∗n ), fω denotes fω1 ◦ fω2 ◦ . . . ◦ fωm , where fωm is applied first and Hω denotes fω (H) for a subset H ⊂ X. By Hλ we will understand the set H. In particular Aω = fω (A ). Moreover, we denote by 0∞ = 0 0 0 . . . ∈ ({0, 1} ) and 1∞ = 111 . . . ∈ ({0, 1} ). The main results concerning the relation between the attractor of an iterated function system and the shift space is contained in the following theorem. The function π : −→ A = A(S ) from the theorem below is called the canonical projection from the shift space onto the attractor of an iterated function system S. Theorem 1.2 ([11]). Let (X, d) be a complete metric space. If A = A(S ) is the attractor of an iterated function system S = (X, { fk }k=1,n ) with c = max Lip( fk ) < 1, then: k=1,n
(1) For every ω ∈ = (N∗n ) we have A[ω]m+1 ⊂ A[ω]m and d (A[ω]m ) −→ 0 when m −→ ∞. More precisely d (A[ω]m ) ≤ cm d (A ), where d (M ) = sup d (x, y ) is the diameter of a set M. x,y∈M (2) If aω is defined by {aω } = A[ω]m , then d (e[ω]m , aω ) −→ 0 m≥1
when m −→ ∞, where e[ω]m is the unique fixed point of f[ω]m .
In general the dendrites represent the topological analogue of the trees. Characterizations of dendrite-type attractors can be found in [3] and [5]. Definition 2.1. a) X is arcwise connected if for every x, y ∈ X there exists a continuous function ϕ : [0, 1] −→ X such that ϕ (0 ) = x and ϕ (1 ) = y. A continuous function ϕ as above is called a path between x and y. We say that two continuous and injective functions ϕ , ψ : [0, 1] −→ X are equivalent if there exists a function u : [0, 1] −→ [0, 1] continuous, bijective and increasing such that ϕ ◦ u = ψ . A class of equivalence is named a curve. b) If X is compact, connected and locally connected, then X is called a dendrite if for every x, y ∈ X there exists a unique equivalence class of continuous and injective functions ϕ : [0, 1] −→ X such that ϕ (0 ) = x and ϕ (1 ) = y (i.e. there exists a unique injective curve joining x with y). We remark that two equivalent, continuous and injective functions have the same images. We also consider that the empty set is a dendrite. c) Let (Ai )i ∈ I be a family of nonempty subsets of X. Then the graph (I, G), where G = {(i, j ) | i, j ∈ I such that Ai ∩ Aj = ∅ and i = j} is called the graph of intersections associated to the family (Ai )i ∈ I . d) A graph (I, G) is called connected if for every i, j ∈ I there exist (ik )k=1,n ⊂ I such that i1 = i, in = j and ik , ik+1 ∈ G for every k ∈ {1, 2, . . . , n − 1}. A family of vertices (i1 , . . . , im ) is called a cycle if ik , ik+1 ∈ G for every k ∈ {1, . . . , m} and ik ∈ / {ik+1 , ik+2 } for every k ∈ {1, . . . , m}, where by im+1 we understand i1 and by im+2 we understand i2 . A graph (I, G) is called a tree if it is connected and has no cycles. The following result gives a characterization of dendrites as attractors of some iterated function systems. Theorem 2.1 ([3]). Let (X, d) be a complete metric space and S = (X, { fk }k=1,n ) an iterated function system with c = max Lip( fk ) < 1. k=1,n
We denote by A = A(S ) the attractor of S, by Ak the set fk (A) for every k ∈ {1, . . . , n} and by ( (1, . . . , n ), G ) the graph of intersections associated with the family of sets (Ak )k=1,n . We suppose that the following conditions are true: (a) The set Ai ∩ Aj is totally disconnected for every i, j ∈ {1, 2, . . . , n} different. (b) Ai ∩ A j ∩ Ak = ∅ for every i, j, k ∈ {1, 2, . . . , n} pairwise different. (c) fk : X −→ X is an injective function for every k ∈ {1, 2, . . . , n}. Then the following statements are equivalent: (1) A is a dendrite. (2) The graph ( (1, . . . , n ), G ) is a tree and card(Ai ∩ Aj ) ∈ {0, 1} for every i, j ∈ {1, 2, . . . , n} different. In [10] and [11] were studied the critical points of an attractor of an iterated function system, some connections with the canonical projection and attractors that are homeomorphic to a quotient space of the shift space by some equivalence relation. We consider the following properties for Hata’s tree and Koch’s curve: Hata’s tree, let us say A, is the attractor of the injective functions defined in the complex plane f0 (z ) = cz and f1 (z ) = (1 − |c|2 )z + |c|2 , where c ∈ C and |c|, |1 − c| ∈ (0, 1 ), A 0 ∩ A 1 = f0 (A ) ∩ f1 (A ) = {|c|2 } and the intersection point satisfies |c|2 = π (001∞ ) =π (10∞ ); Koch’s curve, let us say B, is the attractor of
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
the injective functions defined in the complex plane f 0 (z ) = α z and f1 (z ) = (1 − α )(z − 1 )+ 1, where α ∈ C such that |α|2 + |1 − α|2 < 1, B0 ∩ B1 = f0 (B ) ∩ f1 (B ) = {α} and the intersection point satisfies {α} = π (01∞ ) =π (10∞ ). Starting from these properties we will study in this paper the iterated function systems S = (X, { f0 , f1 }), with f0 , f1 : X −→ X injective and A = A(S ) the attractor of S such that f0 (A ) ∩ f1 (A ) = {a} and {a} = π (0m 1∞ ) = π (1n 0∞ ) with m, n ≥ 1, where π is the canonical projection on the attractor. The purpose of that is to prove there are infinitely-many (countably) non-homeomorphic dendrite-type attractors formed by two injective functions. Such attractors are counted by the number of connected components, after we have removed three distinct points. We start our construction with the following lemma. Lemma 2.1. Let ({0, 1}) be the shift space formed by two symbols and m, n ∈ N∗ be fixed and arbitrarily chosen. Let the relation ∼ be defined by: α , β ∈ ({0, 1}), α ∼ β if and only if α = β or α = β and there exists γ ∈ ∗ ({0, 1}) such that {α , β} = {γ 0m 1∞ , γ 1n 0∞ }, where 0∞ = 0 0 0 . . . ∈ ({0, 1} ) and 1∞ = 111 . . . ∈ ({0, 1} ). Then ∼ is an equivalence relation. Proof. The reflexivity α ∼ α and the symmetry α ∼ β ⇒β ∼ α are obvious. For the transitivity suppose α ∼ β and β ∼ η. If α = β or β = η, then it is clear. Suppose now that α = β and β = η. Then there exist γ 1 , γ 2 ∈ ∗ ({0, 1}) such that {α , β} = {γ 1 0m 1∞ , γ1 1n 0∞ } and {β , η} = {γ2 0m 1∞ , γ2 1n 0∞ }. Suppose also that α = γ 1 0m 1∞ and β = γ1 1n 0∞ . Since {β , η} = {γ 2 0m 1∞ , γ 2 1n 0∞ }, we obtain β = γ2 1n 0∞ and η = γ2 0m 1∞ . But β = γ1 1n 0∞ = γ 2 1n 0∞ implies γ1 = γ2 = γ and hence α = η = γ 0m 1∞ . Similar, if α = γ1 1n 0∞ and β = γ 1 0m 1∞ we obtain γ1 = γ2 = γ and hence α = η = γ 1n 0∞ . Thus α ∼ η and we obtain the transitivity. Remark 2.1. (a) We consider the quotient space A = ({0, 1}) ∼, where ∼ is considered in Lemma 2.1 and the canonical : ({0, 1} ) −→ A, π (α ) = α
for projection on the quotient space π every α ∈ ({0, 1}). We remark that:
αˆ =
⎧ m ∞ n ∞ ⎨ {γ 0 1 , γ 1 0 } ⎩
{α}
γ ∈ ({0, 1}) and α = γ 0 1 or α = γ 1n 0∞ .
, if
∗
m ∞
, otherwise.
(b) Let the functions F0 , F1 : ({0, 1} ) −→ ({0, 1} ) be defined by F0 (x ) = 0x, F1 (x ) = 1x and the functions f0 , f1 : A −→ A be de (x )) = π (F0 (x )) = π (0x ) and f1 (π (x )) = π (F1 (x )) = fined by f0 (π (1x ). We remark that f0 and f1 are well-defined on A. (c) Using π Theorem 2.4 from ([10]), there exists a metric d on A such that f0 and f1 are contractions with respect to the considered metric d . In this way the metric space (A, d ) is Hausdorff, compact being the quotient space of the compact space ({0, 1}) and complete. Lemma 2.2. The functions f0 , f1 : A −→ A are injective, f0 (A ) ∪ f1 (A ) = A and f0 (A ) ∩ f1 (A ) = {a}, where {a} =π (0m 1∞ ) = π (1n 0∞ ). Proof. We have that f0 (π˜ (x )) = f0 (π˜ (y )) ⇒ 0x ∼ 0y. Then, or (x ) = π (y ) and it is clear, or there ex0x = 0y ⇒ x = y ⇒ π ists γ = γ1 γ2 . . . .γ p ∈ ∗ ({0, 1}) such that 0x = γ 0m 1∞ and 0y = γ 1n 0∞ . In the second case we have 0x = γ1 γ2 . . . .γ p 0m 1∞ and 0y = γ1 γ2 . . . .γ p 1n 0∞ . Then necessary γ1 = 0 and hence 0x = 0γ2 . . . .γ p 0m 1∞ and 0y = 0γ2 . . . .γ p 1n 0∞ . This means that x = γ2 . . . .γ p 0m 1∞ and y = γ2 . . . .γ p 1n 0∞ which implies x ∼ y. Hence (x ) = π (y ) and thus f0 is injective. Similar, one can we obtain π prove that f1 is injective. also have the following: f 0 (A ) =
We 0 x | for every x ∈ ({0, 1}) and f 1 (A ) =
1 x | for every x ∈ ({0, 1}) . Thus f (A ) ∪ f (A ) = A. Let {a} = 0
1
435
( 0n 1∞ ) = π (1m 0∞ ). Then {a}⊆f0 (A) ∩ f1 (A). We consider now π b ∈ f0 (A) ∩ f1 (A) arbitrarily chosen. Then there exist x, y ∈ ({0, ( 0x ) = π (1y ). Suppose 0x = γ 0m 1∞ and 1}) such that b = π 1y = γ 1n 0∞ with γ = γ1 γ2 . . . γr ∈ r ({0, 1}), r ≥ 1. Then γ1 = 0 and γ1 = 1 which is a contradiction. Thus γ = λ and b = ( 0n 1∞ ) = π (1m 0∞ ) = a. In this way we have obtained that π f0 (A) ∩ f1 (A)⊆{a}. Hence f0 (A) ∩ f1 (A ) = {a}. Lemma 2.3. The set A is a dendrite. Proof. In the settings of Theorem 2.1 we consider X ≡ A and d ≡ d . Hence the iterated function system S = (X, { f0 , f1 } ), where f0 , f1 : X −→ X becomes in this case S = (A, { f0 , f1 } ), where f0 , f1 : A −→ A. We recall that f0 , f1 are contractions and f0 (A ) ∪ f1 (A ) = A. Then A = A(S ) is the attractor of S. Moreover, f0 , f1 are injective, f0 (A) ∩ f1 (A ) = {a} consists of a single point and the graph ((0, 1), G) of intersections has only one edge, namely (0,1), and thus it is a tree. Hence, using Theorem 2.1, we obtain that A is a dendrite. Taking into consideration the above lemmas and remarks we have the following theorems. Theorem 2.2. If m = n = 1, then A is homeomorphic to the interval [0,1]. Proof. We are going to define a map h : A −→ [0, 1]. Then we prove that it is a homeomorphism. Let α ∈ ({0, 1}).
