#### Abstract

Let be a sequence of continuous maps on a compact metric space which converges uniformly to a continuous map on . In this paper, some equivalence conditions or necessary conditions for the limit map to be distributional chaotic are obtained (where distributional chaoticity includes distributional chaotic in a sequence, distributional chaos of type 1 (DC1), distributional chaos of type 2 (DC2), and distributional chaos of type 3 (DC3)).

#### 1. Introduction

In this paper, a topological dynamical system (shortly, TDS) is a pair , where is a continuous surjective map that acts on a compact metric space with a metric . And let be the set of nonnegative integers.

To find conditions that assure the preservation of any chaotic property under limit operations is an interesting problem (see [1–11]). In [8], the author proved that if are continuous transitive functions on a metric space , converge uniformly to a function , and then, a few sufficient conditions for the limit function to be topologically transitive were presented. In [5], the authors discussed the dynamical behaviour of the uniform limit of a sequence of continuous maps on a compact metric space which satisfy (topological) transitivity or other related properties and presented some conditions different from [8] for the transitivity of a limit map. In [12], the limit behaviour of sequences with the form was studied; the author considered whether the simplicity (respectively, chaoticity) of implies the simplicity (respectively, chaoticity) of (where is a sequence of continuous interval maps converging uniformly to a continuous map ). More recently, [4] considered nonautonomous discrete dynamical systems , which were given by surjective continuous map sequence converged uniformly to a map . It is obtained that the full Lebesgue measure of a distributional scrambled set of the nonautonomous system cannot guarantee the existence of distributional chaos of the limit map. There exists a nonautonomous system with an arbitrarily small distributional scrambled set which converges to a map that is distributional chaotic almost everywhere. As one knows, sensitivity property characterizes the unpredictability of chaotic phenomenon in a system and it is one of the essential conditions of various definitions of a chaotic system. So, when is a system sensitive? This question has gained some attention in more recent papers (see [10, 13–16]). A TDS is sensitive if for any region of the phase space there are two points in and some satisfying that the th iterate of the two points under the map is significantly separated. The size of the set of all satisfying that this significant separation or sensitivity happens can be considered a measure of how sensitive the dynamical system is. In particular, if this set is relatively thin with arbitrarily large gaps between consecutive entries, then we can consider the dynamical system as practically nonsensitive!

In [10], the authors obtained an equivalence condition under which the uniform limit is sensitive. In [7], we obtain an equivalence condition for that the uniform map is topologically transitive (resp., syndetically transitive, topologically weak mixing, and topological mixing). Moreover, necessary conditions for the uniform map to be sensitive or cofinitely sensitive or multisensitive were given. In [17], we present the correct proofs of Theorems 4–8 in [7]. Moreover, for a continuous map sequence on a compact metric space converges uniformly to a continuous map , we present an equivalence condition under which the uniform map is syndetically sensitive or cofinitely sensitive or multisensitive or ergodically sensitive, and a sufficient condition under which the uniform map is totally transitive or topologically weak mixing. In [18], we gave an equivalence condition under which the uniform limit map is -transitive or weakly -sensitive or -sensitive or -sensitive and a necessary condition for the uniform limit map to be weakly -sensitive or -sensitive or -sensitive.

In this paper, on a compact metric space , for a continuous map sequence which converges uniformly to a continuous map , we obtain equivalence conditions under which the uniform map is distributional chaotic or distributional chaotic in a sequence or DC1 or DC2 or DC3 and necessary conditions for the uniform map to be distributional chaotic or distributional chaotic in a sequence or DC1 or DC2 or DC3.

In Section 2, some concepts are recalled. Distributional chaotic properties are obtained and proved in Section 3.

#### 2. Preliminaries

For any dynamical systems and , define , where is a perfect metric space (see [8]).

Let be a sequence of integers such that and for any . For a continuous map of a metric space with metric , any and any , let where is if and otherwise. Let and with . The set is said to be a distributional chaotic set in a sequence, and are said to be a pair of points displaying distributional chaos in a sequence if (1), for some (2), for any

A continuous map is said to be distributional chaotic in a sequence if it has a distributional chaotic set in a sequence which is uncountable. In particular, a continuous map is said to be distributional chaotic if it is distributional chaotic in the sequence of positive integers.

