1. Introduction
Li and Yorke [
1] first presented the term “chaos”. Many scholars have since explored the Li–Yorke chaotic autonomous discrete systems (ADDSs), see, for example [
1,
2,
3,
4,
5,
6,
7,
8]. Schweizer and Smítal [
9] defined another sort of chaos, which is called distributional chaos. Distributionally chaotic ADDSs have been explored greatly by lots of researchers at home and abroad (see [
2,
3,
10,
11]). In 2007, ref. [
2] provided the definition of chaos via
, where
is a fixed Furstenberg family. Research has demonstrated that Li–Yorke chaos and various forms of distributional chaos can be characterized as chaos within Furstenberg families. Moreover, they explored some fundamental and important properties of the set of
-scrambled pairs of a given ADDS
and provided the concepts of the generically
-chaotic ADDS and the generically strongly
-chaotic ADDS. They also determined a sufficient condition under which an ADDS can be considered generally highly chaotic. The concept of chaos, defined as
, was introduced in 2009 by [
3], using a pair of Furstenberg families denoted as
and
. According to the findings of [
3], both Li–Yorke chaos and distributional chaos fall under the umbrella of chaos within the context of Furstenberg families. In addition, they derived multiple sufficient conditions that determine if a system is
-chaotic and constructed an interesting and useful example. Furthermore, it was revealed that while an uncountable scrambled set may be foreshadowing of the second type of distributional chaos, it does not necessarily imply the presence of the first type of distributional chaos.
T-VDDSs were introduced in [
12]. We know that they also appear connected to some time-varying difference equations. Chaos of T-VDDSs has been extensively studied (see [
12,
13,
14,
15,
16,
17,
18]). For a sequence
of continuous selfmaps on
converging uniformly to a continuous selfmap
g on
; in [
14], J.S. Canovas studied the limit behaviour of the T-VDDS
and whether the simplicity (resp. chaoticity) of
g can deduce the simplicity (respectively chaoticity) of
. In [
15], J. Dvorakova discussed a T-VDDS
on
, where
converges uniformly to a continuous selfmap
g on
, and
is a surjective continuous map of
for each integer
, and it is proven that for a given time-varying system
, the full Lebesgue measure of a distributional scrambled set of
does not deduce the existence of distributional chaos of the limit selfmap
g of
and, conversely, that a time-varying system exists with an arbitrarily small distributional scrambled set converging to a map distributional chaotic a.e.
This work extends the results of Xiong et al. in [
2] to T-VDDSs. In particular, for a given Furstenberg family and a given T-VDDS
, we define
-scrambled pairs of points for the system
, which contains well-known scrambled pairs. We also study some properties of the set of
-scrambled pairs of a given T-VDDS
and define the generically
-chaotic T-VDDSs and the generically strongly
-chaotic T-VDDSs. Moreover, we obtain a sufficient condition for a given T-VDDS to be generically strongly
-chaotic.
In
Section 2, some notations and basic concepts are recalled. Moreover, for a T-VDDS
and a given Furstenberg family
, we define
-scrambled pairs of points for the system
. In
Section 3, our main results are established.
2. Preliminaries
Throughout this paper, Y is a complete metric space with a metric , and is assumed to be a T-VDDS given by sequences of continuous selfmaps from the complete metric space to itself. Note that if for any integer , then system is a “classical” ADDS.
For any
with
, let
denote the closure of
C and
denote the set
for any
. Let the diagonal
and
be the product metric on the product space
, which is defined by
for any
.
For any pair
and a given integer
, we provide a distribution function
for a given T-VDDS
by
Then,
is a non-decreasing function,
for
and
for any
, where
is the diameter of
Y. Let
and
Then, both functions
and
are nondecreasing. Clearly,
for any
. If
for all
t in an interval, it is simply written as
.
A pair is called a scrambled pair in the distribution sense for a given T-VDDS if and for some .
For a given , a pair is called a -scrambled pair in the distribution sense for a given T-VDDS if and .
A subset is called a scrambled set in the distribution sense for a given T-VDDS if any two different points in S is a scrambled pair in the distribution sense for this system.
A subset is called a strongly scrambled set in the distribution sense for a given T-VDDS if there is a , satisfying that any is a -scrambled pair in the distribution sense for this system.
A T-VDDS
is called Li–Yorke chaotic if there is an uncountable subset
, such that for any pair
with
,
and
For any given
(resp.
), a pair
is called a
-scrambled pair (resp. scrambled pair) in the sense of Li and Yorke for a given T-VDDS
if
and
A subset is called a scrambled set in the sense of Li and Yorke for a given T-VDDS if any is a scrambled pair in the sense of Li and Yorke for this system.
A subset is called a strongly scrambled set in the sense of Li and Yorke for a given T-VDDS if there is a satisfying that any is a -scrambled pair in the sense of Li and Yorke for this system.
Let and . A collection is said to be a Furstenberg family if it is hereditary upwards, that is, and implies .
