CHAOS ON HYPERSPACES
JUAN LUIS GARCÍA GUIRAO1 , DOMINIK KWIETNIAK2 , MAREK
LAMPART3 , PIOTR OPROCHA4 AND ALFRED PERIS5
Abstract. Let f be a continuous self–map of a compact metric
space X. The transformation f induces in a natural way a self–map
f defined on the hyperspace K(X) of all nonempty closed subsets of
X. We study which of the most usual notions of chaos for dynamical
systems induced by f are inherited by f and vice versa. We consider distributional chaos, Li–Yorke chaos, ω chaos, Devaney chaos,
topological chaos (positive topological entropy), specification property and their variants. We answer questions stated independently
by Roman-Flores and by Banks.
1. Introduction
There are many competing notions of chaos for dynamical systems induced by the action of a continuous map f : X → X on a compact metric
space (X, d). This is a difficulty for the search of a universal definition.
Therefore, as it was stated in [6], one should rather concentrate on relations between various definitions of chaos and fields of applications.
There is a need for a theory of chaos inside which various degrees of
chaos might be accepted and compared.
It is the aim of the present paper to explore different notions of chaos
in the following context: if we consider a dynamical system defined
by a continuous map f : X → X, describing the dynamics of individuals (points) in the state space X, then we can study the induced map
f : K(X) → K(X) given by f (K) = f (K) for a compact set K ⊂ X, as a
form of collective dynamics. This interpretation raises a natural question:
Key words and phrases. distributional chaos, Li–Yorke chaos, ω chaos, Devaney
chaos, topological chaos, specification property.
2000 Mathematics Subject Classification: Primary 37B40, 37D45; Secondary
39B12, 37E10, 54B20.
This research was supported in part by: MEC (Ministerio de Educación y Ciencia,
Spain) and FEDER (Fondo Europeo de Desarrollo Regional), grants MTM2004-02262,
MTM2005-03860, MTM2005-06098-C02-01, MTM2006-26627-E; Fundación Séneca
(Comunidad Autónoma de la Región de Murcia), grant 00684-FI-04; Junta de Comunidades de Castilla-La Mancha, grant PAI06-0114; Grant Agency of the Czech
Republic, grant GA201/06/0318.
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J.L.G. GUIRAO ET AL.
does individual chaos imply collective chaos? and conversely? These
questions have attracted many authors (see, e.g., [3, 20, 25, 26, 24, 29]),
and many partial answers have been obtained. A more general problem
of determining the connection between properties of dynamics of f and
f was stated by Bauer and Sigmund [5] in 1975.
Here we consider some of the most popular definitions of chaos in
the setting of discrete dynamical systems: distributional chaos (dC),
Li–Yorke chaos (LYC), ω chaos (ωC), Devaney chaos (DevC), topological chaos (positive topological entropy, PTE), strong periodic specification property (SPSP) and their variants like, e.g. exact Devaney chaos
(exDevC) and total Devaney chaos (totDevC). All necessary definitions
are presented in Section 3. We survey known results related to our problem and complete the picture with some facts and examples solving some
open questions. Our results can be summarized in the following diagram:
Notions of chaos
f
dC, LYC, ωC, PTE, SPSP, totDevC, exDevC
DevC
f
=⇒
⇐=
=⇒
⇐=
Therefore, individual chaos implies collective chaos and not conversely
for all notions except for the classical Devaney definition of chaos for
which there is no relation between individual and collective chaos.
The paper is divided into five sections. In Section 2 we provide the
basic definitions and notation. In Section 3 we define the notions of chaos
that we will consider. Section 4 is devoted to develop some auxiliary
results needed in Section 5 to prove our main results.
2. Basic definitions and notation
For completeness we recall here some basic definitions and facts from
topology and topological dynamics used in the sequel.
