Distributional Chaoticity of the Minimal Subshift of Shift Operators ()
1. Introduction
The shift operator is a distinct linear operator that displaces one or more bits forward (or backward) for each basis vector in the canonical orthogonal basis of Banach space or Frechet space. It has been widely used in many fields, for example, image processing ( [1] , 2015), chaotic encryption ( [2] , 2020), symbolic dynamical systems ( [3] , 2013), and dynamic physical systems ( [4] , 1993).
The shift operator is generally divided into two categories: unilateral shift operator and bilateral shift operator. The weighted shift operator is a generalization of the shift operator. Since Grosse-Erdmann ( [5] , 1999) closely linked the hypercyclicity of operators to the topological transitivity of dynamical systems, the study of chaotic properties of shift operators has attracted more and more attention, especially in symbolic spaces. In 2010, Queffelec ( [6] ) proved that the shift operator acting on the symbol space is topologically exact, so it is topologically mixing. And “topologically weakly mixing” is equivalent to “topologically transitive”, “topologically transitive” is equivalent to “having a dense orbit”. In 2013, Wu and Zhu ( [7] ) studied chaos generated by a class of weighted shift operators. Firstly, it is proved that the weighted shift operator is weakly mixing, transitive (or hypercyclic), and Devaney chaotic are equivalent to the separability of space. Moreover, this property is preserved under iteration. Then, it is obtained that the weighted shift operator is distributional chaotic and Li-Yorke sensitive. They also studied the dynamical properties of general weighted shift operators ( [8] ). It is proved that the weighted shift operator is uniformly distributional chaotic, and this property is maintained under iteration. In addition, it is proved that the principal measure of the weighted shift operator is equal to 1. Wang ( [9] , 2018) proved that there exists an uncountable invariant distributional irregular set of weighted shift operators on normed linear spaces
, which generalizes the main results of [8] .
The study of shift operators acting on symbolic dynamical systems is often more concerned with the chaotic properties of their subshifts. Constructing counterexamples is a common and highly important method in mathematical research. The unique representation of symbolic space subshifts provides a straightforward tool for constructing counterexamples in the study of dynamical systems. Consequently, symbolic space subshifts play a crucial role in exploring various chaotic processes. It is necessary to delve into the various intriguing mathematical properties of subshifts. Exploring the chaotic properties of minimal subshifts reveals the complex balance between order and unpredictability in dynamic systems. Consider the logistic map—a classic example of chaos. Through simple iterations of a mathematical function, it showcases chaotic behavior, exemplifying the butterfly effect: tiny initial changes lead to vastly different outcomes. These insights transcend conventional mathematics and physics, impacting fields like weather forecasting. By dissecting the chaotic properties of minimal subshifts, we gain effective tools to understand and harness interdisciplinary complexity. Jiang and Fa ( [10] , 1993) studied the subshifts of finite-type symbolic dynamical systems and proved that finite-type subshifts are chaotic in the sense of Li-Yorke. Inspired by [8] and [10] , Liao and Fan construct a minimal subshift of the shift operator in [11] . It is proved that this minimal subshift is distributional chaotic, and its topological entropy is zero, so it is not topological chaotic. Furthermore, it is shown that positive topological entropy and distributional chaotic are not equivalent. Fu ( [12] , 2000) proved that the one-sided subshift generated by aperiodic recursive points is chaotic in the Robinson sense. In addition, if the subshift has a periodic point, then it has an infinite permutation set. Finally, some examples are given to discuss the topological entropy of these sub-displacements. Oprocha and Wilczynski ( [13] , 2007) proved the equivalence between distributional chaotic, chaotic in the sense of Li-Yorke, positive entropy and uncountable of subshifts. Some recent studies about subshifts see ( [14] [15] ) and others.
