SENSITIVITY AND CHAOS OF ITERATED FUNCTION SYSTEMS F. H. GHANE, E. REZAALI, M. SALEH, AND A. SARIZADEH
arXiv:1603.08243v1 [math.DS] 27 Mar 2016
Abstract. The present work is concerned with sensitivity and chaos for iterated function systems (IFSs). First of all, we introduce the concept of S-transitivity for IFSs which is relevant with sensitive dependence on initial condition. That yields to a different example of non-minimal sensitive system which is not an M-system. Also, several sufficient conditions for sensitivity of IFSs are presented. Furthermore, we will study the most two popular definitions of chaos, given by Li-Yorke and Devaney, which are generalized for iterated function systems. At last, we give several conditions that guarantee the occurrence of densely Li-Yorke chaos and Devaney chaos for IFSs.
1. Introduction In deterministic dynamical systems, chaos concludes all random phenomena without any stochastic factors. Also, it is one of the core themes in dynamical systems which its role is undeniable on the development of nonlinear dynamics. On the study of chaos, the concept of sensitivity is a key ingredient. The first formulate of sensitivity (as far as we know) was given by Guckenheimer on the study of interval maps, [10]. Sensitivity and chaos have recently been the subject of considerable interest (e.g. [1, 2, 3, 4, 9, 13, 17, 18]). Now, we recall some recent relevant results about the sensitivity. In [9], Glasner and Weiss prove that if X be a compact metric space and (X, f ) a non-minimal M-system then (X, f ) is sensitive. After that Kontorovich and Megrelishvili, in [17], prove that the following holds for a polish space (X, d) and a C-semigroup S: if (S, X) is an M-system, then the system is either minimal and equicontinuous or sensitive. Recall that an M-system means that the set of almost periodic points is dense in X (the Bronstein condition) and, in addition, the system is topologically transitive. 2010 Mathematics Subject Classification. 37B05; 37B10; 54H20; 58F03. Key words and phrases. iterated function systems, minimality, transitivity, sensitivity, Li-Yorke chaos, Devaney chaos.
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The aim of the present paper is to discuss the sensitivity and chaos (Li-Yorke chaos and Devaney chaos) for iterated function systems (IFS). In our results we establish the sensitivity for non-minimal iterated function systems which yields to a different example of non-minimal sensitive system which is not an M-system. Also, we give an example of sensitive minimal IFS that is not equicontinuous. Moreover, we introduce the concepts of strong transitivity and S-transitivity for IFSs which are relevant with sensitive dependence on initial. Then we will study the most two popular definitions of chaos, given by Devaney and Li-Yorke, which are generalized for iterated function systems. Notice that the definitions of Li-Yorke and Devaney of chaos describe the complexity a dynamical system using the behaviors of points under iterations. Here we give several conditions that guarantee the occurrence of densely Li-Yorke chaos and Devaney chaos for IFSs. The paper is organized as follows: In Section 2, we recall some standard definitions about iterated function systems. We study sensitive IFSs in Section 3 and show that every strongly transitive non-minimal IFS is sensitive. This result can be proposed by a weaker form of dynamical irreducibility which is called S-transitivity. Furthermore, we prove that local expanding property and expanding property in IFSs yield to sensitivity and cofinite sensitivity, respectively. Section 4 is devoted to chaotic IFSs. We will discuss two kinds of chaos, Li-Yorke chaos and Devaney chaos, for IFSs. Then we give sufficient conditions for densely Li-Yorke chaotic IFSs. In Section 5, we give an example of non-minimal S-transitive IFS which is not strongly transitive. This example provides a non-minimal sensitive system which is not an M-system. 2. Preliminaries A dynamical system in the present paper is a triple (Γ, X, ϕ), where Γ is a semigroup, X is a compact metric space and ϕ : Γ × X → X, (γ, x) 7→ γx is a continuous action on X with the property that γ(ηx) = (γη)x holds for every triple (γ, η, x) in Γ × Γ × X. Sometimes we write the dynamical system as a pair (Γ, X). The orbit of x is the set Γx := {γx : γ ∈ Γ}. By A we will denote the closure of a subset A ⊆ X. Now, consider a finite family of continuous maps F = { f1 , . . . , fk } defined on a compact metric space X. We denote by F + the semigroup generated by these maps. If Γ = F + then the dynamical system (Γ, X) is called an iterated function system (IFS) associated to F . We use the usual notations: IFS(X; F ) or IFS(F ). Roughly speaking, an iterated function system
