Chaos, Solitons and Fractals 23 (2005) 1381–1384

www.elsevier.com/locate/chaos

On chaotic set-valued discrete dynamical systems

Alessandro Fedeli

Department of Pure and Applied Mathematics, University of L’Aquila, 67100 L’Aquila, Italy

Accepted 15 June 2004

Abstract

Let (X, d) be a metric space and let f: (X, d)! (X, d) be a continuous map. In this note we investigate the relation-

ships between the chaoticity of some set-valued discrete dynamical systems associated to f (collective chaos) and the

chaoticity of f (individual chaos).

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

Let us consider a metric space (X, d) and a discrete dynamical system (X; f) given by the space X and a continuous

map f: (X, d) !(X, d).

Following Devaney [4], we say that f: (X, d)!(X, d) is chaotic (on X) if

(1) f is (topologically) transitive, i.e., for every pair U and V of non-empty open subsets of (X, d) there is a non-negative

integer j such that f j(U) \ V5 ;;(2) f is periodically dense, i.e., the set of periodic points of f is dense in X;

(3) f has sensitive dependence on initial conditions, i.e., there is some d > 0 such that, for any x 2 X and any neighbour-

hood V of x in (X, d), there exist a y 2 V and some non-negative integer j for which d(fj(x), fj(y)) > d.

It is worth noting that has been shown that condition (3) is redundant in the definition above if X is infinite [2,11].

This fact reveals the topological, rather than metric, nature of chaos.

The importance of chaotic systems cannot be overestimated, in fact it involves processes ranging from chemical reac-

tions to pulsation in gas lasers, and, of course, from many other fields (see, e.g., [5]).

It should be observed that if X is an interval of the real line, then f is chaotic if and only if it is transitive [3,9,12] (see

[1] for a more general result).

See also [10] for recent results on chaos of discrete dynamical systems in complete metric spaces.

In this note we investigate the following interesting question raised in [8]:

let us consider the set-valued discrete dynamical system associated to f given by ðKðX Þ; �f Þ, where �f : KðX Þ !KðX Þ is the natural extension of f and KðX Þ is the family of all non-empty compact subsets of (X, d) endowed with

0960-0779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2004.06.039

E-mail address: afedeli@univaq.it

1382 A. Fedeli / Chaos, Solitons and Fractals 23 (2005) 1381–1384

the Hausdorff metric induced by d. When does the chaoticity of f (individual chaos) imply the chaoticity of �f (collective

chaos)? And conversely?

2. Preliminaries

Let (X, d) be a metric space and let sd, be the topology on X generated by d. The distance from a point x to a set A in

(X, d) is defined by letting

dðx;AÞ ¼ inffdðx; aÞ : a 2 Ag if A 6¼ ;; and dðx; ;Þ ¼ 1:

The Hausdorff metric on the family KðX Þ of all non-empty compact subsets of (X, d) is defined by letting

dH ðA;BÞ ¼ maxfsupfdða;BÞ : a 2 Ag; supfdðb;AÞ : b 2 Bgg

for every A;B 2 KðX Þ.It is worth noting that the topology induced by the Hausdorff metric dH on KðX Þ coincides with the Vietoris topol-

ogy sV on KðX Þ [6, p. 4.5.23].For the sake of completeness, we recall that the Vietoris topology on KðX Þ is the topology generated by the base B

consisting of all sets of the form

VðU 1; . . . ;UjÞ ¼ B 2 KðX Þ : B �[ji¼1

Ui;B \ Ui 6¼ ; for i ¼ 1; . . . ; j

( );

where U1, . . ., Uj 2 sd.Now let f: (X, d)! (X, d) be a continuous mapping. By letting �f ðAÞ ¼ f ðAÞ for every A 2 KðX Þ one defines a con-

tinuous mapping �f : KðX Þ ! KðX Þ.More generally, if S is a non-empty subspace ofKðX Þ and f isS-invariant, i.e., f ðAÞ 2 S for every A 2 S, we may

consider the partial map extension �fS : S ! S given by �fSðAÞ ¼ f ðAÞ for every A 2 S.