) = 12 ∈ [0, 1]. i) If α = 01∞ or α = 10∞ , then we consider h(α ∗ ii) If there exists γ = γ1 . . . γ p ∈ ({0, 1}), p ≥ 1, γ i ∈ {0, 1} for every i ∈ {1, . . . , p} such that α = γ 01∞ = γ1 . . . γ p 01∞ or α = p+1 γi
) = γ 10∞ = γ1 . . . γ p 10∞ , then we consider h(α i ∈ [0, 1], i=1
2
where γ p+1 = 1. iii) If there is no γ ∈ ∗ ({0, 1}) such that α = γ 01∞ or
) = α = γ 10∞ , then if α = γ1 . . . γ p . . . ∈ ({0, 1} ) we consider h(α γi i ∈ [0, 1] (remark that (γ i )i ≥ 1 is not stationary in this case). i≥1
2
Remark that in the choice of the function h we have used the decomposition of every number from the interval [0,1] by the form γi i with γ i ∈ {0, 1} for every i ≥ 1 (this also arises from the binary i≥1
2
writing of a subunitary number: if α = α1 α2 . . . ∈ ({0, 1} ), then, αi
) = in fact, h(α i ). i≥1
2
i) Surjectivity: It follows directly from the definition of h. Indeed, if β ∈ [0, 1], then there exist γ i ∈ {0, 1} for every i ≥ 1 such that γi
β= i . Consider now α = γ1 . . . γ p . . . ∈ ({0, 1} ). Then h (α ) = i≥1
β.
2
and α = α α . . . , β =
) = h β ii) Injectivity: Consider h(α 1 2 βi αi
) =
) = 12 , β1 β2 . . .. Then h(α and h β = . a) If h (α i i then
αi i≥1
2i
=
βi i≥1
2i
i≥1
=
1 2
2
i≥1
2
which implies (α1 = 1, αi = 0, for ev-
ery i ≥ 2 or α1 = 0, αi = 1 for every i ≥ 2) and (β1 = 1, βi = 0 for every i ≥ 2 or β1 = 0, βi = 1 for every i ≥ 2). This means that (α = 01∞ or α = 10∞ ) and (β = 01∞ or β = 10∞ ). Hence p+1 γi
. b) If h(α
=β
) = h β α = i ∈ [0, 1], where γ p+1 = 1, then αi i≥1
2i
=
βi i≥1
2i
=
p+1 i=1
γi 2i
=
p γi i=1
every i ∈ {1, . . . , p} and
2i
i=1
+
2
1 . 2 p+1
αi i≥p+1
2i
This implies that αi = βi = γi for
=
βi i≥p+1
2i
=
1 . 2 p+1
Thus we obtain
(α p+1 = 1, αi = 0 for every i ≥ p + 2 or α p+1 = 0, αi = 1 for every i ≥ p + 2) and (β p+1 = 1, βi = 0 for every i ≥ p + 2 or β p+1 = 0, βi = 1 for every i ≥ p + 2). This means that (α = γ1 . . . γ p 01∞ or α = γ1 . . . γ p 10∞ ) and (β = γ1 . . . γ p 01∞ or β = γ1 . . . γ p 10∞ ). γi
. c) If h(α
=β
) = h β Hence α = i with (γ i )i ≥ 1 not stationary, i≥1
2
436
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
then
αi i≥1
2i
=
βi i≥1
2i
=
γi i≥1
implies αi = βi = γi for every i ≥ 1 and
2i
.
=β thus α = β . Hence α iii) Recall a well-known result from topology which states that if f : X −→ Y is continuous and bijective, where X is compact and Y is Hausdorff, then f is a homeomorphism. Thus, returning to our problem, we only have to prove now that h is continuous. Let now [m, M] ⊂ [0, 1] be a fixed closed interval with m < M. a) If m = 2αp and M = 2βr , α , β , p, r ≥ 1, then we consider n = max{ p, r} ∈ Then:
N∗
and
α∗,
h−1 ([m, M] ) = h−1
β∗
α∗ β 2n
,
∗ ∗ such that m = α2n and M = β2n .
≤ 2n
∗
α ≤ ω|base10 ≤ β − 1
α 2
,q p
h
([m, M]) = h
q,
β
2r
which is a closed set. d) If m = q1 and M = q2 such that q1 < q2 with q1 , q2 = β ∈ [0, 1] for every β , r ≥ 1, β ≤ 2r , then we consider 2r
q −q n0 =min n ∈ N∗ | 21n < 2 2 1 . Then:
h−1 ([m, M] ) = h−1 ([q1 , q2 ] )
ω|base10 ≤ q2 · 2n and n ≥ n0
∩
[n] f10∞ (Y ).
Thus
[n]
[m]
[n]
Proof. We denote by c = 0m , d = 1n , fc = f0 and fd = f1 . Then 0m 1∞ = cd∞ and 1n 0∞ = dc∞ . Let γ : [0, 1] −→ Y be continuous, bijective and with the inverse also continuous given by Theorem 2.3 (i.e.γ is a homeomorphism between Y and [0, 1]). [m−1] We consider the injective paths γ , f0 ◦ γ , . . . , f0 ◦ γ . The endpoints of γ are π (0∞ ) and π (1∞ ), the endpoints of f0 ◦γ [m−1] are π (0∞ ) and π (01∞ ) and so on the endpoints of f0 ◦γ m−1 ∞ ∞ are π (0 ) and π 0 1 . Then these paths have a common point, π (0∞ ), and the rest of their images disjoint. are mutually Hence, the points π (1∞ ), π (01∞ ), . . . ,π 0m−1 1∞ lie in differ-
ent connected components, since the paths joining π (0k 1∞ ) and π (0l 1∞ ) have to pass through the point π (0∞ ) for every k = l, k, l ∈ {0, 1, . . . , m − 1}. Then A {π (0∞ )} has m connected components, more exactly, the connected components of the points π (1∞ ), π (01∞ ), . . . , π 0m−1 1∞ which are:
k≥0
K1 = (A01 ) ∪ (A0m+1 1 ) ∪ (A02m+1 1 ) ∪ . . . = ∪ A0km+1 1 ,
k≥0
K2 = (A02 1 ) ∪ (A0m+2 1 ) ∪ (A02m+2 1 ) ∪ . . . = ∪ A0km+2 1 ,
which is a closed set. In this way we have proved that h−1 ([m, M] ) is closed for every [m, M] ⊂ [0, 1]. Hence h is continuous and thus it is a homeomorphism. 2.3.If we consider the iterated function system S1 = Theorem [m] [n] [k] Y, f0 , f1 , where by fa we understand f a ◦ fa ◦ . . . ◦ fa ,
ktimes
a ∈ {0, 1} and Y is the attractor of S1 , then Y⊆A and Y is homeomorphic to the segment [0,1] (i.e. there exists γ : [0, 1] −→ Y continuous, bijective and with the inverse also continuous). [n]
=
[m] f01∞ (Y )
K0 = (A1 ) ∪ (A0m 1 ) ∪ (A02m 1 ) ∪ . . . = ∪ A0mk 1 ,
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1},
[m]
[n] f10∞ (A )
Theorem 2.4. We have that A{π (0∞ )} has m connected components, A{π (1∞ )} has n connected components and A{π (0m 1∞ ) = π (1n 0∞ )} has m + n connected components.