Let N be the set of positive integers. Through the rest of this paper X
denotes an infinite compact metric space with a metric d. Given n ∈ N
and a map f : X → X by f ×n we denote the n-th Cartesian product
of the map f defined on the n-th Cartesian product X ×n of the space
X. We write f n for n-th iterate of a map f , i.e., f ◦ . . . ◦ f (n-times
composition). A discrete dynamical system is a pair (X, f ) where f : X →
X is a continuous map. If we endow the space K(X) of all nonempty
compact subsets of X with the Hausdorff metric dH given by d, then it
is a compact metric space, and the system (X, f ) induces a set–valued
CHAOS ON HYPERSPACES
3
discrete dynamical system (K(X), f), where f : K(X) → K(X) is defined
as f (K) = {f (a) : a ∈ K} for each K ∈ K(X). For any finite collection
S1 , . . . , Sk of nonempty subsets of X let
k
S1 , . . . , Sk = K ∈ K(X) : K ⊂
Si , K ∩ Si = ∅ for i = 1, . . . , k .
i=1
A basis for the topology of the metric space (K(X), dH ) can be explicitly
described as the collection of all subsets of K(X) of the form U1 , . . . , Uk where U1 , . . . , Uk is an arbitrary finite family of nonempty open subsets
of X. We refer the reader to [17].
Let (X, f ) be a dynamical system. We define the orbit of a point x ∈ X
as the set O(x, f ) = {f n (x) : n ∈ N}. The ω-limit set of x is the set
ω(x, f ) of all limit points of the orbit of x, regarded as a sequence.
We say that x ∈ X is a periodic point if f n (x) = x for some n ∈ N.
The set of all periodic points of the system (X, f ) is denoted as Per(f ).
If x ∈ Per(f ) then the smallest positive integer n such that f n (x) = x
is called a primary period of x. A point x ∈ X is said to be regularly
recurrent if for every neighborhood V of x, there is an n ∈ N such that
f nk (x) ∈ V for every nonnegative integer k.
A map f is:
(1) transitive if for any pair of nonempty open sets U, V ⊂ X there
exists an n ∈ N such that f n (U) ∩ V = ∅;
(2) totally transitive if for each n ∈ N the map f n is transitive;
(3) weakly mixing if the map f ×2 is transitive;
(4) mixing if for any pair of nonempty open sets U, V ⊂ X there exists
an N ∈ N such that f n (U) ∩ V = ∅ for every integer n ≥ N;
(5) locally eventually onto if for every nonempty open set U ⊂ X
there exists an m ∈ N such that f m (U) = X. Since this property
can be regarded as the topological analog of exactness defined in
ergodic theory, it is often called topological exactness. We use the
second name here.
3. Definitions of chaos
We shall consider the most popular notions of chaos for discrete dynamical systems: distributional chaos; Li–Yorke chaos; ω–chaos; Devaney
chaos and the property “f has positive topological entropy”, also known
as topological chaos. In addition we consider the specification property,
which can be regarded as an indicator of fairly strong chaotic behavior
of the system. See the survey articles by Forti [12] and Kolyada [18] for
more details.
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J.L.G. GUIRAO ET AL.
From now on, f denotes a continuous self–map defined on a compact
metric space (X, d).
Distributional chaos
The notion of distributional chaos was introduced in [27] and then
generalized in [2, 28].
For x, y ∈ X; t ∈ R and a positive integer n, let
ξ(x, y, n, t) = #{i; 0 ≤ i < n and d(f i(x), f i (y)) < t},
Put
1
∗
(t) = lim sup ξ(x, y, n, t) and
Fxy
n→∞ n
1
Fxy (t) = lim inf ξ(x, y, n, t).
n→∞ n
∗
∗
≤ 1,
Then both Fxy and Fxy are nondecreasing maps, with 0 ≤ Fxy ≤ Fxy
∗
∗
Fxy
(t) = 0 for t < 0, and Fxy (t) = 1 for t > diam(X). We refer to Fxy
and Fxy as the upper and l ower distribution map of x, y, respectively.
If there exists a pair of points x, y in X such that
∗
(d1 C) Fxy
≡ 1 and Fxy (t) = 0 for some t > 0, or
∗
∗
(d2 C) Fxy
≡ 1 and Fxy (t) < Fxy
(t) for some t > 0, or
∗
(d3 C) Fxy (t) < Fxy (t) for all t ∈ J, where J is a nondegenerate interval,
then we say that the map f is distributionally chaotic of type 1 − 3
(briefly, d1 C, d2 C, or d3 C, respectively).
Obviously, d1 C implies d2 C and d2 C implies d3 C, but not conversely
(see for instance [2, 28]).