SENSITIVITY AND CHAOS OF ITERATED FUNCTION SYSTEMS
3
(IFS) can be thought of as a finite collection of maps which can be applied successively in any order. In particular, if Γ = { f n }n∈N and f : X → X is a continuous function, then the classical dynamical system (Γ, X) is called a cascade and we use the standard notation: (X, f ). Throughout this paper, we assume that (X, d) is a compact metric space with at least two distinct points and without any isolated point. Also, we assume that IFS(F ) is an iterated function system generated by a finite family { f1 , . . . , fk } of homeomorphisms on X. For the semigroup F + and x ∈ M the total forward orbit of x is defined by O+F (x) = {h(x) : h ∈ F + }. Analogously, the total backward orbit of x is defined by O−F (x) = {h−1 (x) : h ∈ F + }. Now, consider an iterated function system IFS(F ) generated by a finite family of homeomorphisms F = { f1 , . . . , fk } on X. (i) IFS(F ) is called symmetric, if for each f ∈ F it holds that f −1 ∈ F . (ii) IFS(F ) is called forward minimal or F + acts minimally on X, if O+F (x) = X for every x ∈ X. (iii) IFS(F ) is called topologically transitive, if for any two non-empty open sets U and V in X, there exists h ∈ F + such that h(U) ∩ V , ∅. (iv) IFS(F ) is called strongly transitive or backward minimal, if IFS(F −1 ) is minimal where F −1 = { f1−1 , . . . , fk−1 }, i.e. O−F (x) = X for every x ∈ X. It is not hard to show that strong transitivity implies topological transitivity. Symbolic dynamic is a way to represent the elements of F + . Indeed, consider the product space Σ+k = {1, . . . , k}N . For any sequence ω = (ω1 ω2 . . . ωn . . . ) ∈ Σ+k , take fω0 := Id and fωn (x) = fωn1 ...ωn (x) = fωn ◦ fωn−1 (x); ∀ n ∈ N. Obviously, fωn = fωn1 ...ωn = fωn ◦ fωn−1 ∈ F + , for every n ∈ N. 3. Sensitive iterated function systems In this section, we give several sufficient conditions for sensitivity of IFSs. Before to state our results, we introduce the concept of sensitivity for IFSs which is a generalized version of the existing definition for cascades (see [8, 9]). The IFS(F ) is sensitive on X if there exists δ > 0 (sensitivity constant) such that for every point x ∈ X and every r > 0 there are y ∈ B(x, r), ω ∈ Σk+ and n ∈ N with d( fωn (x), fωn (y)) > δ.
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Following [5], we know that there exists an IFS of homeomorphisms of the circle that is forward minimal but not backward minimal. That is equivalent to the existence of a strongly transitive (backward minimal) IFS of homeomorphisms of the circle which is not forward minimal. The following theorem insures that such systems are sensitive. Theorem 3.1. If IFS(F ) is strongly transitive but not minimal then IFS(F ) is sensitive. Proof. Let y ∈ X with O+F (y) , X and z ∈ X \ O+F (y). Take V = B(z, δ) where δ = 41 d(z, O+F (y)). Also, let U ⊂ X be an arbitrary neighborhood of X. Since IFS(F ) is strongly transitive, there exist T1 , . . . , Tℓ in F + so that the following holds: S (H1 ) X ⊆ ℓi=1 Ti (U). (i) (i) Furthermore, for every i ∈ {1, . . . , ℓ}, there exist T1 , . . . , Tℓ so that i Si (i) (H2 ) X ⊆ ℓj=1 T j (Ti (U)); ∀ 1 ≤ i ≤ ℓ. (i)
Take t := ℓ + max Ξ where Ξ = {|T j | : 1 ≤ i ≤ ℓ & 1 ≤ j ≤ ℓi } and |T| is the length of T. Choose a neighborhood W around y such that diam( fρi (W)) < δ, for every ρ ∈ Σ+k and for every i = 0, 1, . . . , t. Moreover, by the choice of δ we may assume that d( fρi (W), V) ≥ 2δ, for i = 0, 1, . . . , t and for each ρ ∈ Σ+k . T The condition (H1 ) ensures that for some 1 ≤ s ≤ ℓ, Ts (U) W , ∅. On the other T (s) hand, according to (H2 ), there exists 1 ≤ j0 ≤ ℓs so that T j (Ts (U)) V , ∅. Also, one 0 T (s) (s) (s) has that T j (Ts (U)) T j (W) , ∅. By the choice of W, we have d(T j (W), V) ≥ 2δ. Thus, 0
(s) diam(T j (Ts (U))) 0
0
0
> δ which completes the proof.