With the development of chaos, three types of distributional chaotic (DC1,
DC2, DC3) are proposed by Balibrea in [16] . Subsequently, DC
,
-DC,
uniformly distributional chaotic, distributional chaotic in a sequence
, and distributional chaotic in a sequence
have been proposed. Therefore, a natural problem arises. Do the subshifts constructed above is uniformly distributional chaotic or the above type of distributional chaotic? This paper answers the above questions. To address disturbances or illusions in dynamic systems, the concept of measure centrality is proposed, dividing chaos into three different levels of complexity. The perspective of hierarchical chaos will contribute to a deeper understanding of chaotic systems. It indicates that all significant dynamical states of a system are manifested in its measure centrality, where the measure centrality of a minimal system is itself, hence discussing issues on such minimal systems is meaningful. In contrast to other studies on the chaotic properties of subshifts, this paper focuses on constructed subshifts, namely the study of the distribution chaotic properties of minimal subshifts. This work aims to comprehensively explore the distribution patterns of minimal subshifts, revealing their chaotic behavior and aiding in a further understanding of the chaotic properties of subshifts from a distributional chaos perspective. In Section 2, some basic concepts and definitions are introduced. In Section 3, some necessary lemmas are given first. Then, it is proved that the subshift
is uniformly
distributional chaotic. So,
is DC1, DC2, DC
, DC3 and
-DC. In
addition, it is proved that
is distributional chaotic in a sequence
and a sequence
.
2. Preliminaries
Let
be a Banach space on the real number field
. And
.
Suppose that ~ be an equivalence relation on X. A family of sets consisting of all different ~ equivalence classes of X is called the quotient set of X with respect to ~, denoted by X/~.
A metric
is defined as
for any
.
Obviously, d is a metric on
and
is a compact metric space. The backward shift operator
is defined by
for any
. In other words, the backward shift operator is to move the symbol sequence x on the space
to the left one by one. If M is a closed set and
, then
is called a subshifts of
.
A finite arrangement
of symbols in X is called a symbol segment on X, and n is the length of A, denoted by
. If
is another symbol segment, write
,
then AB is also a symbol segment. A is said to appear in B (or A appears in B), denoted as
, if
and there is
, such that
,
.
In fact, a mapping g from a symbol segment
to
can be established as
.
A subset
is called a cylinder of
.
For a given unit element
, denote
.
Let
be the inverse of a symbol segment
on D, where
Obviously,
and
. Now take a symbol segment
, let
be an arrangement of
and
, that is,
or
. Inductively, the symbol segments
can be defined. And for any
,
exactly a finite arrangement of all members in the symbol segment set
For any
, denote
Let
, then
. Denote
, M is the
-limit set of
, obviously M is a closed set and
, i.e., through the aforementioned construction, M is obtained, where
acts as a shift operator on the closed set M. So
is a subshift of
. In fact, Liao and Fan ( [11] , 1998) proved that it is also a minimal subshift of
.
The concepts of several types of distributonal chaos are introduced below.
Let f be a continuous self-map on a metric space X. For any pair
and for each
, the distribution function
is defined by
where card
denotes the cardinality of the set A.
Put
Then
is called the upper distribution function, and
the lower distribution function of x and y.
If the pair
satisfies
(DC1)
and
for some
, or
(DC2)
and
for any
in an interval, or
(DC2
) there exist
and
such that
for any
, or
(DC3)
for any
in an interval, or
((p, q)-DC) there exist
and
such that
for any
, then,
is called a distributional chaotic pair of type k (
) or type
for f. The mapping f is said to be distributional chaotic of type k (
) or type
if there exists an uncountable set
such that every pair
of distinct points in F is a distributional chaotic pair of type k (
) or type
for f. f is said to be
uniformly distributional chaotic if there exist an uncountable set
and an
such that for every pair
of distinct points in F,
and
. It follows by the definition that, for a continuous map f of a compact metric space, DC1 implies DC2, and DC2 implies DC3.
Next, the definition of distributional chaotic in a sequence is given.
Definition 2.1 [17] Let X be a compact metric space,
be a strictly increasing sequence of positive integers, and
be a continuous map.
Then f is said to be distributional chaotic in a sequence
if there is an uncountable subset
such that for any two distinct points
,
for any
and
for some
.
And, a stronger chaotic description than distributional chaotic in a sequence is introduced, which will use the following concepts about density.
Definition 2.2 [17] Let
be a strictly increasing positive integer sequence. If
exists, then the limit is called the density of the sequence J, denoted by
.
For a given positive integer sequence
, and a positive integer m, write
Obviously, if the density of the sequence is meaningful, then
.
Definition 2.3 [17] Let
be a strictly increasing sequence of positive integers and
is a subsequence of J. The upper limit
is the upper density of the sequence
relative to the sequence J, denoted by
.
Definition 2.4 [17] A sequence
is said converge weakly to j if there exists a subsequence
of natural number sequence with upper density 1 such that
, denoted by
.