Li–Yorke chaos
A pair of points x, y ∈ X is called a Li-Yorke pair if
(1) lim supn→∞ d(f n (x), f n (y)) > 0
(2) lim inf n→∞ d(f n (x), f n (y)) = 0.
A set S ⊂ X is called a LY-scrambled set for f (Li-Yorke set) if #S ≥ 2
and every pair of different points in S is a LY-pair where # means the
cardinality . For continuous self–maps on the interval [0, 1], Li and Yorke
[23] suggested that a map should be called “chaotic” if it admits an uncountable scrambled set. This was subsequently accepted as a formal
definition. Hence, we say that a map f is Li and Yorke chaotic (briefly,
LYC) if it has an uncountable LY-scrambled set. One may consider
weaker variants of chaos in the sense of Li and Yorke based on the cardinality of scrambled sets (see for instance [16]).
ω chaos
CHAOS ON HYPERSPACES
5
The following notion was proposed by Li [22]. A set S ⊂ X containing
at least two points is called an ω-scrambled set for f if for any two different
points x, y in S the following conditions are fulfilled:
(1) ω(x, f ) \ ω(y, f ) is uncountable,
(2) ω(x, f ) ∩ ω(y, f ) = ∅ and
(3) ω(x, f ) \ Per(f ) = ∅.
The map f is ω-chaotic (briefly, ωC) if there is an uncountable ωscrambled set. Again one may consider weaker variants of ω-chaos changing the condition on cardinality of the ω-scrambled set (see [21]).
Devaney chaos
A map f is called Devaney chaotic (briefly, DevC) if it satisfies the
following two properties:
(1) f is transitive,
(2) the set Per(f ) is dense in X.
The original definition given by Devaney [10] contained an additional condition on f , which reflects unpredictability of chaotic systems: sensitive
dependence on initial conditions. However, it was proved see, e.g., [4] or
[14] that sensitivity is a consequence of transitivity and dense periodicity
under the assumption that X is an infinite set.
We can replace the first condition in the above definition by some
stronger transitivity property such as total transitivity or topological
exactness to obtain new variants of Devaney chaos. We say that a map f
is totally Devaney chaotic (briefly, totDevC), (or exactly Devaney chaotic
(briefly, exDevC), respectively) if it has dense set of periodic points and
it is totally transitive (topologically exact, respectively).
Totally Devaney chaotic systems were studied by Furstenberg [13] before Devaney’s definition appeared, and they were called F -systems. Definition of exact Devaney chaos was stated in [19].
Positive topological entropy: topological chaos
An attempt to measure the complexity of a dynamical system is based
on a computation of how many points are necessary in order to approximate (in some sense) with their orbits all possible orbits of the system.
A formalization of this intuition leads to the notion of topological entropy of the map f , which is due to Adler, Konheim and McAndrew [1].
We recall here the equivalent definition formulated by Bowen [7], and
independently by Dinaburg [11]: the topological entropy of a map f is a
number h(f ) ∈ [0, ∞] defined by
h(f ) = lim lim sup #E(n, f, ε),
ε→0
n→∞
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J.L.G. GUIRAO ET AL.
where E(n, f, ε) is a (n, f, ε)–span with minimal possible number of
points, i.e., a set such that for any x ∈ X there is y ∈ E(n, f, ε) satisfying d(f j (x), f j (y)) < ε for 1 ≤ j ≤ n.
A map f is topologically chaotic (briefly, PTE) if its topological entropy
h(f ) is positive.
Specification properties
In spite of their rather complicated definition, the specification properties, due to Bowen [8], are satisfied by many discrete dynamical systems
arising naturally in applications.
We follow here the terminology used in [5] for the specification property
and its variants. A more extensive study of these properties can be found
in [9]. However the terminology used there differs from ours.
We also define two new versions of specification properties, recurrent
strong specification property and recurrent specification property, which
we need in the sequel.
A map f satisfies strong periodic specification property (briefly, SPSP)
if for every ε > 0 there exists an integer Nε > 0 such that for any s ≥ 2,
any points y1 , . . . , ys ∈ X and any integers 0 = a1 ≤ b1 < a2 ≤ b2 < . . . <
as ≤ bs with aj − bj−1 ≥ Nε for j = 2, . . . , s there exists a point x ∈ X
such that the following two conditions hold:
(1) d(f i (x), f i (yl )) < ε for al ≤ i ≤ bl and l = 1, . . . , s,
(2) f (x)bs +Nε = x.