Theorem 3.1 can be true by the following weaker assumptions. Definition 3.2. We say that IFS(F ) is S-transitive if for every open set U, there exist T1 , . . . , Tℓ ∈ S F + so that X ⊆ ℓi=1 Ti (U). Remark 3.3. Notice that S-transitivity follows by strong transitivity. For the proof of Theorem 3.1 it is enough to have the S-transitivity condition. Hence, in the assumptions of Theorem 3.1, the strong transitivity condition may be replaced by the S-transitivity condition. Obviously, one can see that strong transitivity =⇒ S-transitivity =⇒ transitivity In Section 5, we give an example of non-minimal S-transitive IFS which is not strongly transitive. This example is important from another point of view. In fact, it provides a non-minimal sensitive system which is not an M-system, see [17].
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By the above remark, the proof of the following corollary is the same as the proof of Theorem 3.1 and so, it is omitted. Corollary 3.4. If IFS(F ) is S-transitive but not minimal then IFS(F ) is sensitive. Let f be a homeomorphism on a compact metric space (X, d) and x ∈ X be a fixed point of f . Then the stable set W s (x) and unstable set W u (x) are defined, respectively, by W s (x) = {y ∈ X : f n (x) → y, whenever n → +∞}, W u (x) = {y ∈ X : f n (x) → y, whenever n → −∞}. Definition 3.5. We say that x is an attracting (or repelling) fixed point of f if the stable set W s (x) (or unstable set W u (x)) contains an open neighborhood B of x. Then B is called the basin of attraction (or repulsion) of x. Theorem 3.6. If IFS(F ) is strongly transitive and the associated semigroup F + has a map with a repelling fixed point then IFS(F ) is sensitive on X. Proof. Suppose that IFS(F ) is strongly transitive and the associated semigroup F + has a map h with a repelling fixed point q and the repulsion basin B. Suppose that IFS(F ) is not sensitive. Then for each n ∈ N, there is a non-empty open subset Un of X such that the following holds: 1 ; ∀ω ∈ Σk+ ∀i ∈ N. (1) n Take n large enough, then for each y, z ∈ B there exists m ∈ N sufficiently large so that diam( fωi (Un ))
0 such that Bδ (q) ⊂ B and T−1 (Bδ (q)) ⊂ Un which implies that Bδ (q) ⊂ T(Un ). Let us take y′ = T−1 (y) and z′ = T−1 (z). Then y′ , z′ ∈ Un and hence by (2) d(hm (y), hm (z)) ≥
d(hm ◦ T(y′ ), hm ◦ T(z′ )) = d(hm (y), hm (z)) ≥ which contradicts (1).
1 n
It is a well known fact that each Morse-Smale circle diffeomorphisom with a fixed point possesses a repelling fixed point. Consequently, we get the following result.
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Corollary 3.7. Let F be a finite family of Morse-Smale diffeomorphisoms on the circle S1 . If IFS(S1 ; F ) is strongly transitive and F has a map with a fixed point then IFS(F ) is sensitive on X. Suppose that the generators fi , i = 1, . . . , k, of IFS(F ) are C1 diffeomorphisms defined on a compact smooth manifold M. We say that IFS(F ) is local expanding if for each x ∈ M there exist h ∈ F + so that m(Dh(x) > 1, where m(A) = kA−1 k−1 denote the co-norm of a linear transformation A. Moreover, we say that IFS(F ) is an expanding iterated function system if the generators fi , i = 1, . . . , k, are C1 expanding maps. Obviously, if IFS(F ) is expanding, by compactness of M, there exists 0 < η < 1 so that kD fi (x)−1 k < η for every x ∈ M, and i = 1, . . . , k.