Observe that every continuous map f: (X, d)!(X, d) is KðX Þ-invariant and �fKðX Þ simply �f .Let us recall that a topological space X has the fixed point property (in short, f.p.p.) if every continuous map f:

X! X has a fixed point.

We will denote the family of all non-empty compact subsets of X which have the f.p.p. and the family of all non-

empty finite subsets of X by Sðp;X Þand SxðX Þ respectively.Observe that (X, sd) is homeomorphic to the subspace S1ðX Þ of kðX Þ consisting of all one-point subsets of X.

All spaces considered here are assumed to be infinite.

We refer the reader to [4] and [6] for notations and terminology not explicitly given.

3. Collective chaos versus individual chaos

Now we are ready to state our results on the relationships between the chaoticity of f and �f .

Theorem 3.1. Let (X, d) be a metric space, f: (X, d)!(X, d) a continuous mapping, and S a subspace of KðX Þ such that

S1ðX Þ � S. Let us assume that f is S-invariant. Then:

(1) If �fS is transitive, then f is transitive.

(2) If S � Sðp;X Þ and �fS is chaotic, then f is chaotic.

(3) If f is periodically dense and SxðX Þ � S, then �fS is periodically dense.

(4) If S � SxðX Þ and fS is chaotic, then f is chaotic.

Proof. To simplify the notation let us denote �fS by g. Moreover, for every C � X setVSðCÞ ¼ fK 2 S : K � Cg. SinceS1ðX Þ � S, it follows that VSðCÞ ¼ ; if and only if C = ;.

(1) Let us consider a pair A and B of non-empty open subsets of X. Since A, B 2 sd it follows that VSðAÞand VSðBÞare non-empty (basic) open sets of the subspaceS ofKðX Þ. Since g is transitive, there is some non-negative integer

j such that gjðVSðAÞÞ \VSðBÞ 6¼ ;.

A. Fedeli / Chaos, Solitons and Fractals 23 (2005) 1381–1384 1383

Now observe that gjðVSðAÞÞ � VSðf jðAÞÞ, in fact, let D 2 gjðVSðAÞÞ, then there is some C 2 VSðAÞ such that

gj(C) = fj(C) = D, since C � A, it follows that D � fj(A), i.e., D 2 VSðf jðAÞÞ.Therefore ; 6¼ gjðVSðAÞÞ \VSðBÞ � VSðf jðAÞÞ \VSðBÞ ¼ VSðf jðAÞ \ BÞ. Hence fj(A) \ B 5 ;, and f is

transitive.

(2) Now let us show that f is chaotic.

Since g is transitive, it follows, by (1), that f is transitive, so it is enough to show that f is periodically dense.

Let us take a non-empty open subset U of X, so VSðUÞ is a non-empty open subset of S.

Since g is periodically dense, there is some B 2 VSðUÞ and some positive integer j such that gj(B) = f j(B) = B. So

the map h: B ! B given by h(b) = f j(b) for every b 2 B is a continuous map.

Since B has the fixed point property (recall that S � Sðp;X ÞÞ, it follows that h has a fixed point, i.e., there is some

p 2 B such that h(p) = p.

So f j(p) = p and p is a periodic point of f contained in U.

Therefore f is periodically dense, hence chaotic.

(3) We have to show that g is periodically dense.

Let VSðU 1; . . . ;UnÞ ¼ VðU 1; . . . ;UnÞ \S be a non-empty basic open set of S. Since f is periodically dense and

every Ui, i = 1, . . ., n, is a non-empty open subset of X, there is a periodic point pi of period ji of f such that pi 2 Ui

for every i 2 {1, . . ., n}.Now set j = j1 . . . jn and B = {p1, . . ., pn}, then gj(B) = B.

So B is a periodic point of g such that B 2 VSðU 1; . . . ;UnÞ (recall that SxðX Þ � S). Therefore g is periodically

dense.

(4) By (1) we already know that f is transitive. Let us show that f is periodically dense. Let U be a non-empty open

subset of X. Then VSðUÞ is a non-empty open subset of S. Since g is periodically dense, there is a B 2 VSðUÞand some positive integer j such that gj(B) = B. Let h: B ! B be the map given by h(b) = f j(b) for every b 2 B.

