Chaos, Solitons and Fractals 25 (2005) 681–685

www.elsevier.com/locate/chaos

Chaos for induced hyperspace maps

John Banks

Department of Mathematics, La Trobe University, Bundoora 3086, Australia

Accepted 26 November 2004

Abstract

For (X,d) be a metric space, f :X! X a continuous map and ðKðX Þ;HÞ the space of non-empty compact subsets of

X with the Hausdorff metric, one may study the dynamical properties of the induced map

0960-0

doi:10.

E-m

�f : KðX Þ ! KðX Þ : A7!f ðAÞ: ð�Þ

H. Roman-Flores [A note on in set-valued discrete systems. Chaos, Solitons & Fractals 2003;17:99–104] has shown that

if �f is topologically transitive then so is f, but that the reverse implication does not hold. This paper shows that the

topological transitivity of �f is in fact equivalent to weak topological mixing on the part of f. This is proved in the more

general context of an induced map on some suitable hyperspace H of X with the Vietoris topology (which agrees with

the topology of the Hausdorff metric in the case discussed by Roman-Flores.

� 2005 Elsevier Ltd. All rights reserved.

1. Definitions and preliminary results

Let f :X! X be a continuous map on a metric space X. We say that f is (topologically) transitive if for any pair of

non-empty open sets U and V there is a kP 1 such that f k(U) \ V 5 ;. We say that f is totally transitive if each iterate

fm is transitive. We say that f is weakly (topologically) mixing if for all non-empty open sets U1, U2, V1 and V2 there

exists kP 1 such that f k(U1) \ V1 5 ; and f k(U2) \ V2 5 ;. We say that f is (topologically) mixing if for any pair

of non-empty open sets U and V there exists N P 1 such that for all k P N one has f k(U) \ V5 ;.A point x is periodic if f k(x) = x for some kP 1. The least such k is called the period of x. We say that f has sensitive

dependence on initial conditions if there is a constant d > 0 such that for every point x and every open set U about x,

there is a y 2 U and a k P 1 such that d(f k(x), f k(y)) P d. A map that is transitive, has a dense set of periodic points

and has sensitive dependence on initial conditions is called chaotic in [5, §1.8]. It turns out, however, that sensitive

dependence on initial conditions is a consequence of transitivity together with a dense set of periodic points [3,11].

We will consider dynamical properties of induced maps on various hyper-spaces, i.e., subspaces of the space 2X of all

non-empty closed subsets of X with the Vietoris or exponential topology. A basis for this topology on 2X is given by the

collection of sets of the form

hU 1; . . . ;Uni :¼ A 2 2X : A �[ni¼1

Ui and A \ Ui 6¼ ; for all i 6 n

( ); ð1Þ

779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

1016/j.chaos.2004.11.089

ail address: j.banks@latrobe.edu.au

682 J. Banks / Chaos, Solitons and Fractals 25 (2005) 681–685

where U1, . . .,Un are non-empty open subsets of X and the topology of any hyperspace H � 2X is just the subspace

topology induced by the Vietoris topology on 2X. We make use of the easily verified fact that if U = Ui for some

i 6 n, then for any mP 1

hU 1; . . . ;Uni ¼ hU 1; . . . ;Un;U ; . . . ;U|fflfflfflfflffl{zfflfflfflfflffl}m

i ð2Þ

We also make use of the facts that 2X contains the collection JðX Þ of all finite subsets of X (so every basic open set

hU1, . . .,Uni is non-empty) and JðX Þ is dense in 2X.

The set of all non-empty compact subsets of X is denoted by KðX Þ. It is well known that the Hausdorff metric in-

duces the Vietoris topology on KðX Þ and that KðX Þ inherits many useful properties possessed by X including com-

pactness, completeness and connectedness [9]. This space is the setting for the iterated function schemes of

Hutchison [8] and Barnsley [4]. KðX Þ is dense in 2X since JðX Þ � KðX Þ.We say that f :X! X is compatible with a subset H of 2X provided f ðAÞ 2 H for every A 2 H. Every map is com-

patible withJðX Þ and every continuous map is compatible withKðX Þ. For other subspaces of 2X, it may be possible to

impose extra conditions on f which ensure compatibility. For example, compatibility with 2X itself is equivalent to f

being a closed mapping and continuous closed mappings are always compatible with the space CðX Þ of all closed con-

nected subsets of X. Provided f is compatible with H � 2X , the induced map �f : H ! H : A7!f ðAÞ is well defined.