(3)
It is not hard to see that IFS(F ) is local expanding if and only if there are h1 , . . . , hℓ ∈ F + , open balls V1 , . . . , Vℓ in M and a constant σ < 1 such that M = V1 ∪ . . . ∪ Vℓ , and kDhi (x)−1 k < σ for every x ∈ Vi , i = 1, . . . , ℓ.
(4)
Fix the mappings h1 , . . . , hℓ ∈ F + and the open subsets V1 , . . . , Vℓ of M that satisfy (4). Given x ∈ M, we say that the sequence ω = (ω0 ω1 . . . ωn . . .) ∈ Σ+ℓ is an admissible itinerary of x if the following holds: x ∈ Vω0 and for each j ≥ 1, fω j ◦ · · · ◦ fω0 (x) ∈ Vω j+1 . Let δ be the Lebesgue number of the covering {V1 , . . . , Vℓ }. Then, for each point x ∈ M, every open neighborhood U of x and each admissible itinerary ω of x, the subset NF + (U, ω, δ) is nonempty, where NF + (U, ω, δ) = {n ∈ N : there exist x, y ∈ U such that d( fωn (x), fωn (y)) > δ}. IFS(F ) is called cofinitely sensitive if there exists a constant δ > 0 such that for any nonempty open set U of X, there exist a branch ω ∈ Σ+k and an N ≥ 1 such that NF + (U, δ, ω) ⊃ [N, +∞)∩ N, while δ is called a sensitive constant. Hence the next result follows immediately. Proposition 3.8. The following statements hold. i) Each local expanding IFS is sensitive. ii) Each expanding IFS is cofinitely sensitive. Now we apply the techniques used in Theorems 3.1 and 3.2 of [13] for IFSs to establish the next two results. Before it, we need to introduce the following notations. For any two nonempty open sets U, V of X and ω ∈ Σ+k we denote NF + (U, V, ω) = {n ∈ N : fωn (U) ∩ V , ∅}.
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Take Nn = {0, 1, . . . , n − 1} and denote |A| for the cardinal number of the set A. Theorem 3.9. Let X be a compact metric space and IFS(F ) be an iterated function system generated by a finite family { f1 , . . . , fk } of homeomorphisms on X. If for every two open sets U, V of X there is a branch ω ∈ Σ+k such that IFS(F ) satisfies lim sup n→+∞
|(NF + (U, V, ω) ∩ Nn | |(NF + (U, V, ω) ∩ Nn | + lim inf >1 n→+∞ n n
(5)
then IFS(F ) is sensitive. Proof. To get a contradiction, let for each δ > 0 there exist a point x ∈ X and 0 < r < δ such that for each ω ∈ Σ+k sup sup d( fωn (x), fωn (y)) ≤ δ,
(6)
n≥0 y∈Br
where Br is the ball with center x and radius r. Take Cδ := Cl(B4δ+r ). Since X is nontrivial, Uδ = X \ Cδ is nonempty open subset of X. By (5), NF + (Br , Br , ω) is nonempty for each ω. Then for each fixed ω and n ∈ NF + (Br , Br , ω) there exists y ∈ Br such that d( fωn (y), x) < r. It follows from (6) that for any z ∈ Br , d( fωn (y), fωn (z)) ≤ d( fωn (z), fωn (x)) + d( fωn (x), fωn (y)) ≤ 2δ. Hence, for any w ∈ Br d( fωn (z), w) ≤ ( fωn (z), fωn (y)) + ( fωn (y), w) < 2δ + 2r < 4δ which implies that fωn (z) ∈ Cδ . This yields that fωn (Br ) ⊂ Cδ . Thus, for each n ∈ NF + (Br , Br , ω), one has that fωn (Br ) ∩ Uδ = ∅, and then lim sup n→+∞
|(Nn \ NF + (Br , Br , ω)) ∩ Nn | |(NF + (Br , Uδ , ω) ∩ Nn | ≤ lim sup n n n→+∞
|(NF + (Br , Uδ , ω) ∩ Nn | n which contradicts (5). Therefore, the assertion holds. = 1 − lim inf n→+∞
IFS(F ) is called topologically strong ergodic, if for any two nonempty open sets U, V of X there exists a branch ω ∈ Σ+k so that the upper density of NF + (U, V, ω) equals 1 where the upper density of NF + (U, V, ω) is lim sup n→+∞
|(NF + (U, V, ω) ∩ Nn | . n
With a similar argument to that used in the proof of Theorem 3.9, one can easily show the following result:
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Theorem 3.10. If IFS(F ) is topologically strongly ergodic and satisfies that for any nonempty open set U there exists a branch ω ∈ Σ+k so that lim inf n→+∞