Definition 2.5 [17] Let X be a compact metric space,
be a strictly increasing sequence of positive integers, and
be a continuous map. f is said to be distributional chaotic in a sequence
if there exists an uncountable set
such that for any positive integer l, there exist l distinct points
such that for any finite subset
of l points in S, and for any mapping
, the
for any
.
From the definition of several distributional chaos mentioned above, uniformly distributional chaotic is stronger than DC1, because the number
that appears in its definition does not depend on the pair
. And DC1 implies
DC2, DC2 implies DC
, DC
implies DC3, i.e.,
,
which in turn does not hold. Distributional chaos is obviously distributed in accordance with the natural number sequence, otherwise it does not necessarily hold.
3. Main Result
Lemma 3.1 [11] For any
,
.
Proof. According to the definition, it can be directly verified.
Lemma 3.2 [11] For any
,
is an infinite arrangement of symbol segments in
.
Proof. This lemma can be found in [11] . For the completeness of the paper, we provide its proof. The following primarily consists of proving Lemma 3.2 through mathematical induction.
For any given
, by definition,
and
are finite arrangement of symbol segments in
. Suppose for some
, it has been proved that
,
,
,
,
,
and
are all the finite symbol segments in
has been proved. Since
and
are both the finite symbol segments of the form
in
, where
they are also the finite symbol segments in
. In this way, it is proved that for each
,
and
are finite symbol segments in
. By the definition of a, one can get that Lemma 3.2 holds.
Lemma 3.3 There is an uncountable subset
such that for any two distinct points
,
for infinitely many k and
for infinitely many l.
Proof. The key to proving Lemma 3.3 lies in defining an equivalence relation on the set
, thereby obtaining uncountably many equivalence classes. Selecting one element from each equivalence class results in the construction of an uncountable set S satisfying the given conditions.
For any
, define a relation on the set
, denoted by
, if only for a finite number of k,
, or only for a finite number of l,
. It can be verified that ~ is an equivalence relation on the set
. Let
, according to the above equivalence relation, it is not difficult to get a countable set
, so the quotient set
is uncountable.
Therefore, one can take a representative element in each equivalence class of an uncountable set
to form a subset of
, denoted by S. Then, S is an uncountable set satisfying the condition.
Lemma 3.4 [18] Let X be a compact metric space and
be a continuous map. If there are two nonempty descending closed set sequences
,
in X and a positive integer sequence
such that
1)
and
;
2)
for any
,
then f is distributional chaotic in the sequence
, where
.
Proof. This conclusion can be found in [18] . For the completeness of the paper, we provide its proof. The proof is given by constructing a sequence of positive integers
and defining a mapping
from a set E to
.
Let
, then for any
, there exists
, such that for any
,
holds, where
.
The positive integer sequence
is selected, where
,
,
. Let
be uncountable, such that for any different points two distinct points
, there are infinitely many m such that
, and there are infinitely many n such that
, by Lemma 3.3, such a set E exists. Define
by
for each
, where
and for any
, if
, the
For each
, it can be seen that there must be
such that when
,
holds. Let
. Since
is injective, E is uncountable, so
is uncountable, and then F is uncountable. Therefore, it is only necessary to verify that the points in F satisfy the two conditions of Definition 2.1. Let
,
, there exists
, such that when
, there are
and
.
On the one hand, there exists
, such that for each i, when
,
and
are always different. Take
, when i sufficiently large,
and
,
holds. In particular,
holds. Then, for sufficiently large i, there is
This indicates that
.
On the other hand,
also implies that there exists
such that for each j, when
, there is always
, that is,
and
are either
or
at the same time. For any
, there exists a sufficiently large j such that when
,
holds. In particular,
holds. Then, for sufficiently large
, one can get
Thus
. In summary, the points in the uncountable set F satisfy the two conditions in Definition 2.1. Therefore, f is distributional chaotic in the
sequence
, where
.
Lemma 3.5 [17] Let X be a compact metric space and
be a continuous map. If there are l closed set sequences
in X and a positive integer sequence
, such that
1)
and
for
,
2)
, for any
.
Let
, then f is distributional chaotic in the sequence
.
Proof. This result can be found in [17] . For the completeness of the paper, we provide its proof. Constructing an appropriate sequence of positive integers and defining a key mapping are the difficulties and key points of the whole proof.