Let , where denotes the cardinality of a set B. Clearly, is a Furstenberg family.
Let
be an infinite subset, and let
then,
is the upper density of
. It is clear that
.
For any
, let
Clearly,
.
For a given T-VDDS
, any
and any
, we write
For a given Furstenberg family
and
with
, a point
is an
-attaching point of
B for a given T-VDDS
if
, and that
is the
-attaching set of
B for the system
. Obviously,
is the set of all
-attaching points of
B for the system
. A point
is an
-adherent point of
B for a given T-VDDS
, if
for any
. Let
A point
is called an
-adherent point of a given set
for a given T-VDDS
if
y is an
-attaching point of the set
for this system. For a given
, a point
is called an
-
-escape point of a given set
for a given T-VDDS
,
y is an
-attaching point of the set
for this system. A point
is called an
-escape point of a given set
for a given T-VDDS
if
y is an
-attaching point of the set
for some
and this system. A point
is called an
-reciprocating point of a given set
for a given T-VDDS
if it is both an
-adherent point and an
-escape point of the set
B for this system. A point
is said to be an
-
-reciprocating point of a given set
for a given T-VDDS
if it is both an
-adherent point and an
-
-escape point of the set
B for this system. Let
(resp.
,
,
or
) be the set of
-adherent points (resp.
-escape points,
-
-escape points,
-reciprocating points or
-
-reciprocating points) of a given set
for a given T-VDDS
. Clearly, from the definitions, one has
and
For a given T-VDDS
, an
-reciprocating point (resp.
-
-reciprocating point) of the diagonal
for the product system
is called an
-scrambled pair (resp.
-
-scrambled pair) of the system
. A subset
is called an
-scrambled scrambled set for a given T-VDDS
if any
is an
-scrambled pair of this system. A subset
is called a strongly
-scrambled scrambled set for a given T-VDDS
if there is a
such that any two different points in
S is an
-
-scrambled pair of this system. A given T-VDDS
is called generically
-chaotic if the set of all
-scrambled pairs of this system is a
set, which is dense in
. A given T-VDDS
is called generically strongly
-chaotic if there is a
, satisfying that the set of all
-
-scrambled pairs of this system is a
set, which is dense in
.
3. Main Results
In this section, some results of Xiong et al. in [
2] are extended to T-VDDSs.
3.1. Furstenberg Families Compatible with a T-VDDS
For a given T-VDDS
and any
, we write
If
, we write
. For a given Furstenberg family
and
with
, a point
is an
-attaching point of
B for a given T-VDDS
if
, and
is the
-attaching set of
B for the system
.
Theorem 1. Assume that is a Furstenberg family, and that is a T-VDDS and . Then, we have Proof. Clearly,
, if and only if for any
, one can find
with
, if and only if
. So,
Assume that
Then,
, if and only if there is a
with
for any
, if and only if there is a
such that
. So,
Thus, the proof is finished. □
A Furstenberg family is called compatible with a given T-VDDS , if for any given open set , the set is a set.
Theorem 2. Assume that is a Furstenberg family, and that is a T-VDDS. If is countably generated and proper or if for some , then the family is compatible with the system .
Proof. Assume that
is open. If
is countably generated and proper, from the definition, there is
, which generates
. From Theorem 1,
This means
is a
set.
Assume that
for some
. Clearly, the theorem holds for
. For any
and any open set
, we have
Obviously, for any
and any
there are positive integers
and
such that
. So,
implies that there are positive integers
with
and
for any
. Consequently,
where
and
This means that
is a
set. Thus, the proof is finished. □
3.2. The Sets of -Reciprocating Points
for a T-VDDS
A point
is called an
-adherent point of a given set
for a given T-VDDS
, if for any
,
y is an
-attaching point of the set
for this system. For a given
, a point
is called an
-
-escape point of a given set
for a given T-VDDS
,
y is an
-attaching point of the set
for this system. A point
is called an
-escape point of a given set
for a given T-VDDS
if
y is an
-attaching point of the set
for some
and this system. A point
is called an
-reciprocating point of a given set
for a given T-VDDS
if it is both an
-adherent point and an
-escape point of the set
B for this system. A point
is called an
-
-reciprocating point of a given set
for a given T-VDDS
if it is both an
-adherent point and an
-
-escape point of the set
B for this system. Let
(resp.
,
,
or
) be the set of
-adherent points (resp.
-escape points,
-
-escape points,
-reciprocating points or
-
-reciprocating points) of a given set
for a given T-VDDS
. Clearly, from the definitions, we have
and
Theorem 3. Assume that is a Furstenberg family, and that is a T-VDDS. If is compatible with the system , then the following hold:
- (1)
is a set for any .
- (2)
is a set for any and any .
Proof. From the condition, the set
is a
set for any open set
. As
is a
set. As
and
is a
set. Thus, the proof is ended. □
Let . C and D are said to be positively disjoint if is positive.