|(NF + (U, U, ω) ∩ Nn | >0 n
holds, then IFS(F ) is sensitive. 4. Iterated function systems and chaos In this section we will generalize the most two popular definitions of chaos in deterministic systems that given by Li-Yorke and Devaney for iterated function systems. Consider an iterated function system IFS(F ) generated by F = { f1 , . . . , fk }. We say that x and y are proximal if there exists ω ∈ Σk+ such that lim inf d( fωn (x), fωn (y)) = 0. n→∞
Then (x, y) ∈ X × X is called a proximal pair. The iterated function system IFS(F ) is called proximal iterated function system if each x, y ∈ X are proximal. Recall that x and y are asymptotic if for each ω ∈ Σk+ , lim d( fωn (x), fωn (y)) = 0.
n→∞
The pair (x, y) ∈ X × X is called a Li-Yorke pair whenever there exist ω, ω′ ∈ Σk+ satisfying lim inf d( fωn (x), fωn (y)) = 0 and lim sup d( fωn′ (x), fωn′ (y)) > 0. n→∞
n→∞
This means that x and y are proximal but not asymptotic. A subset S ⊂ X is a scrambled set (for IFS(F )), if any different points x and y from S are Li-Yorke pair. We say IFS(F ) is chaotic in the sense of Li-Yorke, if there exists an uncountable scrambled set. Denote the asymptotic relation by AR, the proximal relation by PR and Li-Yorke relation by LYR. Clearly, LYR = PR \ AR. Moreover, if R is a relation on X, denote the relation class of x ∈ X by R(x) = {y ∈ X : (x, y) ∈ R}. The following facts follow immediately. Lemma 4.1. Let l, n ∈ N, ω ∈ Σk+ and Al,n,ω
∞ \ = ( fω × fω )−i ∆n , i=l
where ( fω × fω )−i = fω−11 × fω−11 ◦ . . . ◦ fω−1i × fω−1i and let ∆n = {(x, y) ∈ X × X : d(x, y) < n1 }. Also let \ Al,n,ω . Al,n = ω∈Σk+
SENSITIVITY AND CHAOS OF ITERATED FUNCTION SYSTEMS
Then
∞ [ ∞ \ AR = ( Al,n );
9
(7)
n=1 l=1
AR(x) =
∞ [ ∞ ∞ [ ∞ ∞ \ \ \ \ 1 −j j ( Al,n (x)); ( fω (B( fω (x, )))) = n k n=1 l=1 ω∈Σ+ j=l ∞ [ \
PR =
(
l=1
ω∈Σk+
(8)
n=1 l=1
∞ [ ( fω × fω )−n ∆l ).
(9)
n=1
Clearly, PR is a Gδ subset. Theorem 4.2. If the iterated function system IFS(F ) is sensitive on X then AR has the first category in X × X. Proof. Let δ > 0 be the constant in the definition of sensitivity of IFS(F ) and n ∈ N with 2 n < δ. Then for all l ∈ N and x ∈ X, Al,n (x) has empty interior, where Al,n (x) = {y ∈ X : (x, y) ∈ Al,n }. Indeed, if U is a non-empty open subset of X contained in Al,n (x) then for each y, z ∈ U, d( fωi (y), fωi (z)) ≤ n2 , for all i ≥ l, ω ∈ Σk+ and so by shrinking U if necessary for all i ≥ 0, contradicting the definition of δ. Then for each x ∈ X, AR(x) has the first category and therefore AR has the first category by the previous lemma, as Al,n has empty interior too. Now we need the following lemmas. Lemma 4.3 ([14],Lemma 3.1). Assume that X is a complete separable metric space without isolated points. If R is a symmetric relation with the property that there is a dense Gδ subset A of X such that for each x ∈ A, R(x) contains a dense Gδ subset, then there is a dense subset B of X with uncountably many points such that B × B \ ∆ ⊂ R. Lemma 4.4 ([14],Lemma 3.2). Let R be a relation on a complete separable metric space X, which contains a dense Gδ subset of X × X. Then, there is a dense Gδ subset A of X such that for each x ∈ A, there exists a dense Gδ subset Ax of X for which the following holds: {(x, y) : x ∈ A, y ∈ Ax } ⊂ R. Definition 4.5. We say that IFS(F ) is densely Li-Yorke chaotic if it admits a dense uncountable scrambled set. Theorem 4.6. Let IFS(F ) be a sensitive IFS on a compact metric space X. If PR is a dense Gδ set then LYR contains a dense Gδ subset of X × X. In particular, IFS(F ) is densely Li-Yorke chaotic.