ω|base10 ≤ β and n ≥ n0
[m] = f01∞ (A )∩
ntimes
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1},
q1 · 2n <
ktimes
f01∞ (Y ) ∩ f10∞ (Y ) = {a} and so {a} =π (01∞ ) = π (10∞ ), where π : ({0, 1}) −→ Y is the canonical projection on Y. Hence Y is satisfying the conditions of Theorem 2.2 and thus Y is homeomorphic to the segment [0,1] (in fact Y is the image of the injective path joining the points π (0∞ ) and π (1∞ ), which are the fixed points of f0 and f1 ). [m]
q · 2n <
[n]
f10∞ (A ) =
mtimes
which is a closed set. c) If m = q and M = 2βr , β , r ≥ 1, β ≤ 2r such that q = 2αr ∈ [0, 1] n for every α , r ≥ 1, α ≤ 2r , then we consider ω0 = ω01 . . . ω00 ∈ ∗ ({0, 1}) such that β = ω0|base 10 . Then:
and
f1n 0∞ (A ) = ∩ f1n 00 . . . 0 (A ). k≥1
f11 . . . 10∞ (A )
α ≤ ω|base10 < q · 2n and n ≥ n0
−1
k≥1
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1},
−1
f0m 1∞ (A ) = ∩ f0m 11 . . . 1 (A )
ktimes
which is a closed set. b) If m = 2αp , α , p ≥ 1, α < 2p and M = q such that q = 2βr ∈ [0, 1] n for every β , r ≥ 1, β ≤ 2r , then we consider ω0 = ω01 . . . ω00 ∈ ∗ ({0, 1}) such that α = ω0|base10 . Then:
h−1 ([m, M] ) = h−1
m≥1
[m]
f01∞ (A ) =
by
Since {a} = π (0m 1∞ ) =π (1n 0∞ ), we have {a} = f00 . . . 01∞ (A ) ∩
∗
m≥1
A, where A = A(S ) is the attractor of the iterated function system S = (A, { f0 , f1 }), with f0 , f1 : A −→ A. For ω = ω1 ω2 . . . ωm . . . we denote by f ω (A ) = fω1 ω2 ...ωm ... (A ) = ∩ fω1 ω2 ...ωm (A ). Hence we denote now
2n
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1} and ∗
fractal operator of S = (A, { f0 , f1 }), with f0 , f1 : A −→ A. In this way we obtained Y ⊂ FS (Y ) and if we define the sequence A0 = Y, Am = FS (Am−1 ), then Am−1 ⊆ Am for every m ≥ 1. Thus Y ⊆ Am =
Proof. Let g0 = f0 , g1 = f1 and consider the iterated function system S1 = (Y, {g0 , g1 }), where g0 , g1 : Y −→ Y . Then Y = g0 (Y ) ∪ [m] [n] g1 (Y ) = f0 (Y ) ∪ f1 (Y ) ⊂ f0 (Y ) ∪ f1 (Y ) = FS (Y ), where FS is the
k≥0
. . . . . . . . . . . . . . . . . . .............. . . .
Km−1 = (A0m−1 1 ) ∪ (A0m+(m−1) 1 ) ∪ . . . = ∪ A0km+(m−1) 1 k≥0
[n−1]
We consider the injective paths γ , f1 ◦ γ , . . . , f1 ◦ γ . The endpoints of γ are π (0∞ ) and π (1∞ ), the endpoints of f1 ◦γ are π (10∞ ) and π (1∞ ) and so on the endpoints of f1[n−1] ◦ γ are π 1n−1 0∞ and π (1∞ ). Then these paths have a common point, π (1∞ ), and the rest of their images are mutually disjoint. The points π (0∞ ), π (10∞ ), . . . , π 1n−1 0∞ lie in different connected
components, since the paths joining π (1k 0∞ ) and π (1l 0∞ ) have to pass through the point π (1∞ ) for every k = l, k, l ∈ {0, 1, . . . , n − 1}. Then A {π (1∞ )} has n connected components, more exactly, the connected components of the points π (0∞ ), π (10∞ ), . . . ,
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
π 1n−1 0∞ which are:
From Theorem 2.1 A is a dendrite and A \ { 12 } has two connected components. In the conditions of Theorem 2.4 we have m = 1 and n = 1. Hence we obtain the following connected components: (I) Aࢨ π (0∞ ) has 1 connected component K0 = ∪ A0k 1 =
K 0 = (A0 ) ∪ (A1n 0 ) ∪ (A12n 0 ) ∪ . . . = ∪ A1nk 0 , k≥0
K 1 = (A10 ) ∪ (A1n+1 0 ) ∪ (A12n+1 0 ) ∪ . . . = ∪ A1kn+1 0 , k≥0
K 2 = (A12 0 ) ∪ (A1n+2 0 ) ∪ (A12n+2 0 ) ∪ . . . = ∪ A1kn+2 0 ,
k≥0
A1 ∪ A01 ∪ A001 ∪ . . .. = ∪A k = (II) Aࢨ π (1∞ ) has 1 connected component K 0 1 0
k≥0
....................................................
k≥0
K n−1 = (A1n−1 0 ) ∪ (A1n+(n−1 ) 0 ) ∪ . . . = ∪ A1kn+(n−1 ) 0 k≥0
We consider the injective paths fc ◦γ , fc ◦ f1 ◦ γ , fc ◦ f1 ◦ γ , . . . , fc ◦ [2]
[n−1]
[m−1]
f1 ◦ γ , fd ◦γ , fd ◦f0 ◦γ , fd ◦ f0 ◦ γ , . . . , fd ◦ f0 ◦ γ . The endpoints of fc ◦γ are π (0∞ ) and π (0m 1∞ ), the endpoints of fc ◦f1 ◦γ [2] are π (0m 10∞ ) and π (0m 1∞ ), the endpoints of fc ◦ f1 ◦ γ are [2]
437
π (0m 12 0∞ ) and π (0m 1∞ ) and so on the endpoints of fc ◦ f1[n−1] ◦ γ are π 0m 1n−1 0∞ and π (0m 1∞ ). Also the endpoints of fd ◦γ are π (1n 0∞ ) and π (1∞ ), the endpoints of fd ◦f0 ◦γ are π (1n 0∞ ) [2] and π (1n 01∞ ), the endpoints of fd ◦ f0 ◦ γ are π (1n 0∞ ) and π (1n 021∞ ) and so on the endpoints of fd ◦ f0[m−1] ◦ γ are π (1n 0∞ ) and π 1n 0m−1 1∞ . Then these paths have a common point, π (0m 1∞ ) = π (1n 0∞ ), and the rest of their images are mu tually disjoint. The points π (0∞ ), π (0m 10∞ ), π 0m 12 0∞ , . . . , π 0m 1n−1 0∞ , π (1∞ ), π (1n 01∞ ), π 1n 02 1∞ , . . . , π 1n 0m−1 1∞ lie in different connected components, since the paths joining then have to pass through π (0m 1∞ ) = π (1n 0∞ ). Then A {π (0m 1∞ ) = π (1n 0∞ )} has m + n connected components, ∞ more exactly, the connected components m n−1 ∞ of the∞points πn (0∞ ), m ∞ m 2 ∞ π (0 10 ), π 0 1 0 , . . . , π 0 1 0 , π (1 ), π (1 01 ),
π 1n 02 1∞ , . . . , π 1n 0m−1 1∞ which are:
K 0 = (A0m+1 ) ∪ (A0m 1n 0 ) ∪ . . . = ∪ A0m 1kn 0 , k≥0
K 1 = (A0m 10 ) ∪ (A0m 1n+1 0 ) ∪ . . . = ∪ A0m 1kn+1 0 , k≥0
K 2 = (A0m 12 0 ) ∪ (A0m 1n+2 0 ) ∪ . . . = ∪ A0m 1kn+2 0 , k≥0
............................................................ K n−1 = (A0m 1n−1 0 ) ∪ (A0m 1n+(n−1 ) 0 ) ∪ ... = ∪ A0m 1kn+(n−1 ) 0 , k≥0
K n = (A1n+1 ) ∪ (A1n 0m 1 ) ∪ (A1n 02m 1 ) ∪ . . . = ∪ A1n 0km 1 ,
A0 ∪ A10 ∪ A110 ∪ . . .. (III) Aࢨ {π (01∞ ) = π (10∞ )} has 2 connected = ∪ A k =A ∪A ponents K 0 00 010 ∪ A0110 ∪ . . . 01 0 k≥0
comand
= ∪ A k =A ∪ A K 1 11 101 ∪ A1001 ∪ . . .. 10 1 k≥0
Fig. 1.
The next example is Koch’s curve which is homeomorphic to the unit interval. Example 3.2 (Koch’s curve). ([11]) Let X = C and α ∈ C such that |α|2 + |1 − α|2 < 1. We consider the injective functions f0 (z ) = α z and f1 (z ) = (1 − α )(z − 1 )+ 1. Let D be a triangle domain with the vertices {0, 1, α } including the boundary. Then f0 (D) ∪ f1 (D)⊆D and f0 (D ) ∩ f1 (D ) = {α }. The attractor of the iterated function systems S =(X, { f0 , f1 } ) is called Koch’s curve and it is denoted by A = K (α ). We denote by A0 = f0 (A ), A1 = f1 (A ) and we have A ⊂ D and f0 (A ) ∩ f1 (A ) = A0 ∩ A1 ={α }. Moreover, π (0∞ ) = {0}, π (1∞ ) = {1}, π (01∞ ) = π (10∞ ) = {α}, π (02 1∞ ) = π (010∞ ) = {|α|2 }, π (101∞ ) = π (12 0∞ ) = {1 − |1 − α|2 }, π ( 03 1∞ ) = 2 ∞ 2 ∞ 2 ∞ π (0 10 ) = {α|α| }, π (0101 ) = π (01 0 ) = {α (1 − |1 − α|2 )}, π (102 1∞ ) = π (1010∞ ) = {1 − (1 − |α|2 )(1 − α )}, π (12 01∞ ) = π (13 0∞ ) = {1 − |1 − α|2 (1 − α )}. Thus from Theorem 2.1 A is a dendrite and in the conditions of Theorem 2.4 we have m = 1 and n = 1. Using Theorem 2.2 we deduce the well-known result that Koch’s curve is homeomorphic to the unit interval. Similar to Example 2.1 we have the following connected components: I) Aࢨ π (0∞ ) has 1 connected component K0 = ∪ A0k 1 = k≥0
k≥0
K n+1 = (A1n 01 ) ∪ (A1n 0m+1 1 ) ∪ . . . = ∪ A1n 0km+1 1 ,
A1 ∪ A01 ∪ A001 ∪ . . .. = ∪A k = II) Aࢨ π (1∞ ) has 1 connected component K 0 1 0
K n+2 = (A1n 02 1 ) ∪ (A1n 0m+2 1 ) ∪ . . . = ∪ A1n 0km+2 1 ,
A0 ∪ A10 ∪ A110 ∪ . . .. III) Aࢨ {π (01∞ ) = π (10∞ )} has 2 connected components = ∪ A k =A ∪A = K and K ∪ A10k 1 = 0 00 010 ∪ A0110 ∪ . . . 1 01 0
k≥0
k≥0
k≥0
...................................................... K n+m−1 = (A1n 0m 1 ) ∪ (A1n 0m+(m−1 ) 1 ) ∪ . . . = ∪ A1n 0km+(m−1 ) 1 k≥0
k≥0
k≥0
A11 ∪ A101 ∪ A1001 ∪ . . ..