Theorem 4. Assume that , A, and B are positively disjoint, is a Furstenberg family, and is a T-VDDS. If is compatible with the system , then the following holds:
- (1)
for some .
- (2)
for some .
Proof. As
A and
B are positively disjointed, there is some
with
. So, one has
This implies that
and
. □
Theorem 5. Assume that is nonempty, is a Furstenberg family, and bis a T-VDDS. If is compatible with the system , then one can find with and , satisfying that and are dense in Y, if and only if is a dense set in Y for some .
Proof. Assume that there is a with , , and and are dense in Y. From Theorem 3, and are dense sets in Y. From Theorem 4, for some . So, from Theorem 3, is a dense set in Y for the above . □
Now, we assume that is a dense set in Y for some . As , is a dense set in Y. Let . Then, , and . So, is a dense set in Y.
3.3. Chaos with Respect to a Furstenberg Family for a T-VDDS
For a given T-VDDS , an -reciprocating point (resp. --reciprocating point) of the diagonal for the product system is called an -scrambled pair (resp. --scrambled pair) of the system . A subset is called an -scrambled scrambled set for a given T-VDDS if any is an -scrambled pair of this system. A subset is called a strongly -scrambled scrambled set for a given T-VDDS if there is a , such that any two different points in S is an --scrambled pair of this system. A given T-VDDS is called generically -chaotic if the set of all -scrambled pairs of this system is a set, which is dense in . A given T-VDDS is called generically strongly -chaotic if there is a that satisfies that the set of all --scrambled pairs of this system is a set, which is dense in .
Theorem 6. Assume that and are a T-VDDS. Then, the following hold:
- (1)
A pair of points in Y is an -scrambled pair (resp. -τ-scrambled pair) of the system if and only if the pair is a scrambled pair (resp. τ-scrambled pair) in the Li–Yorke sense for the system .
- (2)
A set is an -scrambled (resp. strongly -scrambled) set for the system , if and only if the set is a scrambled set (resp. strongly scrambled set) in the Li–Yorke sense for the system .
- (3)
A pair of points in Y is an -scrambled pair (resp. -τ-scrambled pair) of the system , if and only if the pair is a scrambled pair (resp. τ-scrambled pair) in the distribution sense for the system .
- (4)
A set is an -scrambled (resp. strongly -scrambled) set for the system , if and only if the set is a scrambled set (resp. strongly scrambled set) in the distribution sense for the system .
Proof. The proof is immediate and omitted. □
3.4. Criteria for a Generically Strongly
-Chaotic T-VDDS
Theorem 7. Assume that is a Furstenberg family and is a T-VDDS. If is compatible with the system , then the system is generically strongly -chaotic, if and only if the following hold:
- (1)
is dense in , and
- (2)
There exists a nonempty that is positively disjoint with , satisfying that is dense in .
Proof. The proof is direct and omitted for brevity. □
Theorem 8. Assume that is a Furstenberg family, is a T-VDDS, and . Then, Proof. For any
, choose
with
. Let
Then,
and
. This means
and
From the hypothesis and the definition, one has
Consequently,
. □
Theorem 9. Assume that is a full Furstenberg family, and that is a T-VDDS, and let be compatible with the product system . If there are a point and a with and , such that and are dense, then the system is generically strongly -chaotic.
Proof. From the hypothesis and Theorem 8, one can see that
is dense in
, and that
is dense in
. As
is full,
Therefore,
is dense in
. Clearly,
for any
. So,
Hence,
is dense in
. From Theorem 7, the system
is generically strongly
-chaotic. □
Theorem 10. Assume that is a T-VDDS, is a fixed point such that is dense, and is invariant with , , and is dense. If is a full Furstenberg family that is compatible with the system , then the system is generically strongly -chaotic. Consequently, the system is generically strongly -chaotic.
Proof. From the hypothesis, we can see that
is dense in
Y, and that
is dense in
Y. As
is full,
Therefore,
is dense in
Y. From Theorem 9, the system
is generically strongly
-chaotic. From Theorems 2 and 9, the system
is generically strongly
-chaotic. □
For any given integer , we let the set be endowed with the discrete topology and the space be endowed with the product topology. The compact metric space is said to be the symbolic space generated by m different elements. The shift map on the space is defined by for any . Obviously, and are the fixed points of the shift map , where for every integer . Then, we obtain a T-VDDS , given by sequences of continuous maps . As an application, we have the following example.
Example 1. If is a full Furstenberg family that is compatible with the product system , then the system is generically strongly -chaotic. Consequently, the system is generically strongly -chaotic.
Proof. Clearly, and are the fixed points of the system . Also, it is easy to show that the conditions of Theorem 10 are satisfied for these two fixed points. From Theorem 10, the result of Example 1 holds. □
Remark 1. For any countably infinite set of functions , in order for results to be independent of the numbering, we need a condition on the family , namely that if , then for any permutation of the set of positive integers.