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Proof. Since IFS(F ) is sensitive, AR is of the first category by Theorem 4.2. As LYR = PR\AR the first statement follows immediately, i.e. LYR contains a dense Gδ subset of X × X. By Lemma 4.4, there is a dense Gδ subset A of X such that for each x ∈ A, there is a dense Gδ subset Ax with {(x, y) : x ∈ A, y ∈ Ax } ⊂ LYR. Now according to Lemma 4.3, we can find a dense uncountable scrambled set for IFS(F ). Thus IFS(F ) is densely Li-Yorke chaotic. Corollary 4.7. Let IFS(F ) be a strongly transitive in which the associated semigroup F + has a map h with exactly two fixed points, one attracting and one repelling. Then IFS(F ) is densely Li-Yorke chaotic. Proof. By Proposition 3.6, since IFS(F ) is strongly transitive and the associated semigroup has a map with a repelling fixed point then IFS(F ) is sensitive. Fix two points x, y ∈ X and take K = {x, y}. Then U := X \ K is an open subset of X. By the backward minimality there exist T1 , . . . , Ts ∈ F + so that s [ X= Ti (U). i=1
Let p and q be, respectively, the attracting and the repelling fixed points of h. Then there is i ∈ {1, . . . , s} so that q ∈ Ti (U). Therefore, q does not belong to Ti (K) and hence the diameter of hn ◦ Ti (K) converges to zero. This shows that (x, y) is a proximal pair. Since the points x, y are arbitrary, then PR = X × X. Sensitivity of IFS(F ) and Theorem 4.6 ensure that IFS(F ) is densely Li-Yorke chaotic. Now, we recall the fixed point theorem of Jachymski [11]. This theorem is a generalization of Banach’s fixed point theorem. Definition 4.8. Let (X, d) be a metric space. We say that a map g : X → X is an asymptotic contraction if d(gn (x), gn (y)) → 0 whenever n → +∞, for all x, y ∈ X and there exists δ > 0 such that this convergence is uniform if d(x, y) ≤ δ. Theorem 4.9 (Jachymski, [15]). Suppose that (X, d) is a complete metric space and g : X → X is a continuous asymptotic contraction. Then there exists x ∈ X such that: d(gn (y), x) → 0 whenever n → +∞ for all y ∈ X. Proposition 4.10. Let IFS(F ) be a symmetric forward minimal IFS on a compact metric space X. If the associated semigroup F + has a map with an attracting or a repelling fixed point then the following set {x ∈ X : x is an attracting (or a repelling) fixed point for some h ∈ F + } is dense in X.