If this definition is fulfilled only for the special case s = 2, then we say
that f satisfies the periodic specification property (briefly, PSP). If we
omit condition (2) in the definition of SPSP (PSP, respectively) we obtain the notion of strong specification property (SSP) (weak specification
property (WSP), respectively).
If we replace conditions (1) and (2) in the definition of SPSP by
(1 )
d(f i+k(bs +N ) (x), f i+k(bs +N ) (yl )) < ε
for k ∈ N, al ≤ i ≤ bl , and l = 1, . . . , s, then we obtain the concept
of recurrent strong specification property (briefly, RSSP). If the system
fulfils the definition of RSSP only for the special case s = 2, then we say
that f holds recurrent weak specification property (briefly, RWSP).
4. Auxiliary results
We provide in this section some lemmas and an example needed in
Section 5.
Let (X, f ) be a dynamical system.
CHAOS ON HYPERSPACES
7
Lemma 1. If the set of all regularly recurrent points of f is dense in X,
then the induced map f has dense set of periodic points.
Proof. By [29, Thm. 2.2.] it is sufficient to prove that for every nonempty
open set U ⊂ X there is a periodic point for f in U
. Compactness
of X implies that there is a nonempty open subset V of U such that
V ⊂ U. By our assumption we can find a regularly recurrent point x ∈ V.
Hence there is a positive integer n such that O(x, f n ) ⊂ V. Therefore
O(x, f n ) ⊂ V, in particular ω(x, f n ) ⊂ V. Since ω(x, f n ) ∈ K(X) and
n
f (ω(x, f n )) = ω(x, f n ), we have found a periodic point for f in U
and
the proof is complete.
The proof of the following result follows directly from the definitions.
Lemma 2. If {(Xn , fn )}∞
n=1 is any sequence of discrete dynamical systems such that for every positive integer n the set of all regularly recurrent
∞
points of fn is dense in Xn , then the product system ( ∞
n=1 Xn ,
n=1 fn )
has a dense set of regularly recurrent points.
of dynamical
systems with
Lemma 3. If {(Xn , fn )}∞
n=1 is any sequence
∞
X
,
f
)
has
RSSP or
SPSP or PSP then the product system ( ∞
n=1 n
n=1 n
RWSP respectively.
Proof. Let us set any ε > 0. There exists a positive integer M such that
∞
1
ε
< .
i
2
4
i=M +1
Let Nn be the constant given by PSP for fn and 4ε , where 1 ≤ n ≤ M
and let N = max Nn . Let x, y ∈ ∞
n=1 Xn and let a1 ≤ b1 < a2 ≤ b2
be such that b1 − a1 > N and b2 − a2 > N. Let p1 , . . . , pM be periodic
points such that each point pj is obtained by PSP for fj , xj , yj and times
a1 ≤ b1 < a2 ≤ b2 . Let us define the point
p = (p1 , . . . , pM , qM +1 , qM +2 , . . . ) ∈
∞
Xn
n=1
where qj ∈ Xj are any points. Direct calculations show that p fulfills the
conditions of the RWSP definition.
In the case of RSSP the proof is similar.
Lemma 4. If map f has RSSP ( RWSP) then the induced map f has
SPSP ( PSP).
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J.L.G. GUIRAO ET AL.
Proof. Let assume that f has RSSP. Let set any ε > 0 and let N be
the constant given by RSSP for 2ε . Let us fix s ≥ 2, sets A1 , . . . , As and
a1 ≤ b1 < a2 ≤ b2 < · · · < as ≤ bs with bj − aj > N.
i
l
l
, . . . , Ui,r
form an open cover of the set f (Al ), given by balls
Let Ui,1
of diameters less than 4ε and centers Ai where al ≤ i ≤ bl . We may also
l
∈ Al be
assume that the number r is the same for any i and l. Let yi,j
l
i,j
the center of the ball Ui,1 . Let x be the point given by RSSP for the
1
s
, . . . , yi,j
and a1 ≤ b1 < a2 ≤ b2 < · · · < as ≤ bs . By similar
points yi,j
arguments to those in the proof of Lemma 1 we may find periodic points
Pi,j ∈ K(X) with period bs + N and such that
ε
k
k l )) ≤ for al ≤ i ≤ bl and l = 1, . . . , s.
dH (f (Pi,j ), f ( yi,j
2
If we set
Pi,j ,
P =
i,j
then P is desired periodic point.