Remark 2.2. From Theorem 2.4 it follows that there are infinitely many (countably indexed by m, n ∈ N∗ ) non-homeomorphic dendrite-type attractors of iterated function systems formed by two injective functions. 3. Examples Example 3.1 (unit interval). ([18]) Let R be endowed with the Euclidean metric and the injective functions f0 , f1 : R −→ R be defined by f0 (x ) = 2x and f1 (x ) = x+1 2 . Then A = [0, 1] is the attractor of the iterated function system S = (X, { f0 , f1 } ) since FS ([0, 1] ) =f0 ([0, 1]) ∪ f1 ([0, 1] ) = [0, 12 ] ∪ [ 12 , 1] = [0, 1]. We have A0 = f0 ([0, 1] ) = [0, 12 ], A1 = [ 21 , 1] and A0 ∩ A1 ={ 12 }. Moreover, π (01∞ ) = π (10∞ ) = { 12 }, π (0∞ ) = {0}, π (1∞ ) = {1}, π (02 1∞ ) = π (010∞ ) = { 14 } and π (101∞ ) = π (12 0∞ ) = { 34 }.
Fig. 2.
The next example is Hata’s tree-like set which is not homeomorphic to the unit interval. Example 3.3 (Hata’s tree-like set). ([8,11]) Let X = C. We consider the injective functions f0 (z ) = cz and f1 (z ) = (1 − |c|2 )z + |c|2 , where c ∈ C and |c|, |1 − c| ∈ (0, 1 ). Let M = {t ∈ [0, 1]} ∪ {tc | t ∈
438
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
[0, 1]}. Then M ⊂ f0 (M) ∪ f1 (M). Hence, if Mk =
ω ∈ k ( { 0 , 1 } )
fω (M ) for
every k ∈ N, then (Mk )k ≥ 0 is an increasing sequence and the set A = Mk is the attractor A = A(S ) of the iterated function system k≥0
S = (X, { f0 , f1 } ). The attractor A = A(S ) is called Hata’s tree-like set. According to [11] all Hata’s tree are homeomorphic for every |c|, |1 − c| ∈ (0, 1 ). We consider now A0 = f0 (A ) and A1 = f1 (A ). To prove the conditions of Theorem 2.1 are fulfilled, we can easily observe that A0 ∩ A1 = f0 (A ) ∩ f1 (A ) = {|c|2 }, Ai ∩ A j ∩ A j = ∅ for i, j, k ∈ {0, 1} pairwise different, because we cannot choose such indices, and f0 , f1 are injective functions. Also the graph ((0, 1), G), where G = {(i, j ) ∈ {0, 1} × {0, 1}| Ai ∩ Aj = ∅} consists of a single edge (0,1), thus it is a tree and A0 ∩ A1 = {|c|2 } implies card (A0 ∩ A1 ) = 1. Hence, using Theorem 2.1, A is a dendrite. Moreover we remark that f0 (0 ) = 0, f0 (1 ) = c, f0 (c ) = |c|2 , f0 (|c|2 ) = c|c|2 , f 1 ( 0 ) = |c |2 , f 1 ( 1 ) = 1, f 1 ( c ) = ( 1 − |c |2 )c + |c|2 , f 1 ( |c |2 ) = ( 1 − |c|2 )|c|2 + |c|2 , f0 ( f0 (0 )) = 0, f0 ( f0 (1 )) = f1 (0 ) = |c|2 . Hence we have the following correspondence: {0} = π (0∞ ), {1} = π (1∞ ), {|c|2 } = π (001∞ ) =π (10∞ ), {(1 − |c|2 )|c|2 + |c|2 } = π (1001∞ ) = π (110∞ ) and also {(1 − |c|2 )c + |c|2 } = π (101∞ ), {(1 − |c|2 )2 c + (1 − |c|2 )|c|2 + |c|2 } = π (1101∞ ), {c|c|2 } = π (03 1∞ ) =π (010∞ ), {(1 − |c|2 )c2 + c|c|2 } = π (0101∞ ), {c} = π (01∞ ). In the conditions of Theorem 2.4 we have m = 2 and n = 1. Thus we obtain the following connected components: I) Aࢨ π (0∞ ) has 2 connected components K0 = ∪ A02k 1 = A1 ∪ A001 ∪ A00001 ∪ . . . and K1 = ∪ A02k+1 1 = k≥0
k≥0
A01 ∪ A0001 ∪ A000001 ∪ . . .. = ∪A k = II) Aࢨ π (1∞ ) has 1 connected component K 0 1 0 k≥0
A0 ∪ A10 ∪ A110 ∪ . . .. III) Aࢨ {π (02 1∞ ) = π (10∞ )} has 3 connected = ∪A k =A ponents K 0 0 0 0 ∪ A0010 ∪ A00110 ∪ ..., 001 0 k≥0
∪ A102k 1 = A11 ∪ A1001 ∪ A100001 ∪ . . .
k≥0
A101 ∪ A10001 ∪ A1000001 ∪ . . ..
Fig. 3.
and
com = K 1
= ∪ A 2k+1 = K 2 10 1 k≥0
Acknowledgement The author would like to thank the referees for their useful suggestions to improve the paper. References [1] Barnsley MF. Fractals everywhere. New York: Academic Press; 1998. [2] Dumitru D. Attractors of topological iterated function systems. Annals of Spiru Haret University: Mathematics - Informatics series 2012;8(2):11–16. [3] Dumitru D, Mihail A. Some remarks concerning the attractors of iterated function systems. Rocky Mountain J Math 2014;44(2):479–96. [4] Dumitru D. About the attractors of infinite iterated function systems. Fixed Point Theory 2017;18(1):203–12. [5] Croydon D. Random fractal dendrites. St. Cross College, University of Oxford, Trinity; 2006. Thesis. [6] Falconer KJ. The geometry of fractal sets. Cambridge Univ Press; 1985. [7] Falconer KJ. Fractal geometry - Foundations and applications. John Wiley; 1990. [8] Hata M. On the structure of self-similar sets. Japan J Appl Math 1985(2):381–414. [9] Hutchinson J. Fractals and self-similarity. Indiana Univ J Math 1981;30(5):713–47. [10] Kameyama A. Distances on topological self-similar sets and the kneading determinants. J Math Kyoto Univ 20 0 0;40(4):603–74. [11] Kigami J. Analysis on fractals. Cambridge Univ Press; 2001. [12] Miculescu R, Mihail A. Lipscomb’s space ϖA is the attractor of an iterated function system containing affine transformation of l2 (a). Proceedings of AMS 136 2008(2):587–92. [13] Miculescu R, Mihail A. A sufficient condition for a finite family of continuous functions to be transformed into ψ -contractions. Ann Acad Sci Fenn, Math 2016;41(1):51–65. [14] Miculescu R, Mihail A. Remetrization results for possibly infinite self-similar systems. Topol Methods Nonlinear Analysis 2016;47(1):333–45. [15] Mihail A. The hutchinson measure for generalized iterated function systems. Rev Roumaine Math Pures Appl 2009;54(4):297–316. [16] Mihail A. A topological version of iterated function systems. An Stiint Univ Al I Cuza Iasi, Seria Noua, Matematica 2012;58(1):105–20. [17] Mihail A. The canonical projection between the shift space of an IIFS and its attractor as a fixed point. Fixed Point Theory Appl 2015;75:15. Article ID, electronic only [18] Secelean NA. Masura si fractali. Sibiu: Lucian Blaga Univ; 2002. [19] Secelean NA. Generalized f -iterated function systems on product of metric spaces. J Fixed Point Theory Appl 2015;17(3):575–95. [20] Secelean NA, Wardowski D. ψ f-Contractions: not necessarily nonexpansive picard operators. Results in Mathematics 2016;70(3–4):415–31. November [21] Tetenov A. Semigroups satisfying p-condition and topological self-similar sets. Sib Elektron Mat Izv 2010;7:461–4. [22] Yamaguchi M, Hata M, Kigami J. Mathematics of fractals. American Mathematical Society, Translations of mathematical monographs 1997;167.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Dendrite-type attractors of IFSs formed by two injective functions Dan DUMITRU Faculty of Engineering, Computer Science and Geography, Spiru Haret University of Bucharest, 46G Fabricii Str, Bucharest, Romania
a r t i c l e
i n f o
Article history: Received 29 October 2017 Revised 11 September 2018 Accepted 17 September 2018
MSC: 28A80
a b s t r a c t The aim of this paper is to study the dendrite-type attractors of an iterated function system formed by two injective functions. We consider (X, d) a complete metric space and S = (X, {f0 , f1 }) an iterated function system (IFS), where f0 , f1 : X −→ X are injective functions and A is the attractor of S. Moreover, we suppose that f0 (A ) ∩ f1 (A ) = {a} and {a} = π (0m 1∞ ) = π (1n 0∞ ) with m, n ≥ 1, where π is the canonical projection on the attractor. We compute the connected components of the sets A\{π (0∞ )}, A\{π (1∞ )}, A\{π (0m 1∞ ) = π (1n 0∞ )} and deduce there are infinitely-many (countably) non-homeomorphic dendritetype attractors of iterated function systems formed by two injective functions. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Attractor Iterated function system Dendrite Connected component
1. Introduction Iterated function systems were conceived in the present form by J. Hutchinson in [9], popularized by M. Barnsley in [1] and are one of the most common and general ways to generate fractals. Many of the important examples of functions and sets with special and unusual properties turn out to be fractal sets or functions whose graphs are fractal sets and a great part of them are attractors of iterated function systems. There is a current effort to extend the classical Hutchinson’s framework to more general spaces and infinite iterated function systems or, more generally, to multifunction systems and to study them ([10,11,14,15,17]). Such example can be found in [12], where the Lipscomb’s space, which is an important example in dimension theory, can be obtained as an attractor of an infinite iterated function system defined in a very general setting. In those settings the attractor can be a closed bounded set, in contrast with the classical theory where only compact sets are considered. Although the fractal sets are defined with measure theory, being sets with non-integer Hausdorff dimension ([6–8,18,22]), it turns out that they have interesting topological properties ([3–5]). The topological properties of fractal sets have a great importance in analysis on fractals as we can see in ([5,10,11]). Generalized iterated function systems can be found in ([13,18–20]). Topological versions of an iterated function system have been studied in ([2,16,21]). In this article we intend to characterize the dendrites which are attractors of iterated function systems composed by two injective
functions. For a metric space (X, d), we denote by K∗ (X ) the set of nonempty compact subsets of X. Definition 1.1. Let (X, d) be a metric space. The function h : K∗ (X ) × K∗ (X ) −→ [0, +∞ ) defined by h(A, B ) = max (d(A, B), d(B, A)), where d(A, B) = sup d (x, B ) = sup infd (x, y ) x∈A
x∈A
y∈B
is called the
Hausdorff-Pompeiu metric. Remark 1.1 ([18]). The space (K∗ (X ), h ) is complete if (X, d) is complete, compact if (X, d) is compact and separable if (X, d) is separable. Definition 1.2. Let (X, d) be a metric space. For a function f : X −→ X let us denote by Lip( f ) ∈ [0, +∞] the Lipschitz constant associd ( f (x ), f (y )) ated to f, which is Lip( f ) = sup . We say that f is d (x, y ) x,y∈X ; x=y a Lipschitz function if Lip( f ) < +∞ and a contraction if Lip(f) < 1. Definition 1.3. An iterated function system (IFS) on a complete metric space (X, d) consists of a finite family of contractions { fk }k=1,n , fk : X −→ X for every k ∈ {1, 2, . . . , n} and it is denoted by S = (X, { fk }k=1,n ). Definition 1.4. For an iterated function system S = (X, { fk }k=1,n ), n the function FS : K∗ (X ) −→ K∗ (X ) defined by FS (B ) = fk (B ) is k=1
E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2018.09.031 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
called the fractal operator associated to the iterated function system S.