Where we wish to emphasize the particular hyperspace H to which �f is restricted we denote the induced map by �fH.

Notice that �fkðBÞ ¼ f kðBÞ for any B 2 H. We conclude this section with a theorem that collects together some

previously known results needed in the main parts of the paper.

Theorem 1. For any continuous map f

(a) If f is weakly mixing and U1, . . .,Un, V1, . . .,Vn are non-empty open sets, there is a k P 1 such that f k(Ui) \ Vi 5 ;for all i 6 n.

(b) f is weakly mixing iff for any non-empty open sets U and V there is a kP 1 such that f k(U) \ V5 ; and

f k(V) \ V5 ;.(c) f is weakly mixing iff for any non-empty open sets U, V and W there is a kP 1 such that f k(U) \ V5 ; and

f k(V) \W 5 ;.(d) If f is totally transitive with a dense set of periodic points, f is weakly mixing.

(e) If f weakly mixing then f is totally transitive.

Proof. (a) See [7].

(b) See [2].

(c) See [2].

(d) See [1, p. 508].

(e) Let m P 1 and let U and V be non-empty open sets. It is readily verified that the open set f�i(V) is non-empty for

0 6 i 6 m, so by (a) there is a kP 1 such that f k(U) \ f�i(V)5 ; for 0 6 i 6 m. Writing k = qm + r for some qP 0 and

0 6 r 6 m � 1 gives 0 < m � r 6 m. Thus ;5 f k(U) \ f�(m�r)(V), so

; 6¼ U \ f �ðkþm�rÞðV Þ ¼ U \ f �ðqþ1ÞmðV Þ ð3Þ

and hence fm(q+1)(U) \ V5 ;. h

One may define the Vietoris topology on KðX Þ for any topological space X, metrizable or otherwise. We can ensure

that JðX Þ � 2X and that every continuous map is compatible with KðX Þ by assuming that X is Hausdorff.

In the absence of a metric, the definition of sensitive dependence and hence Devaney�s definition of chaos have no

meaning. In view of the main theorem of [3]. one could regard transitivity together with a dense set of periodic points as

generalizing the definition of chaos. Although the term chaos is not used in this way here, the reader is invited to make

the obvious adjustments to definitions and statements of theorems.

2. Transitivity results

Throughout this section we assume f is compatible with H. Since f is continuous, so is �fH by [9, p. 170] and hence

Theorem 1 applies to �fH. We are mainly interested in the situation where H is dense in 2X. We have already noted that

J. Banks / Chaos, Solitons and Fractals 25 (2005) 681–685 683

JðX Þ is dense in 2X an this enables us to see that many other hyperspaces are also dense in 2X. For example, KðX Þ, thespace LKðX Þ of closed locally compact subsets of X and the space LCðX Þ of closed locally connected subsets of X are

all dense in 2X since they all containJðX ÞOn the other hand,CðX Þ is closed in 2X and so is only dense in trivial cases [12].

Theorem 2 (Main theorem). Suppose H is dense in 2X and let �f ¼ �fH. Then the following are equivalent.

(a) f is weakly mixing,

(b) �f is weakly mixing,

(c) �f is transitive.