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Proof. It is enough to prove the density of attracting periodic points. The conclusion for repelling periodic points follows in a similar way. Suppose that IFS(F ) is forward minimal and the map h belongs to the associated semigroup F + with an attracting fixed point p. Let Vp be an open ball of center p and radius r > 0 that is contained in the stable set of p. Let us fix any open subset U of X. Since F + acts forward minimally on X, there exists a map T ∈ F + so that T(p) ∈ U. By continuity there exist an open subset V ′ with p ∈ V ′ ⊂ Vp and n ∈ N such that U′ := T(V ′ ) ⊂ U and T ◦ hn ◦ T−1 (U′ ) ⊂ U. Take g := T ◦ hn ◦ T−1 . Since e ⊂ Cl(U) e ⊂ U′ so that Cl(U) e is compact. Clearly, g is a IFS(F ) is symmetric, g ∈ F + . Let U e and hence it has a unique fixed continuous asymptotic contraction on a compact set Cl(U) e ⊂ U which is attracting by the Jachymski Theorem. This terminates the point x ∈ Cl(U) proof of the proposition. Definition 4.11. Let (X, d) be a metric space. An iterated function system IFS(F ) is said to be chaotic in the sense of Devaney if i) IFS(F ) is topologically transitive; ii) the set of the periodic points of IFS(F ) is dense in X; iii) IFS(F ) is sensitive. The next result follows by Propositions 4.10 and 3.6. Corollary 4.12. Let IFS(F ) be a symmetric forward minimal IFS on a compact metric space X so that the associated semigroup F + admits a map h with a repelling or an attracting fixed point. Then IFS(F ) chaotic in the sense of Devaney. Theorem 4.13. There exists an iterated function system IFS(F ) of C1 diffeomorphisms on the n-sphere Sn , n ≥ 1, for which the following statements hold: i) IFS(F ) is C1 robustly densely Li-Yorke chaotic; ii) IFS(F ) admits a dense subset of attracting (repelling) periodic points. In particular, IFS(F ) is C1 robustly chaotic in the sense of Devaney. Proof. Consider Morse-Smale diffeomorphisms f1 , . . . , fn+1 on the n-sphere Sn so that they satisfy the following conditions: i) fi has exactly two fixed points, one attracting pi and one repelling qi , i = 1, . . . , n + 1, so that the points pi are affine independent; ii) D fi (pi ) is a multiple of the identity, i. e. D fi (pi ) = λi I, where λi , i = 1, . . . , n + 1, are real numbers with 21 < λi < 1 and I is the n × n identity matrix.
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Take ∆ the general n-simplex on Sn with vertices pi . Let h be a Morse-Smale diffeomorphism on Sn with exactly two fixed points, one attracting p and one repelling q so that q ∈ int(∆). If we apply the technique used in the proof of Theorem 1 of [6] for the iterated function −1 , h, h−1 }, we provide the C1 robust system IFS(F ) generated by F = { f1 , . . . , fn+1 , f1−1 , . . . , fn+1 backward and forward minimality of IFS(F ). Now the statement (1) follows by the previous corollary. The statement (2) is a consequence of Proposition 4.10. 5. Examples It is a well known fact that for ordinary dynamical systems, the minimality of a map f is equivalent to that of f −1 . Nevertheless this is not the case for iterated function systems as Kleptsyn and Nalskii pointed at [16, pg. 271]. However, they omitted to include these examples of forward but not backward minimal IFSs. Recently, the authors in [5], illustrated an example of forward but not backward minimal IFS which we explain it in below. Example 5.1. Consider a symmetric semigroup S of homeomorphisms of the circle. Then, one of the following possibilities holds [19, 7]: i) there is a finite S-orbit; ii) every S-orbit is dense on the circle; or iii) there is a unique S-invariant minimal Cantor set K. The Cantor set K in the above third conclusion is usually called exceptional minimal set. This set is S-invariant and minimal, that is, g(K) = K for all g ∈ S and
K = S(x) for all x ∈ K.
By [19, Exer. 2.1.5], one can ensure that there exists a symmetric semi-group G generated by homeomorphisms f1 , . . . , fk of S1 which admits an exceptional minimal set K such that the S-orbit of every point of S1 \ K is dense in S1 . For this symmetric semi-group G, the closed invariant subsets of S1 by G are ∅, K and S1 . Now, consider any homeomorphism h of S1 such that h(K) strictly contains K. Then the IFS generated by f1−1 , . . . , fn−1 , h−1 is backward minimal but not forward minimal. Therefore, the IFS generated by f1−1 , . . . , fn−1 , h−1 is sensitive. Also, it is not hard to check that the IFS generated by f1−1 , . . . , fn−1 , h−1 is an M-system. Example 5.2. Suppose f is a north-south pole C1 -diffeomorphism of the circle S1 possessing an attracting fixed point p as a north pole and a repelling fixed point q as a south pole with multipliers 1/2 < f ′ (p) < 1 & 1/2 < ( f −1 )′ (q) < 1.