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D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
Remark 1.2 ([1]). The function FS is a contraction satisfying Lip(FS ) ≤ max Lip( fk ). k=1,n
Using Banach’s contraction theorem there exists, for an iterated function system S = (X, { fk }k=1,n ), a unique set A ∈ K∗ (X ) such that FS (A ) = A, which is called the attractor of the iterated function system S. More precisely we have the following well-known result.
(3) A =
{aω }, Aα =
ω ∈
{aαω } for every α ∈ ∗ , A =
ω ∈
for every m ∈ N∗ and, more general, Aα =
ω ∈ m
ω ∈ m
Aω
Aαω for every α ∈ ∗
and every m ∈ N∗ . (4) The set {e[ω]m | ω ∈ and m ∈ N∗ } is dense in A. (5) The function π : −→ A defined by π (ω ) = aω is continuous and surjective. 2. Main results
Theorem 1.1 ([1,8,18]). Let (X, d) be a complete metric space and S = (X, { fk }k=1,n ) an iterated function system with c = max Lip( fk ) < 1. k=1,n
Then there exists a unique set A = A(S ) ∈ K∗ (X ) such that FS (A ) = A. Moreover, for any H0 ∈ K∗ (X ) the sequence (Hn )n ≥ 1 defined by Hn+1 = FS (Hn ) is convergent to A. For the speed of the convergence cn we have the following estimate: h(Hn , A ) ≤ h(H0 , H1 ) for every 1−c n ≥ 1. Now we briefly present the shift space of an iterated function system. For more details one can see [11]. We start with some set notations: N denotes the set of natural numbers, N∗ = N \ {0}, N∗n = {1, 2, . . . , n}. For two nonempty sets A and B, BA denotes the set of functions from A to B. By = (B ) we will under∗ stand the set BN and by n = n (B ) we will understand the set ∗ ∗ BNn . The elements of = (B ) = BN will be written as infinite words ω = ω1 ω2 . . . ωm ωm+1 . . . , where ωm ∈ B and the elements of ∗ n = n (B ) = BNn will be written as finite words ω = ω1 ω2 . . . ωn . By λ we will understand the empty word. Let us remark that ∗ (B ) we will understand the set of all finite 0 (B) = {λ}. By ∗ = words ∗ = ∗ (B ) = n ( B ) . n≥0
We denote by |ω| the length of the word ω. An element of = (B ) is said to have length +∞. If ω = ω1 ω2 . . . ωm ωm+1 . . . or if ω = ω1 ω2 . . . ωn and n ≥ m, then [ω]m : = ω1 ω2 . . . ωm . More generally, if l < m then [ω]lm = ωl+1 ωl+2 . . . ωm and we have [ω]m = [ω]l [ω]lm for every ω ∈ n (B), where n ≥ m > l ≥ 1. For two words α , β ∈ ∗ (B) ∪ (B), α ≺β means that |α | ≤ |β | and [β ]|α| = α . For α ∈ n (B) and β ∈ m (B) or β ∈ (B) by αβ we will understand the joining of the words α and β , namely αβ = α1 α2 . . . αn β1 β2 . . . βm and respectively αβ = α1 α2 . . . αn β1 β2 . . . βm βm+1 . . .. ∗ On = (N∗n ) = (N∗n )N , we can consider the metric ds (α , β ) = ∞
β 1−δα k
1, if x = y y k , where δx = and α = α1 α2 . . . , β = β1 β2 . . .. 3k 0, if x = y k=1 Let (X, d) be a complete metric space, S = (X, { fk }k=1,n ) an iterated function system on X and A = A(S ) the attractor of the iterated function system S. For ω = ω1 ω2 . . . ωm ∈ m (N∗n ), fω denotes fω1 ◦ fω2 ◦ . . . ◦ fωm , where fωm is applied first and Hω denotes fω (H) for a subset H ⊂ X. By Hλ we will understand the set H. In particular Aω = fω (A ). Moreover, we denote by 0∞ = 0 0 0 . . . ∈ ({0, 1} ) and 1∞ = 111 . . . ∈ ({0, 1} ). The main results concerning the relation between the attractor of an iterated function system and the shift space is contained in the following theorem. The function π : −→ A = A(S ) from the theorem below is called the canonical projection from the shift space onto the attractor of an iterated function system S. Theorem 1.2 ([11]). Let (X, d) be a complete metric space. If A = A(S ) is the attractor of an iterated function system S = (X, { fk }k=1,n ) with c = max Lip( fk ) < 1, then: k=1,n
(1) For every ω ∈ = (N∗n ) we have A[ω]m+1 ⊂ A[ω]m and d (A[ω]m ) −→ 0 when m −→ ∞. More precisely d (A[ω]m ) ≤ cm d (A ), where d (M ) = sup d (x, y ) is the diameter of a set M. x,y∈M (2) If aω is defined by {aω } = A[ω]m , then d (e[ω]m , aω ) −→ 0 m≥1
when m −→ ∞, where e[ω]m is the unique fixed point of f[ω]m .
In general the dendrites represent the topological analogue of the trees. Characterizations of dendrite-type attractors can be found in [3] and [5]. Definition 2.1. a) X is arcwise connected if for every x, y ∈ X there exists a continuous function ϕ : [0, 1] −→ X such that ϕ (0 ) = x and ϕ (1 ) = y. A continuous function ϕ as above is called a path between x and y. We say that two continuous and injective functions ϕ , ψ : [0, 1] −→ X are equivalent if there exists a function u : [0, 1] −→ [0, 1] continuous, bijective and increasing such that ϕ ◦ u = ψ . A class of equivalence is named a curve. b) If X is compact, connected and locally connected, then X is called a dendrite if for every x, y ∈ X there exists a unique equivalence class of continuous and injective functions ϕ : [0, 1] −→ X such that ϕ (0 ) = x and ϕ (1 ) = y (i.e. there exists a unique injective curve joining x with y). We remark that two equivalent, continuous and injective functions have the same images. We also consider that the empty set is a dendrite. c) Let (Ai )i ∈ I be a family of nonempty subsets of X. Then the graph (I, G), where G = {(i, j ) | i, j ∈ I such that Ai ∩ Aj = ∅ and i = j} is called the graph of intersections associated to the family (Ai )i ∈ I . d) A graph (I, G) is called connected if for every i, j ∈ I there exist (ik )k=1,n ⊂ I such that i1 = i, in = j and ik , ik+1 ∈ G for every k ∈ {1, 2, . . . , n − 1}. A family of vertices (i1 , . . . , im ) is called a cycle if ik , ik+1 ∈ G for every k ∈ {1, . . . , m} and ik ∈ / {ik+1 , ik+2 } for every k ∈ {1, . . . , m}, where by im+1 we understand i1 and by im+2 we understand i2 . A graph (I, G) is called a tree if it is connected and has no cycles. The following result gives a characterization of dendrites as attractors of some iterated function systems. Theorem 2.1 ([3]). Let (X, d) be a complete metric space and S = (X, { fk }k=1,n ) an iterated function system with c = max Lip( fk ) < 1. k=1,n
We denote by A = A(S ) the attractor of S, by Ak the set fk (A) for every k ∈ {1, . . . , n} and by ( (1, . . . , n ), G ) the graph of intersections associated with the family of sets (Ak )k=1,n . We suppose that the following conditions are true: (a) The set Ai ∩ Aj is totally disconnected for every i, j ∈ {1, 2, . . . , n} different. (b) Ai ∩ A j ∩ Ak = ∅ for every i, j, k ∈ {1, 2, . . . , n} pairwise different. (c) fk : X −→ X is an injective function for every k ∈ {1, 2, . . . , n}. Then the following statements are equivalent: (1) A is a dendrite. (2) The graph ( (1, . . . , n ), G ) is a tree and card(Ai ∩ Aj ) ∈ {0, 1} for every i, j ∈ {1, 2, . . . , n} different. In [10] and [11] were studied the critical points of an attractor of an iterated function system, some connections with the canonical projection and attractors that are homeomorphic to a quotient space of the shift space by some equivalence relation. We consider the following properties for Hata’s tree and Koch’s curve: Hata’s tree, let us say A, is the attractor of the injective functions defined in the complex plane f0 (z ) = cz and f1 (z ) = (1 − |c|2 )z + |c|2 , where c ∈ C and |c|, |1 − c| ∈ (0, 1 ), A 0 ∩ A 1 = f0 (A ) ∩ f1 (A ) = {|c|2 } and the intersection point satisfies |c|2 = π (001∞ ) =π (10∞ ); Koch’s curve, let us say B, is the attractor of
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
the injective functions defined in the complex plane f 0 (z ) = α z and f1 (z ) = (1 − α )(z − 1 )+ 1, where α ∈ C such that |α|2 + |1 − α|2 < 1, B0 ∩ B1 = f0 (B ) ∩ f1 (B ) = {α} and the intersection point satisfies {α} = π (01∞ ) =π (10∞ ). Starting from these properties we will study in this paper the iterated function systems S = (X, { f0 , f1 }), with f0 , f1 : X −→ X injective and A = A(S ) the attractor of S such that f0 (A ) ∩ f1 (A ) = {a} and {a} = π (0m 1∞ ) = π (1n 0∞ ) with m, n ≥ 1, where π is the canonical projection on the attractor. The purpose of that is to prove there are infinitely-many (countably) non-homeomorphic dendrite-type attractors formed by two injective functions. Such attractors are counted by the number of connected components, after we have removed three distinct points. We start our construction with the following lemma. Lemma 2.1. Let ({0, 1}) be the shift space formed by two symbols and m, n ∈ N∗ be fixed and arbitrarily chosen. Let the relation ∼ be defined by: α , β ∈ ({0, 1}), α ∼ β if and only if α = β or α = β and there exists γ ∈ ∗ ({0, 1}) such that {α , β} = {γ 0m 1∞ , γ 1n 0∞ }, where 0∞ = 0 0 0 . . . ∈ ({0, 1} ) and 1∞ = 111 . . . ∈ ({0, 1} ). Then ∼ is an equivalence relation. Proof. The reflexivity α ∼ α and the symmetry α ∼ β ⇒β ∼ α are obvious. For the transitivity suppose α ∼ β and β ∼ η. If α = β or β = η, then it is clear. Suppose now that α = β and β = η. Then there exist γ 1 , γ 2 ∈ ∗ ({0, 1}) such that {α , β} = {γ 1 0m 1∞ , γ1 1n 0∞ } and {β , η} = {γ2 0m 1∞ , γ2 1n 0∞ }. Suppose also that α = γ 1 0m 1∞ and β = γ1 1n 0∞ . Since {β , η} = {γ 2 0m 1∞ , γ 2 1n 0∞ }, we obtain β = γ2 1n 0∞ and η = γ2 0m 1∞ . But β = γ1 1n 0∞ = γ 2 1n 0∞ implies γ1 = γ2 = γ and hence α = η = γ 0m 1∞ . Similar, if α = γ1 1n 0∞ and β = γ 1 0m 1∞ we obtain γ1 = γ2 = γ and hence α = η = γ 1n 0∞ . Thus α ∼ η and we obtain the transitivity. Remark 2.1. (a) We consider the quotient space A = ({0, 1}) ∼, where ∼ is considered in Lemma 2.1 and the canonical : ({0, 1} ) −→ A, π (α ) = α
for projection on the quotient space π every α ∈ ({0, 1}). We remark that:
αˆ =
⎧ m ∞ n ∞ ⎨ {γ 0 1 , γ 1 0 } ⎩
{α}
γ ∈ ({0, 1}) and α = γ 0 1 or α = γ 1n 0∞ .