Let
, then for any
, there exists
, such that for any
,
holds, where
. The positive integer sequence
is selected, for any positive integer s, there must be an uncountable subset
(by [17] ) of the symbol space E with s symbols
such that for any s distinct points
in
and any mapping
.
Obviously, there are
such mappings, denoted by
. Then there exists a sequence
such that
Define
by
for each
, where
and for any
, if
, the
. For each
, it can be seen that there must be
such that when
,
holds. Let
. Since
is injective,
is uncountable, so
is uncountable, and then D is uncountable.
Now we assert that D is distributional scrambled set in the sequence
.
Indeed, for any subset
with s different points in D, there exists
such that for any
,
holds. For
any
, take
, such that
. Then, when
, there are
.
Thus,
for any
. By Definition 2.5, f is distributional chaotic in the sequence
.
The following are the main conclusions of this study.
Theorem 3.1 The subshift operator
is uniformly distributional chaotic.
Proof. The key uncountable set S can be obtained by lemma 3.3, and a mapping f satisfying certain conditions is defined on S, and then the set F is constructed according to the following construction method. It can be proved that F
is a distributional
-scrambled set of
with
.
Let S be an uncountable subset of
such that for any two distinct points
,
for infinitely many k and
for infinitely many l.
Denote a mapping f, such that
for any
, where
Let
and
. For a fixed
and arbitrarily selected
, there always have
. Therefore, there exists
, such that the first
symbols of
are
. That is to say, for a fixed
, there exists a
such that the front
components of
are
. This shows that for any
,
. Therefore,
, and because h is injective, S is an uncountable set, so S is also an uncountable set.
Assume that F is a distributional
-scrambled set of
with
. For
any pair
with
, let
and
. Without loss of generality, we assume
and
, where
,
. According to the construction of S, there exist a subsequence
such that
and a subsequence
such that
.
First, it is easy to see that the first
components of
and
coincide correspondingly for
. So
Thus, for a given
, there exists a positive integer N such that
for any
and any
. Then,
Second, it can be obtained that
for
. Then
So,
Hence,
is uniformly distributional chaotic.
This proof has been completed.
Theorem 3.2
is distributional chaotic of type k (
) and is
-distribution (where
).
Proof. This theorem can be obtained by Theorem 3.1. Since
is uniformly distributional chaotic, then
is DC1. So,
is also DC2 and DC3.
According to the Theorem 3.1, there exist an uncountable set
and a
such that, for every pair
of distinct points in F,
and
. Take
, for any
,
Then, for every pair
of distinct points in F, there exist
and
such that
for any
. Thus,
is DC
. Take
and
, for every pair
of distinct points in F,
and
for any
. Thus,
is
-DC.
This proof has been completed.
Theorem 3.3 The subshift operator
is distributional chaotic in a sequence
.
Proof. It is proved that only the sequence satisfying the condition can be constructed. For a given
, let
,
, and
, where
Obviously,
. Denote
and
,
where
, then it is easy to see that
and
Let
, according to Lemma 3.4,
is distributional chaotic in the sequence
.
This proof has been completed.
Theorem 3.4 The subshift operator
is distributional chaotic in a sequence
.
Proof. Similar to the proof of Theorem 3.3, it is sufficient to construct a sequence that satisfies the condition.
For a given
, let
and
,
,
where
Obviously,
. Denote
where
, then it is easy to see that
,
for
and
for any
.
Let
, according to Lemma 3.5,
is distributional chaotic in a sequence
.
This proof has been completed.
4. Conclusion
This study investigated distributional chaoticity of minimal subshift of shift operators. The conclusions involved include DC1, DC2, DC3, DC
, (p,
q)-distributional chaos, uniformly distributional chaos, and distributional chaotic in a sequence. Compared with other literatures, the contents in this paper are more comprehensive.
Acknowledgements
There are many thanks to the experts for their valuable suggestions.
Authors and Affiliations
The authors Yuanlin Chen and Jiazheng Zhao is in Sichuan University of Science and Engineering. The corresponding author Tianxiu Lu has two affiliations, that is, Sichuan University of Science and Engineering and Key Laboratory of Bridge Non-destruction Detecting and the Engineering Computing.
Funding
This work was supported by Natural Science Foundation of Sichuan Province (No. 2023NSFSC0070) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and the Engineering Computing (Nos. 2023QYJ06, 2023QYJ07).