Proof. (a) ) (b). Suppose f is weakly mixing. By Theorem 1(b), it suffices to show that for any non-empty open sets

U ¼ hU 1; . . . ;Uri \H and V ¼ hV 1; . . . ; V si \ C; ð4Þ

there is a k P 1 such that f k(U) \ V5 ; and f k(V) \ V5 ;. Letting n = max(r, s) gives

U ¼ hU 1; . . . ;Uni \H and V ¼ hV 1; . . . ; V ni \H; ð5Þ

where Ui = Ur for i > r and Vi = Vs for i > s. By Theorem 1(a), there is a kP 1 such that f k(Ui) \ Vi 5 ; and

f k(Vi) \ Vi 5 ; for each i 6 n whence

G ¼ hU 1 \ f �kðV 1Þ; . . . ;Un \ f �kðV nÞi ð6Þ

and

H ¼ hV 1 \ f �kðV 1Þ; . . . ; V n \ f �kðV nÞi; ð7Þ

are non-empty and open in 2X. Since H is dense there exist elements A 2 G \H � U and B 2 H \H � V . It is easy to

check that �fkðAÞ 2 V and �f

kðBÞ 2 V .(b) ) (c). Follows directly from the definitions.

(c) ) (a). Suppose �f is transitive. By Theorem 1(b), it suffices to show that for any non-empty open sets U, V and W

in X there is a k P 1 such that f k(U) \ V 5 ; and f k(U) \W 5 ;. By transitivity of �f there is a kP 1 such that

G ¼ ðhUi \HÞ \ �f�kðhV ;W i \HÞ 6¼ ;. For any B 2 G, we have B � U, f k(B) \ V5 ; and f k(B) \W 5 ;. h

The situation for mixing maps is quite straightforward. The technique of Theorem 2 is easily adapted to prove the

following.

Theorem 3. Suppose H is dense in 2X. Then f is mixing iff �f ¼ �fH is mixing.

Of the subspaces of 2X mentioned so far, only CðX Þ is not dense in 2X. Indeed, if X is non-trivial, CðX Þ is nowheredense in 2X. We briefly consider the extent to which the results of the previous section apply to �f CðxÞ. If X is totally dis-

connected, then fCðX Þ is topologically conjugate to f. Apart from this entirely uninteresting case, there seems little pros-

pect of finding reasonably general conditions which would ensure that �f CðX Þ has a dense set of periodic points. On the

other hand, there are more interesting obstacles to the transitivity of �f CðX Þ. At least one part of Theorem 2 still holds for

connected spaces.

Theorem 4. Let X be connected. If �f ¼ �f CðX Þ is transitive then f is weakly mixing.

Proof. Let U, V and W be non-empty open subsets of X. Then hUi \ CðX Þ is non-empty since it contains singleton sets

and hV ;W ;X i \ CðX Þ is non-empty since it contains X. By the transitivity of �f , there is an A 2 hUi \ CðX Þ and a k P 1

such that f kðAÞ 2 hV ;W ;X i \ CðX Þ. Thus fk(A) \ V5 ; and fk(A) \W 5 ; which shows that f is weakly mixing by

Theorem 1(c). h

This theorem enables us to exhibit cases where �f CðX Þ can never be transitive. Suppose �f CðIÞ is transitive, where I is a

non-degenerate interval. Since f is weakly mixing by Theorem 4 and hence transitive, it has a dense set of periodic points

by [11, p. 357]. There is therefore a periodic orbit P for f with period greater than 1 such that neither s = minP nor

t = maxP is an endpoint of I. Thus s < t so J = {x 2 I :x < s}, (s, t) and K = {x 2 I :x > t} are all open and non-empty.

Let C be any element of the non-empty open set hJ ;K; Ii \ CðIÞ. Since C is connected and contains s and t, we must

have [s, t] � C. By definition of s and t, we must have fk(C) � fk([s, t]) � [s, t] for all kP 1 whence �fkCðIÞðCÞ 62 hðs; tÞi. But

this implies that

�fk ðhJ ;K; IiÞ \ hðs; tÞi ¼ ;; ð8Þ

CðIÞ

684 J. Banks / Chaos, Solitons and Fractals 25 (2005) 681–685

which contradicts the transitivity of fCðIÞ. A slightly more complicated argument shows that fCðS1Þ can never be transi-

tive. These considerations suggest that non-trivial examples for which X is connected and �f CðX Þ transitive may be dif-

ficult to find. Of course, it may be that the specifically one-dimensional connectivity properties used in the above

argument make this case exceptional.