, if
∗
m ∞
, otherwise.
(b) Let the functions F0 , F1 : ({0, 1} ) −→ ({0, 1} ) be defined by F0 (x ) = 0x, F1 (x ) = 1x and the functions f0 , f1 : A −→ A be de (x )) = π (F0 (x )) = π (0x ) and f1 (π (x )) = π (F1 (x )) = fined by f0 (π (1x ). We remark that f0 and f1 are well-defined on A. (c) Using π Theorem 2.4 from ([10]), there exists a metric d on A such that f0 and f1 are contractions with respect to the considered metric d . In this way the metric space (A, d ) is Hausdorff, compact being the quotient space of the compact space ({0, 1}) and complete. Lemma 2.2. The functions f0 , f1 : A −→ A are injective, f0 (A ) ∪ f1 (A ) = A and f0 (A ) ∩ f1 (A ) = {a}, where {a} =π (0m 1∞ ) = π (1n 0∞ ). Proof. We have that f0 (π˜ (x )) = f0 (π˜ (y )) ⇒ 0x ∼ 0y. Then, or (x ) = π (y ) and it is clear, or there ex0x = 0y ⇒ x = y ⇒ π ists γ = γ1 γ2 . . . .γ p ∈ ∗ ({0, 1}) such that 0x = γ 0m 1∞ and 0y = γ 1n 0∞ . In the second case we have 0x = γ1 γ2 . . . .γ p 0m 1∞ and 0y = γ1 γ2 . . . .γ p 1n 0∞ . Then necessary γ1 = 0 and hence 0x = 0γ2 . . . .γ p 0m 1∞ and 0y = 0γ2 . . . .γ p 1n 0∞ . This means that x = γ2 . . . .γ p 0m 1∞ and y = γ2 . . . .γ p 1n 0∞ which implies x ∼ y. Hence (x ) = π (y ) and thus f0 is injective. Similar, one can we obtain π prove that f1 is injective. also have the following: f 0 (A ) =
We 0 x | for every x ∈ ({0, 1}) and f 1 (A ) =
1 x | for every x ∈ ({0, 1}) . Thus f (A ) ∪ f (A ) = A. Let {a} = 0
1
435
( 0n 1∞ ) = π (1m 0∞ ). Then {a}⊆f0 (A) ∩ f1 (A). We consider now π b ∈ f0 (A) ∩ f1 (A) arbitrarily chosen. Then there exist x, y ∈ ({0, ( 0x ) = π (1y ). Suppose 0x = γ 0m 1∞ and 1}) such that b = π 1y = γ 1n 0∞ with γ = γ1 γ2 . . . γr ∈ r ({0, 1}), r ≥ 1. Then γ1 = 0 and γ1 = 1 which is a contradiction. Thus γ = λ and b = ( 0n 1∞ ) = π (1m 0∞ ) = a. In this way we have obtained that π f0 (A) ∩ f1 (A)⊆{a}. Hence f0 (A) ∩ f1 (A ) = {a}. Lemma 2.3. The set A is a dendrite. Proof. In the settings of Theorem 2.1 we consider X ≡ A and d ≡ d . Hence the iterated function system S = (X, { f0 , f1 } ), where f0 , f1 : X −→ X becomes in this case S = (A, { f0 , f1 } ), where f0 , f1 : A −→ A. We recall that f0 , f1 are contractions and f0 (A ) ∪ f1 (A ) = A. Then A = A(S ) is the attractor of S. Moreover, f0 , f1 are injective, f0 (A) ∩ f1 (A ) = {a} consists of a single point and the graph ((0, 1), G) of intersections has only one edge, namely (0,1), and thus it is a tree. Hence, using Theorem 2.1, we obtain that A is a dendrite. Taking into consideration the above lemmas and remarks we have the following theorems. Theorem 2.2. If m = n = 1, then A is homeomorphic to the interval [0,1]. Proof. We are going to define a map h : A −→ [0, 1]. Then we prove that it is a homeomorphism. Let α ∈ ({0, 1}).
) = 12 ∈ [0, 1]. i) If α = 01∞ or α = 10∞ , then we consider h(α ∗ ii) If there exists γ = γ1 . . . γ p ∈ ({0, 1}), p ≥ 1, γ i ∈ {0, 1} for every i ∈ {1, . . . , p} such that α = γ 01∞ = γ1 . . . γ p 01∞ or α = p+1 γi
) = γ 10∞ = γ1 . . . γ p 10∞ , then we consider h(α i ∈ [0, 1], i=1
2
where γ p+1 = 1. iii) If there is no γ ∈ ∗ ({0, 1}) such that α = γ 01∞ or
) = α = γ 10∞ , then if α = γ1 . . . γ p . . . ∈ ({0, 1} ) we consider h(α γi i ∈ [0, 1] (remark that (γ i )i ≥ 1 is not stationary in this case). i≥1
2
Remark that in the choice of the function h we have used the decomposition of every number from the interval [0,1] by the form γi i with γ i ∈ {0, 1} for every i ≥ 1 (this also arises from the binary i≥1
2
writing of a subunitary number: if α = α1 α2 . . . ∈ ({0, 1} ), then, αi
) = in fact, h(α i ). i≥1
2
i) Surjectivity: It follows directly from the definition of h. Indeed, if β ∈ [0, 1], then there exist γ i ∈ {0, 1} for every i ≥ 1 such that γi
β= i . Consider now α = γ1 . . . γ p . . . ∈ ({0, 1} ). Then h (α ) = i≥1
β.
2
and α = α α . . . , β =
) = h β ii) Injectivity: Consider h(α 1 2 βi αi
) =
) = 12 , β1 β2 . . .. Then h(α and h β = . a) If h (α i i then
αi i≥1
2i
=
βi i≥1
2i
i≥1
=
1 2
2
i≥1
2
which implies (α1 = 1, αi = 0, for ev-
ery i ≥ 2 or α1 = 0, αi = 1 for every i ≥ 2) and (β1 = 1, βi = 0 for every i ≥ 2 or β1 = 0, βi = 1 for every i ≥ 2). This means that (α = 01∞ or α = 10∞ ) and (β = 01∞ or β = 10∞ ). Hence p+1 γi
. b) If h(α
=β
) = h β α = i ∈ [0, 1], where γ p+1 = 1, then αi i≥1
2i
=
βi i≥1
2i
=
p+1 i=1
γi 2i
=
p γi i=1
every i ∈ {1, . . . , p} and
2i
i=1
+
2
1 . 2 p+1
αi i≥p+1
2i
This implies that αi = βi = γi for
=
βi i≥p+1
2i
=
1 . 2 p+1
Thus we obtain
(α p+1 = 1, αi = 0 for every i ≥ p + 2 or α p+1 = 0, αi = 1 for every i ≥ p + 2) and (β p+1 = 1, βi = 0 for every i ≥ p + 2 or β p+1 = 0, βi = 1 for every i ≥ p + 2). This means that (α = γ1 . . . γ p 01∞ or α = γ1 . . . γ p 10∞ ) and (β = γ1 . . . γ p 01∞ or β = γ1 . . . γ p 10∞ ). γi
. c) If h(α
=β
) = h β Hence α = i with (γ i )i ≥ 1 not stationary, i≥1
2
436
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
then
αi i≥1
2i
=
βi i≥1
2i
=
γi i≥1
implies αi = βi = γi for every i ≥ 1 and
2i
.