The proof for the RWSP case is identical.
Lemma 5. For a dynamical system (X, f ) the following conditions are
equivalent:
(1) f is topologically exact,
(2) f is topologically exact.
Proof. Assume that f is exact. Let U be a nonempty open subset of X.
Since U
is nonempty and open in K(X), there exists an N ∈ N such
N
N
that f (U
) = K(X). In particular X ∈ f (U
) ⊂ f N (U)
which
gives X = f N (U).
Conversely, suppose that f is exact. We have to show that for every
nonempty open set U = U1 , . . . , Uk ⊂ K(X) there is an N ∈ N such
N
that f (U) = K(X). By compactness of X for every j = 1, . . . , k there
is a nonempty open set Vj ⊂ Uj such that Vj ⊂ Uj . Since f is exact,
we can find positive integers N1 , . . . , Nk such that f Nj (Vj ) = X for every
j = 1, . . . , k. Let N = max{N1 , . . . , Nk }, then
N
f (V1 , . . . , Vk ) = f N (V1 ), . . . , f N (Vk )
= K(X).
N
Hence f (U) = K(X).
Example 6. To present our next example we need to fix some notation.
Let n ∈ N and let Zn+1 be a cyclic group with n + 1 elements. We endow
Zn+1 with the discrete topology. From now on ”+” and ”−” denote
CHAOS ON HYPERSPACES
9
addition and subtraction mod (n+1). Let Xn = (Zn+1 )∞ = {(xm )∞
m=1 :
xm ∈ Zn+1 , m ∈ N} be the product topological space of countably infinite
copies of Zn+1 . It is well known that Xn is homeomorphic to the Cantor
set, that is, Xn is an compact, perfect and has countable base containing
clopen sets. This basis can be chosen to consist of cylinder sets, i.e., sets
of the form
[z1 , . . . , zk ] = {(xm )∞
m=1 ∈ Xn : x1 = z1 , . . . , xk = zk },
where k ∈ N and z1 , . . . , zk is an arbitrary sequence of elements of Zn+1
of length k.
∞
We define the map fn : Xn → Xn , by fn ((xm )∞
m=1 ) = (ym )m=1 , where
if x1 = xn+1
xm+1
ym =
1 + xm+1 if x1 = xn+1
for all m ∈ N.
Lemma 7. Let n ∈ N. Then
(1) the map fn is continuous,
(2) fn does not admit any periodic point with primary period equal to
n,
(3) if n ≥ 3, then fn has SPSP,
(4) the map fn is topologically exact.
Proof.
(1) We need to show that, given z ∈ Xn , the preimages under fn of
arbitrary small open neighbourhoods of z are open. Indeed, let
[z1 , . . . , zk ] (k ≥ n), we have
⎛
⎞
fn−1 ([z1 , . . . , zk ]) = ⎝
[a, z1 , . . . , zk ]⎠ ∪
a∈
n+1\{zn }
[zn − 1, z1 − 1, . . . , zk − 1]
which is open. If k < n we have disjoint decomposition
[z1 , . . . , zk , a1 , . . . , an−k ].
[z1 , . . . , zk ] =
a1 ,...,an−k
(2) Suppose that there is a sequence (xm )∞
m=1 ∈ Xn such that
n
∞
∞
(ym )∞
m=1 = fn ((xm )m=1 ) = (xm )m=1 .
By definition of fn we see that k + xm+n = xm for all m ∈ N,
where k = #{j ∈ {1, . . . , n} : xj = xj+n }. Clearly 0 ≤ k ≤ n.
Consider two cases:
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J.L.G. GUIRAO ET AL.
(a) If k > 0 then there exists an j ∈ {1, . . . , n} such that xj =
xj+n and taking into account the equality k + xj+n = xj we
see that k = 0, which is a contradiction.