3. Dense periodic points

Together with the following lemma and the main theorem of [3], Theorem 2 shows that in cases where JðX Þ � H

andH is metrizable, �f will be chaotic provided f is weakly mixing and has a dense set of periodic points. This covers the

case H ¼ KðX Þ considered in [10].

Lemma 1. Suppose JðX Þ � H. If f has a dense set of periodic points, then so does �f ¼ �fH.

Proof. It suffices to show that there is an element of any non-empty basic open set hU 1; . . . ;Uni \H which is periodic

under fH. There is a periodic point ai 2 Ui for each i 6 n and A = {a1, . . .,an} is an element of hU 1; . . . ;Uni \H. Letting

m be the least common multiple of the periods of a1, . . .,an yields �fmðAÞ ¼ A. h

In general, it is not obvious under what conditions a dense set of periodic points for �f guarantees the existence of a

dense set of periodic points for f. It should be noted that a point A of period n for �f satisfies f n(A) = A. In other words A

is an exactly invariant subset for f n. If �f is transitive, it is weakly mixing by Theorem 2 and hence f n is transitive by

Theorem 1. It is well known that the proper closed invariant subsets for transitive maps are nowhere dense, so for tran-

sitive �f all periodic points other than the domain X are nowhere dense subsets of X.

We now give an example of a transitive f such that �f has a dense set of periodic points but f has none. In this case f is

not weakly mixing so �f is not transitive. We give the Cantor space f0; 1gN an Abelian topological group structure (that

of the 2-adic integers) by defining addition of sequences to be

ðx1; x2; x3; . . .Þ þ ðy1; y2; y3; . . .Þ ¼ ðz1; z2; z3; . . .Þ; ð9Þ

where zi = xi + yi + ci (mod2), c1 = 0 and ci+1 = xi + yi + ci (div2) for i > 1. Thus sequences are added pointwise (mod 2)

with carry to the right. With this definition of addition, it may be shown that the group translation

T ðx1; x2; x3; . . .Þ ¼ ðx1; x2; x3; . . .Þ þ ð1; 0; 0; 0; . . .Þ ð10Þ

is a homeomorphism for which every orbit is dense, so T is transitive but has no periodic points. However, T is not

totally transitive since T 2n fails to be transitive for each n P 1. Maps like T that are obtained as translations groups

of p-adic integers are often called adding machines.

Since f0; 1gN is compact KðX Þ ¼ 2X is a natural domain for an induced map T . It may be shown that the so called

cylinder sets

C½x1; x2 . . . xn� ¼ ðy1; y2; . . .Þ 2 f0; 1gN : yi ¼ xi for i 6 nn o

; ð11Þ

where x1,x2, . . .,xn 2 {0,1} form a basis of clopen sets for f0; 1gN. It turns out that every cylinder set C[x1x2 � � �xn] isperiodic for T since

T 2n ðC½x1x2 � � � xn�Þ ¼ C½x1x2 � � � xn�: ð12Þ

Detailed proofs of the preceding claims about T may be found in [1, pp. 519–520]. By an argument similar to that of

Lemma 1, the set of finite unions of cylinder sets forms a dense set of periodic points for T . It would be interesting to

find an example of a transitive �f with a dense set of periodic points such that f does not.

4. Edalat�s generalization

The upper or sub-open topology on 2X is coarser than the Vietoris topology. It is generated by basic open sets of the

form hUi where U is a non-empty open subset of X. If X is a metric space, KðX Þ with the upper topology is quasi-metr-

izable, with quasi metric hu given by

huðK; LÞ ¼ inf r > 0 : K �[x2K

BrðxÞ( )

: ð13Þ

J. Banks / Chaos, Solitons and Fractals 25 (2005) 681–685 685

If f is chaotic then �f ¼ �fKðX Þ is ‘‘chaotic’’, where one interprets the definition of chaos with respect to hu in the obvi-

ous way [6]. Indeed, Edalat [6] shows that each of the three conditions in Devaney�s definition transfers in directly from

f to �f with respect to the upper topology on KðX Þ. The proofs are easily adapted to show that if f is transitive with a

dense set of periodic points and JðX Þ � H, then �f also has these properties with respect to the upper topology on H.

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