=β thus α = β . Hence α iii) Recall a well-known result from topology which states that if f : X −→ Y is continuous and bijective, where X is compact and Y is Hausdorff, then f is a homeomorphism. Thus, returning to our problem, we only have to prove now that h is continuous. Let now [m, M] ⊂ [0, 1] be a fixed closed interval with m < M. a) If m = 2αp and M = 2βr , α , β , p, r ≥ 1, then we consider n = max{ p, r} ∈ Then:
N∗
and
α∗,
h−1 ([m, M] ) = h−1
β∗
α∗ β 2n
,
∗ ∗ such that m = α2n and M = β2n .
≤ 2n
∗
α ≤ ω|base10 ≤ β − 1
α 2
,q p
h
([m, M]) = h
q,
β
2r
which is a closed set. d) If m = q1 and M = q2 such that q1 < q2 with q1 , q2 = β ∈ [0, 1] for every β , r ≥ 1, β ≤ 2r , then we consider 2r
q −q n0 =min n ∈ N∗ | 21n < 2 2 1 . Then:
h−1 ([m, M] ) = h−1 ([q1 , q2 ] )
ω|base10 ≤ q2 · 2n and n ≥ n0
∩
[n] f10∞ (Y ).
Thus
[n]
[m]
[n]
Proof. We denote by c = 0m , d = 1n , fc = f0 and fd = f1 . Then 0m 1∞ = cd∞ and 1n 0∞ = dc∞ . Let γ : [0, 1] −→ Y be continuous, bijective and with the inverse also continuous given by Theorem 2.3 (i.e.γ is a homeomorphism between Y and [0, 1]). [m−1] We consider the injective paths γ , f0 ◦ γ , . . . , f0 ◦ γ . The endpoints of γ are π (0∞ ) and π (1∞ ), the endpoints of f0 ◦γ [m−1] are π (0∞ ) and π (01∞ ) and so on the endpoints of f0 ◦γ m−1 ∞ ∞ are π (0 ) and π 0 1 . Then these paths have a common point, π (0∞ ), and the rest of their images disjoint. are mutually Hence, the points π (1∞ ), π (01∞ ), . . . ,π 0m−1 1∞ lie in differ-
ent connected components, since the paths joining π (0k 1∞ ) and π (0l 1∞ ) have to pass through the point π (0∞ ) for every k = l, k, l ∈ {0, 1, . . . , m − 1}. Then A {π (0∞ )} has m connected components, more exactly, the connected components of the points π (1∞ ), π (01∞ ), . . . , π 0m−1 1∞ which are:
k≥0
K1 = (A01 ) ∪ (A0m+1 1 ) ∪ (A02m+1 1 ) ∪ . . . = ∪ A0km+1 1 ,
k≥0
K2 = (A02 1 ) ∪ (A0m+2 1 ) ∪ (A02m+2 1 ) ∪ . . . = ∪ A0km+2 1 ,
which is a closed set. In this way we have proved that h−1 ([m, M] ) is closed for every [m, M] ⊂ [0, 1]. Hence h is continuous and thus it is a homeomorphism. 2.3.If we consider the iterated function system S1 = Theorem [m] [n] [k] Y, f0 , f1 , where by fa we understand f a ◦ fa ◦ . . . ◦ fa ,
ktimes
a ∈ {0, 1} and Y is the attractor of S1 , then Y⊆A and Y is homeomorphic to the segment [0,1] (i.e. there exists γ : [0, 1] −→ Y continuous, bijective and with the inverse also continuous). [n]
=
[m] f01∞ (Y )
K0 = (A1 ) ∪ (A0m 1 ) ∪ (A02m 1 ) ∪ . . . = ∪ A0mk 1 ,
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1},
[m]
[n] f10∞ (A )
Theorem 2.4. We have that A{π (0∞ )} has m connected components, A{π (1∞ )} has n connected components and A{π (0m 1∞ ) = π (1n 0∞ )} has m + n connected components.
ω|base10 ≤ β and n ≥ n0
[m] = f01∞ (A )∩
ntimes
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1},
q1 · 2n <
ktimes
f01∞ (Y ) ∩ f10∞ (Y ) = {a} and so {a} =π (01∞ ) = π (10∞ ), where π : ({0, 1}) −→ Y is the canonical projection on Y. Hence Y is satisfying the conditions of Theorem 2.2 and thus Y is homeomorphic to the segment [0,1] (in fact Y is the image of the injective path joining the points π (0∞ ) and π (1∞ ), which are the fixed points of f0 and f1 ). [m]
q · 2n <
[n]
f10∞ (A ) =
mtimes
which is a closed set. c) If m = q and M = 2βr , β , r ≥ 1, β ≤ 2r such that q = 2αr ∈ [0, 1] n for every α , r ≥ 1, α ≤ 2r , then we consider ω0 = ω01 . . . ω00 ∈ ∗ ({0, 1}) such that β = ω0|base 10 . Then:
and
f1n 0∞ (A ) = ∩ f1n 00 . . . 0 (A ). k≥1
f11 . . . 10∞ (A )
α ≤ ω|base10 < q · 2n and n ≥ n0
−1
k≥1
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1},
−1
f0m 1∞ (A ) = ∩ f0m 11 . . . 1 (A )
ktimes
which is a closed set. b) If m = 2αp , α , p ≥ 1, α < 2p and M = q such that q = 2βr ∈ [0, 1] n for every β , r ≥ 1, β ≤ 2r , then we consider ω0 = ω01 . . . ω00 ∈ ∗ ({0, 1}) such that α = ω0|base10 . Then:
h−1 ([m, M] ) = h−1
m≥1
[m]
f01∞ (A ) =
by
Since {a} = π (0m 1∞ ) =π (1n 0∞ ), we have {a} = f00 . . . 01∞ (A ) ∩
∗
m≥1
A, where A = A(S ) is the attractor of the iterated function system S = (A, { f0 , f1 }), with f0 , f1 : A −→ A. For ω = ω1 ω2 . . . ωm . . . we denote by f ω (A ) = fω1 ω2 ...ωm ... (A ) = ∩ fω1 ω2 ...ωm (A ). Hence we denote now
2n
= ∪ Aω=ω1 ...ωn | ω1 , . . . , ωn ∈ {0, 1} and ∗
fractal operator of S = (A, { f0 , f1 }), with f0 , f1 : A −→ A. In this way we obtained Y ⊂ FS (Y ) and if we define the sequence A0 = Y, Am = FS (Am−1 ), then Am−1 ⊆ Am for every m ≥ 1. Thus Y ⊆ Am =
Proof. Let g0 = f0 , g1 = f1 and consider the iterated function system S1 = (Y, {g0 , g1 }), where g0 , g1 : Y −→ Y . Then Y = g0 (Y ) ∪ [m] [n] g1 (Y ) = f0 (Y ) ∪ f1 (Y ) ⊂ f0 (Y ) ∪ f1 (Y ) = FS (Y ), where FS is the
k≥0
. . . . . . . . . . . . . . . . . . .............. . . .
Km−1 = (A0m−1 1 ) ∪ (A0m+(m−1) 1 ) ∪ . . . = ∪ A0km+(m−1) 1 k≥0
[n−1]
We consider the injective paths γ , f1 ◦ γ , . . . , f1 ◦ γ . The endpoints of γ are π (0∞ ) and π (1∞ ), the endpoints of f1 ◦γ are π (10∞ ) and π (1∞ ) and so on the endpoints of f1[n−1] ◦ γ are π 1n−1 0∞ and π (1∞ ). Then these paths have a common point, π (1∞ ), and the rest of their images are mutually disjoint. The points π (0∞ ), π (10∞ ), . . . , π 1n−1 0∞ lie in different connected
components, since the paths joining π (1k 0∞ ) and π (1l 0∞ ) have to pass through the point π (1∞ ) for every k = l, k, l ∈ {0, 1, . . . , n − 1}. Then A {π (1∞ )} has n connected components, more exactly, the connected components of the points π (0∞ ), π (10∞ ), . . . ,
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
π 1n−1 0∞ which are:
From Theorem 2.1 A is a dendrite and A \ { 12 } has two connected components. In the conditions of Theorem 2.4 we have m = 1 and n = 1. Hence we obtain the following connected components: (I) Aࢨ π (0∞ ) has 1 connected component K0 = ∪ A0k 1 =
K 0 = (A0 ) ∪ (A1n 0 ) ∪ (A12n 0 ) ∪ . . . = ∪ A1nk 0 , k≥0
K 1 = (A10 ) ∪ (A1n+1 0 ) ∪ (A12n+1 0 ) ∪ . . . = ∪ A1kn+1 0 , k≥0
K 2 = (A12 0 ) ∪ (A1n+2 0 ) ∪ (A12n+2 0 ) ∪ . . . = ∪ A1kn+2 0 ,
k≥0
A1 ∪ A01 ∪ A001 ∪ . . .. = ∪A k = (II) Aࢨ π (1∞ ) has 1 connected component K 0 1 0
k≥0
....................................................