(b) If k = 0, then in particular xn+1 = x1 , therefore k ≥ 1, which
is a contradiction.
(3) Rewriting the definition of SPSP for fn and using elementary
properties of the compact metric space Xn we see that it is enough
to prove the following: there is N ∈ N such that for every
s, t ∈ N and every two finite sequences u1 , . . . , us and v1 , . . . , vt
of elements of Zn+1 we can find a sequence w1 , . . . , wN such that
fns+N ([u1 , . . . , us , w1 , . . . , wN , v1 , . . . , vt ]) = [v1 , . . . , vt ].
Fix s, t and two finite sequences u1 , . . . , us and v1 , . . . , vt as
above. We claim that N = 2n. To see this let k = #{j ∈
{1, . . . , s − n} : xj = xj+n } mod (n + 1). If k > 0 then define wi = us−n+i for i = 1, . . . , n − k + 1 and wi = us−n+i + 1
for i = n − k, . . . , n. Otherwise set wi = us−n+i + 1 for i =
1, . . . , n. In either case, for i = n + 1, . . . , 2n define wi such
that wi = wi−n and wi = vi . We can do this since we assumed n ≥ 2. By the construction the condition of equality
fns+N ([u1 , . . . , us , w1 , . . . , wN , v1 , . . . , vt ]) = [v1 , . . . , vt ] holds and
we are done.
(4) It is clear (using the same arguments) that if [u1 , . . . , uk ] is a
nonempty cylinder set, then fnk+2n ([u1 , . . . , uk ]) = Xn .
5. Main results
The following results show the forcing relations for the notions of dC,
LYC, ωC and PTE between the maps f and f . The first example constructed in the proof of Theorem 10 is known (see [15], or [20]) but its
chaotic properties, apart from topological entropy, have not been explored so far. For the convenience of the reader we provide the complete
arguments.
Theorem 8.
(1) If there is a set S ⊂ X which is dj C scrambled (j ∈ {1, 2, 3})
(ω-scrambled, LY-scrambled, respectively) for f then there exists
dj C scrambled (j ∈ {1, 2, 3}) (ω-scrambled, LY-scrambled, respectively) set for f with the same cardinality as S.
(2) If f is dj C ( LYC, ωC, respectively) then the same holds for f .
CHAOS ON HYPERSPACES
11
Proof. Observe that we can regard (X, f ) as a subsystem of (K(X), f )
identifying a point x ∈ X with the set {x} ∈ K(X) , moreover this embedding is an isometry when one considers a metric d on X and a corresponding Hausdorff metric dH on K(X). Note also that f ({x}) = {f (x)}.
Since the existence of the scrambled set fulfilling one of definitions of
chaos listed in the theorem is sufficient for the whole system to be chaotic,
the theorem follows.
Remark 9. The fact that the embedding sending a point x to the set
{x} is an isometry was needed only to prove that the condition d3 C for
f holds for the induced map too, since d3 C is a metric property which is
not a conjugacy invariant (see [2]).
In [5] (see also [20]) it was shown that if the system (X, f ) has positive topological entropy, then the induced system (K(X), f) has infinite
topological entropy.
In the proof of the following theorem we provide two examples of dynamical systems which cannot be regarded as chaotic in any sense, but
their extension to hyperspaces can be regarded as chaotic.
Theorem 10. There exists a compact metric space X with a zero topological entropy map f for which there exists no LY pairs, neither d3 C
scrambled set, nor ω-scrambled set such that f is PTE, d1 C, ωC and
LYC.
Proof. Let X∞ = Z∪{∞} be a one-point compactification of the integers
(considered as a discrete topological space). Define a map T : X∞ → X∞
by
n + 1, if n ∈ Z,
T (n) =
∞,
if n = ∞.
¥
... ...
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J.L.G. GUIRAO ET AL.
Observe that all pairs of points from X∞ are asymptotic, hence the
system (X∞ , T ) can not have any scrambled set and has zero topological
entropy.
To prove that T exhibits chaos, we will show that (K(X∞ ), T ) contains
a subsystem conjugated to the full shift on two symbols (Σ2 , σ). Let K∞
be the family of all subsets of X∞ containing ∞. It is easy to see that the
map g : K∞ → Σ2 which sends the set A ∪ {∞} ∈ K∞ to IA ∈ Σ2 , where
A ⊂ Z, and IA denotes the indicator function of A is a homeomorphism
that conjugate systems (K∞ , T ), and (Σ2 , σ).