Since B is finite (recall that S � SxðX Þ) and h is surjective, it follows that h is bijective, i.e., h is a permutation on B.

Therefore there is some integer m>0 such that hm(b) = b = f jm(b) for every b2B. So every point of B is a periodic point

of f contained U.

Therefore f is periodically dense, hence chaotic. h

If we take S ¼ KðX Þ in the theorem above we obtain the following:

Corollary 3.2. Let (X, d) be a metric space, and let f: (X, d) ! (X, d) be a continuous map. Then:

(i) if �f is transitive, then f is transitive;

(ii) if f is periodically dense, then �f is periodically dense.

Remarks 3.3

(i) Corollary 3.2 (i) gives the main result of [8].

(ii) By Corollary 3.2 it follows that if f is periodically dense and �f is transitive, then �f is chaotic.

Let (X, d) be a connected metric space with a disconnecting interval, namely with a subset A of X which is homeomor-

phic to an open interval of the real line and for each x 2 A;X n fxg has exactly two components. It is shown in [1] that a

continuous map f: (X, d) ! (X, d) is chaotic if and only if it is transitive. Then we have

Corollary 3.4. Let (X, d) be a connected metric space with a disconnecting interval and let f: (X, d) ! (X, d) be a contin-

uous mapping. Then �f is chaotic if and only if f and �f are transitive.

Let X be a metric space, D a dense subset of X and f: X ! X a continuous mapping such that f(D) � D. It is shown in

[7] that f is transitive if and only if fjD: D ! D is transitive. Since SxðX Þ is dense in KðX Þ, we have

Corollary 3.5. Let (X, d) be a metric space, f: (X, d)!(X, d) a continuous map and S ¼ SxðX Þ. Then the following

condition are equivalent:

(i) �fS is chaotic;

(ii) f is chaotic and �fS is transitive;

(iii) f is chaotic and �f is transitive.

1384 A. Fedeli / Chaos, Solitons and Fractals 23 (2005) 1381–1384

Now we will give some examples in order to make more transparent the picture described by the results given above.

Example 3.6. Let S1 be the unit circle in the plane and let f: S1 ! S1 defined by f(eih) = ei(h + 2pk) where k is a fixed

irrational number.

f is transitive [4], nonetheless �f is not transitive [8].

Example 3.7. Let X = [0,1] and let f: X! X be the tent map, namely the map defined by f(x) = 1 � j2x � 1j,f is a chaotic map [4] and �f is transitive [8], so by the results above �f is chaotic.

Now let S be the subspace of KðX Þ consisting of all intervals.

Clearly S ¼ Sðp;X Þ and f is S-invariant. In [8] it is shown that �fS is not transitive, so the chaoticity of f need not

imply the chaoticity of fSðp;X Þ.

Example 3.8. Let f: X! X be the tent map, and let us take the subspace S ¼ S1ðX Þ [ ff0; 1g; f12; 1gg of KðX Þ.

Clearly f is S-invariant.

Although f is chaotic, g ¼ �fS is neither transitive nor periodically dense.

Let B = {0,1}, observe that B is not a periodic point of g, in fact gj(B) = fj(B) = {0} for every positive integer j.Moreover B is an isolated point of S, in fact fBg ¼ VðU ; V Þ \S where U ¼ ½0; 1

4Þ and V ¼ ð3

4; 1�. Therefore g is not

periodically dense. Now let C ¼ f12; 1g and note that C is also an isolated point ofS, in fact fCg ¼ VðG;HÞ \S where

G ¼ ð14; 23Þ and H ¼ ð2

3; 1�. Therefore {B} and {C} are two non-empty open subsets ofS such that gj({B}) \ {C} = ; for

every non-negative integer j. Hence g is not transitive.

4. Conclusions

In this note we have shown some connections between the chaoticity of a discrete dynamical system and the chao-

ticity of several set-valued discrete dynamical systems associated to it. This kind of investigation should be useful in the

study of every phenomenon in which we want to understand the relationship between the dynamics of individual move-

ment and the dynamics of collective movement.

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