k≥0
K n−1 = (A1n−1 0 ) ∪ (A1n+(n−1 ) 0 ) ∪ . . . = ∪ A1kn+(n−1 ) 0 k≥0
We consider the injective paths fc ◦γ , fc ◦ f1 ◦ γ , fc ◦ f1 ◦ γ , . . . , fc ◦ [2]
[n−1]
[m−1]
f1 ◦ γ , fd ◦γ , fd ◦f0 ◦γ , fd ◦ f0 ◦ γ , . . . , fd ◦ f0 ◦ γ . The endpoints of fc ◦γ are π (0∞ ) and π (0m 1∞ ), the endpoints of fc ◦f1 ◦γ [2] are π (0m 10∞ ) and π (0m 1∞ ), the endpoints of fc ◦ f1 ◦ γ are [2]
437
π (0m 12 0∞ ) and π (0m 1∞ ) and so on the endpoints of fc ◦ f1[n−1] ◦ γ are π 0m 1n−1 0∞ and π (0m 1∞ ). Also the endpoints of fd ◦γ are π (1n 0∞ ) and π (1∞ ), the endpoints of fd ◦f0 ◦γ are π (1n 0∞ ) [2] and π (1n 01∞ ), the endpoints of fd ◦ f0 ◦ γ are π (1n 0∞ ) and π (1n 021∞ ) and so on the endpoints of fd ◦ f0[m−1] ◦ γ are π (1n 0∞ ) and π 1n 0m−1 1∞ . Then these paths have a common point, π (0m 1∞ ) = π (1n 0∞ ), and the rest of their images are mu tually disjoint. The points π (0∞ ), π (0m 10∞ ), π 0m 12 0∞ , . . . , π 0m 1n−1 0∞ , π (1∞ ), π (1n 01∞ ), π 1n 02 1∞ , . . . , π 1n 0m−1 1∞ lie in different connected components, since the paths joining then have to pass through π (0m 1∞ ) = π (1n 0∞ ). Then A {π (0m 1∞ ) = π (1n 0∞ )} has m + n connected components, ∞ more exactly, the connected components m n−1 ∞ of the∞points πn (0∞ ), m ∞ m 2 ∞ π (0 10 ), π 0 1 0 , . . . , π 0 1 0 , π (1 ), π (1 01 ),
π 1n 02 1∞ , . . . , π 1n 0m−1 1∞ which are:
K 0 = (A0m+1 ) ∪ (A0m 1n 0 ) ∪ . . . = ∪ A0m 1kn 0 , k≥0
K 1 = (A0m 10 ) ∪ (A0m 1n+1 0 ) ∪ . . . = ∪ A0m 1kn+1 0 , k≥0
K 2 = (A0m 12 0 ) ∪ (A0m 1n+2 0 ) ∪ . . . = ∪ A0m 1kn+2 0 , k≥0
............................................................ K n−1 = (A0m 1n−1 0 ) ∪ (A0m 1n+(n−1 ) 0 ) ∪ ... = ∪ A0m 1kn+(n−1 ) 0 , k≥0
K n = (A1n+1 ) ∪ (A1n 0m 1 ) ∪ (A1n 02m 1 ) ∪ . . . = ∪ A1n 0km 1 ,
A0 ∪ A10 ∪ A110 ∪ . . .. (III) Aࢨ {π (01∞ ) = π (10∞ )} has 2 connected = ∪ A k =A ∪A ponents K 0 00 010 ∪ A0110 ∪ . . . 01 0 k≥0
comand
= ∪ A k =A ∪ A K 1 11 101 ∪ A1001 ∪ . . .. 10 1 k≥0
Fig. 1.
The next example is Koch’s curve which is homeomorphic to the unit interval. Example 3.2 (Koch’s curve). ([11]) Let X = C and α ∈ C such that |α|2 + |1 − α|2 < 1. We consider the injective functions f0 (z ) = α z and f1 (z ) = (1 − α )(z − 1 )+ 1. Let D be a triangle domain with the vertices {0, 1, α } including the boundary. Then f0 (D) ∪ f1 (D)⊆D and f0 (D ) ∩ f1 (D ) = {α }. The attractor of the iterated function systems S =(X, { f0 , f1 } ) is called Koch’s curve and it is denoted by A = K (α ). We denote by A0 = f0 (A ), A1 = f1 (A ) and we have A ⊂ D and f0 (A ) ∩ f1 (A ) = A0 ∩ A1 ={α }. Moreover, π (0∞ ) = {0}, π (1∞ ) = {1}, π (01∞ ) = π (10∞ ) = {α}, π (02 1∞ ) = π (010∞ ) = {|α|2 }, π (101∞ ) = π (12 0∞ ) = {1 − |1 − α|2 }, π ( 03 1∞ ) = 2 ∞ 2 ∞ 2 ∞ π (0 10 ) = {α|α| }, π (0101 ) = π (01 0 ) = {α (1 − |1 − α|2 )}, π (102 1∞ ) = π (1010∞ ) = {1 − (1 − |α|2 )(1 − α )}, π (12 01∞ ) = π (13 0∞ ) = {1 − |1 − α|2 (1 − α )}. Thus from Theorem 2.1 A is a dendrite and in the conditions of Theorem 2.4 we have m = 1 and n = 1. Using Theorem 2.2 we deduce the well-known result that Koch’s curve is homeomorphic to the unit interval. Similar to Example 2.1 we have the following connected components: I) Aࢨ π (0∞ ) has 1 connected component K0 = ∪ A0k 1 = k≥0
k≥0
K n+1 = (A1n 01 ) ∪ (A1n 0m+1 1 ) ∪ . . . = ∪ A1n 0km+1 1 ,
A1 ∪ A01 ∪ A001 ∪ . . .. = ∪A k = II) Aࢨ π (1∞ ) has 1 connected component K 0 1 0
K n+2 = (A1n 02 1 ) ∪ (A1n 0m+2 1 ) ∪ . . . = ∪ A1n 0km+2 1 ,
A0 ∪ A10 ∪ A110 ∪ . . .. III) Aࢨ {π (01∞ ) = π (10∞ )} has 2 connected components = ∪ A k =A ∪A = K and K ∪ A10k 1 = 0 00 010 ∪ A0110 ∪ . . . 1 01 0
k≥0
k≥0
k≥0
...................................................... K n+m−1 = (A1n 0m 1 ) ∪ (A1n 0m+(m−1 ) 1 ) ∪ . . . = ∪ A1n 0km+(m−1 ) 1 k≥0
k≥0
k≥0
A11 ∪ A101 ∪ A1001 ∪ . . ..
Remark 2.2. From Theorem 2.4 it follows that there are infinitely many (countably indexed by m, n ∈ N∗ ) non-homeomorphic dendrite-type attractors of iterated function systems formed by two injective functions. 3. Examples Example 3.1 (unit interval). ([18]) Let R be endowed with the Euclidean metric and the injective functions f0 , f1 : R −→ R be defined by f0 (x ) = 2x and f1 (x ) = x+1 2 . Then A = [0, 1] is the attractor of the iterated function system S = (X, { f0 , f1 } ) since FS ([0, 1] ) =f0 ([0, 1]) ∪ f1 ([0, 1] ) = [0, 12 ] ∪ [ 12 , 1] = [0, 1]. We have A0 = f0 ([0, 1] ) = [0, 12 ], A1 = [ 21 , 1] and A0 ∩ A1 ={ 12 }. Moreover, π (01∞ ) = π (10∞ ) = { 12 }, π (0∞ ) = {0}, π (1∞ ) = {1}, π (02 1∞ ) = π (010∞ ) = { 14 } and π (101∞ ) = π (12 0∞ ) = { 34 }.
Fig. 2.
The next example is Hata’s tree-like set which is not homeomorphic to the unit interval. Example 3.3 (Hata’s tree-like set). ([8,11]) Let X = C. We consider the injective functions f0 (z ) = cz and f1 (z ) = (1 − |c|2 )z + |c|2 , where c ∈ C and |c|, |1 − c| ∈ (0, 1 ). Let M = {t ∈ [0, 1]} ∪ {tc | t ∈
438
D. DUMITRU / Chaos, Solitons and Fractals 116 (2018) 433–438
[0, 1]}. Then M ⊂ f0 (M) ∪ f1 (M). Hence, if Mk =
ω ∈ k ( { 0 , 1 } )
fω (M ) for
every k ∈ N, then (Mk )k ≥ 0 is an increasing sequence and the set A = Mk is the attractor A = A(S ) of the iterated function system k≥0
S = (X, { f0 , f1 } ). The attractor A = A(S ) is called Hata’s tree-like set. According to [11] all Hata’s tree are homeomorphic for every |c|, |1 − c| ∈ (0, 1 ). We consider now A0 = f0 (A ) and A1 = f1 (A ). To prove the conditions of Theorem 2.1 are fulfilled, we can easily observe that A0 ∩ A1 = f0 (A ) ∩ f1 (A ) = {|c|2 }, Ai ∩ A j ∩ A j = ∅ for i, j, k ∈ {0, 1} pairwise different, because we cannot choose such indices, and f0 , f1 are injective functions. Also the graph ((0, 1), G), where G = {(i, j ) ∈ {0, 1} × {0, 1}| Ai ∩ Aj = ∅} consists of a single edge (0,1), thus it is a tree and A0 ∩ A1 = {|c|2 } implies card (A0 ∩ A1 ) = 1. Hence, using Theorem 2.1, A is a dendrite. Moreover we remark that f0 (0 ) = 0, f0 (1 ) = c, f0 (c ) = |c|2 , f0 (|c|2 ) = c|c|2 , f 1 ( 0 ) = |c |2 , f 1 ( 1 ) = 1, f 1 ( c ) = ( 1 − |c |2 )c + |c|2 , f 1 ( |c |2 ) = ( 1 − |c|2 )|c|2 + |c|2 , f0 ( f0 (0 )) = 0, f0 ( f0 (1 )) = f1 (0 ) = |c|2 . Hence we have the following correspondence: {0} = π (0∞ ), {1} = π (1∞ ), {|c|2 } = π (001∞ ) =π (10∞ ), {(1 − |c|2 )|c|2 + |c|2 } = π (1001∞ ) = π (110∞ ) and also {(1 − |c|2 )c + |c|2 } = π (101∞ ), {(1 − |c|2 )2 c + (1 − |c|2 )|c|2 + |c|2 } = π (1101∞ ), {c|c|2 } = π (03 1∞ ) =π (010∞ ), {(1 − |c|2 )c2 + c|c|2 } = π (0101∞ ), {c} = π (01∞ ). In the conditions of Theorem 2.4 we have m = 2 and n = 1. Thus we obtain the following connected components: I) Aࢨ π (0∞ ) has 2 connected components K0 = ∪ A02k 1 = A1 ∪ A001 ∪ A00001 ∪ . . . and K1 = ∪ A02k+1 1 = k≥0
k≥0
A01 ∪ A0001 ∪ A000001 ∪ . . .. = ∪A k = II) Aࢨ π (1∞ ) has 1 connected component K 0 1 0 k≥0
A0 ∪ A10 ∪ A110 ∪ . . .. III) Aࢨ {π (02 1∞ ) = π (10∞ )} has 3 connected = ∪A k =A ponents K 0 0 0 0 ∪ A0010 ∪ A00110 ∪ ..., 001 0 k≥0
∪ A102k 1 = A11 ∪ A1001 ∪ A100001 ∪ . . .
k≥0
A101 ∪ A10001 ∪ A1000001 ∪ . . ..
Fig. 3.
and
com = K 1
= ∪ A 2k+1 = K 2 10 1 k≥0
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