We modify our example to show that even infinite topological entropy
is possible.
Let T be a map defined by T (x) = x + 1 for every real number x. The
one-point compactification of reals is well known to be homeomorphic to
the circle S1 , and we extend T to S1 by setting T (∞) = ∞.
Again every pair of points x, y in S1 is an asymptotic pair, and hence
there is no scrambled set for T and h(T ) = 0. We claim that for
every natural number n the system (K(S1 ), T ) contains a subsystem
×n
conjugated to (Σ×n
). We prove this for the case n = 2, the proof
2 ,σ
for the general case is analogous.
Indeed, consider a pair (γ1 , γ2) consisting of two infinite and disjoint
orbits of T . Let K∞ be a family of all compact subsets of S1 of the
form A ∪ B ∪ {∞} where A is (a possibly empty) subset of γ1 , and
B is (a possibly empty) subset of γ2 . The set K∞ is closed and T invariant. Observe that such sets A and B are in a natural way in oneto-one correspondence with some subsets of the integers. It is clear that,
as above, we can define the desired conjugacy as a map which sends the
set A ∪ B ∪ {∞} ∈ K∞ to (IA , IB ) ∈ Σ×2
2 where IA and IB are indicator
functions of A and B regarded as subsets of Z.
We now analyze the three versions of Devaney chaos. Our purpose is
to study the connection between Devaney chaos for f and f . First, let
us recall some known results.
Theorem 11 (Bauer-Sigmund).
(1) If f is totDevC then f is totDevC.
(2) If f has SPSP ( PSP) then f has SPSP ( PSP).
Proof. See Proposition 3 of [5].
We add the following result which is a consequence of Lemma 5 and
the result mentioned above.
CHAOS ON HYPERSPACES
13
Theorem 12. If f is exDevC then f is exDevC.
Remark 13. If f has dense set of periodic points, then so does f (see
e.g. [5]), but not conversely (see [3]). On the other hand, the property
of weakly mixing for f is equivalent to f being weakly mixing (by Theorem 1 and Proposition 1 of [5]), which is equivalent to transitivity of f
(independently proved by Banks [3] and Peris [24]). Hence, for any map
f which is DevC but not totally transitive, its set–valued extension f is
not DevC.
The following result shows that the converse of Theorems 11 and 12 is
not true, moreover, it solves a problem posed by Banks in [3], that is, it
provides an example of a transitive (even exact), but not DevC map f
such that f is DevC.
Theorem 14. There exists a topologically exact dynamical system (X, f )
for which the set Per(f ) is nowhere dense and the induced map f is
exDevC.
Proof. For n ∈ {2, . . .} let (Xn , fn ) be the dynamical system constructed
in Example 6. By Lemma 7 each fn is exDevC and has SPSP. Therefore
their Cartesian product is a topologically exact map f with RSSP (by
Lemma 3). Again by Lemma 7 one can see that f has no periodic points
with primary period greater than or equal to 2. Since f is transitive,
the set Per(f ) is nowhere dense. Hence f is not exDevC and has no
SPSP. On the other hand, the induced map f is exDevC and has SPSP
by Lemmas 2, 4 and 5.
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1
Departamento de Matemática Aplicada y Estadı́stica. Universidad
Politécnica de Cartagena, Campus Muralla del Mar, 30203-Cartagena,
Spain (corresponding author)
E-mail address: [email protected]
2
Institute of Mathematics. Jagiellonian University. Reymonta 4,
30-059 Kraków, Poland
E-mail address: [email protected]
and [email protected]
3
Mathematical Institute at Opava. Silesian University at Opava, Na
Rybnı́čku 1, 746 01 Opava, Czech Republic
E-mail address: [email protected]
4
Faculty of Applied Mathematics, AGH University of Science and
Technology, al. Mickiewicza 39, 30-059 Kraków, Poland
E-mail address: [email protected]
5
Departamento de Matemática Aplicada. Universidad Politécnica de
Valencia, 46022-Valencia, Spain
E-mail address: [email protected]