SENSITIVITY AND CHAOS OF ITERATED FUNCTION SYSTEMS
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It is observed that S1 \ {p, q} composed of exactly two connected pieces U1 and U2 . Now, consider two homeomorphisms h1 and h2 of S1 such that i) U1 $ h1 (U1 ) and U2 $ h(U2 ). ii) p is a boundary point for h1 (U1 ) and h2 (U2 ). Take F = { f, f −1 , h1 , h2 }. One can check O+F (x) = S1 & O−F (x) = S1 , S for all x ∈ S1 \ {p} and also h(p) = p, for every h ∈ F + (F −1 )+ . Therefore, IFS(S1 ; F ) is not forward and strongly transitive (backward minimal). Two important things are that IFS(S1 ; F ) is S-transitive but it is not an M-system. Indeed, x is not almost periodic point when x , p i.e. O+F (x) is not minimal for x , p. As a consequence of the previous example we get the next result. Corollary 5.3. There exists any S-transitive non minimal IFS which is sensitive but it is not an M-system. References [1] C. Abraham, G. Biau, and B. Cadre. Chaotic properties of mappings on a probability space. J. Math. Anal. Appl. 266 (2002), no. 2, 420–431. MR1880515. [2] E. Akin and E. Kolyada. Li-Yorke sensitivity. Nonlinearity 16 (2003), no. 4, 1421–1433. MR1986303 [3] J. Auslander and J. Yorke. Interval maps, factors of maps, and chaos. Tohoku Math. J. (2) 32 (1980), no. 2, 177–188. MR0580273 [4] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey. On Devaney’s definition of chaos. Amer. Math. Monthly 99 (1992), no. 4, 332–334. MR1157223 [5] P. G. Barrientos, F. H. Ghane, D. Malicet, and A. Sarizadeh. On the chaos game of iterated function systems, (to appear in Topological methods in nonlinear analysis). [6] F. H. Ghane, A .J .Homburg and A. Sarizadeh, C1 robustly minimal iterated function systems, Stochastic Dynam., 10 (2010), pp. 155-160. [7] E. Ghys, Groups acting on the circle, Enseign. Math. (2),47 (2001), 329–407. [8] E. Glasner and M. Megrelishvili, Hereditarily non-sensitive dynamical systems and linear representations. Colloq. Math. 104(2), 223283 (2006) [9] E. Glasner, B. Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), 1067-1075. [10] Guckenheimer, John. Sensitive dependence to initial conditions for one-dimensional maps. Comm. Math. Phys. 70 (1979), no. 2, 133–160. MR0553966. [11] M. Hata, On the structure of self-similar sets, Japa J. Indust. Appl. Math. 2 (1985) no 2, 381-414. [12] A. J. Homburg, and M. Nassiri, Robustly minimal iterated function systems on compact manifolds generated by two diffeomorphisms, Ergodic Theory Dynam. Systems, 34 (6) (2014), 1914-1929.
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[13] Q. Huanga, Y. Shi, L. Zhang, Sensitivity of non-autonomous discrete dynamical systems, Appl. Math. Letters, 39 (2015) 31-34. [14] W. Huang, X. Ye, Devaneys chaos or 2-scattering implies LiYorkes chaos, Topology Appl. 117 (2002) 259272. [15] J. R. Jachymski, An fixed point criterion for continuous self mappings on a complete metric space, Aequations Math., 48 (1994), 163-170. [16] V. A. Kleptsyn and M. B. Nalskii, Contraction of orbits in random dynamical systems on the circle, Functional Analysis and Its Applications, 38 (2004), 267–282. [17] E. Kontorovich and M. Megrelishvili, A note on sensitivity of semigroup actions, Semigroup Forum, 76 (2008), 133141. [18] R. Li and Y. Shi. Several sufficient conditions for sensitive dependence on initial conditions. Nonlinear Anal. 72 (2010), no. 5, 2716–2720. MR2577831 [19] A. Navas, Groups of circle diffeomorphisms, University of Chicago Press, 2011. Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran. E-mail address:
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[email protected] Department of Mathematics, University of Neyshabur, Neyshabur, Iran. E-mail address:
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