Let and for any . For a continuous map on a metric space , if there is an uncountable set , for any with , for every and for some , or for every and for some , or for every , then we say that is , , or , respectively (see [19]).

#### 3. Main Results

In all theorems in this section, is a metric space. be a sequence of continuous maps which converges uniformly to on . And it is always assumed that .

Theorem 1. *Let be a strictly infinitely increasing sequence of positive integers. Then, is a distributional chaotic set in the sequence for if and only if the following conditions are satisfied:
*(1)*For any given with , for some , where**for any .
*(2)*For any given with , for any , where**for any , where is the cardinal number of a set.*

*Proof (necessity). *Assume that is a distributional chaotic set in the sequence for .

On the one hand, for any given , for any . By the assumption that
for any given , there is a positive integer such that
for any . Then, we have
for any . This shows that for any . Hence, condition (1) holds.

On the other hand, for any given , for some . This implies that
By (4), for the above , there is an integer such that
for any and any integer . Therefore,
for any integer . That is,
for any integer . This implies that
That is, for the above . So, condition (2) is true.

(Sufficiency) suppose that conditions (1) and (2) are true. From the above argument, we know that
for any , any , and any .

Since for any and for any , for any and any . By hypothesis, for any given , for some . This means that
By (8),
for any integer . That is,
for any integer . This implies that
which means that for the above . So, is a distributional chaotic set in the sequence for .

Thus, the entire proof is completed.

Theorem 2. * is a distributional chaotic set for if and only if the following conditions are true:
*(1)*For any , for some , where**for any .
*(2)*For any , for any , where**for any .*

*Proof. *Let , and then, by Theorem 1, the conclusion holds.

Theorem 3. * is DC1 if and only if there are two points such that the following conditions are satisfied:
*(1)* for some *(2)* for any *

*Proof. *By Theorem 2, the conclusion is obvious. So it is omitted.

Theorem 4. * is DC2 if and only if there are two points such that the following conditions are satisfied:
*(1)* for all in an interval*(2)* for any *

*Proof (necessity). *Assume that is DC2. By the similar proof as Theorem 1, condition (2) holds.

Let is a distributional chaotic set of DC2 of . Then, for any two points and in , for some interval and any , which means that
for any . By (4), for the above , there is an integer such that (8) holds for any and any integer . Therefore,
for any integer . This implies that
That is, for any . Hence, condition (1) holds.

(Sufficiency) assume that conditions (1) and (2) hold. By the above discussion, we get that (12) holds for any and any .

Since for any and for the above and , then for any and for these two points and . By hypothesis, we obtain that for the above two points and , for some interval and any . This means that
for any . By (8),
for any integer . This implies that
That is, for any . By the definition, is DC2.

Thus, the entire proof is ended.

Theorem 5. *If is DC3, then there are and some interval such that for any ,
*

*Proof. *Suppose that is DC3. By the definition, there is such that for some interval and any . By the assumption that
there is some positive integer such that
for any . Then, we have
for any . This implies that . Clearly, for the above two points and and any ,
By (4), for the above , there is an integer such that (8) holds for any and any integer . Therefore,
for any integer . This implies that
That is, for any . Consequently, for any ,
Thus, the proof is ended.

Now, we will give some necessary conditions for that the limit map is DC1, DC2, or DC3. First, the following lemma which comes from [2] is needed.

Lemma 6. *For any and any integer , there is an integer (possibly depending on ) satisfying that for any integer , for any .*

Theorem 7. *Let be a strictly infinitely increasing sequence of positive integers. If the integer in Lemma 6 is independent of and is distributional chaotic in a sequence , then there is an uncountable distributional chaotic set in the sequence for such that the following conclusions are true:
*(1)*For any and any , there is an integer such that for any integer *(2)*For any , there are some and some integer such that for any integer *

*Proof. *Suppose that is distributional chaotic in a sequence .
(1)By the definition, there is an uncountable distributional chaotic set in the sequence for such that for any given , for any . By hypothesis and Lemma 6, for any given , there is , which is independent of , such that for any ,for any integer and any . Therefore,
for any integer , any integer , and any . Consequently, implies that for any and any .
(2)Since is distributional chaotic in a sequence , then for any given , for some . This means thatBy hypothesis and the above argument, for the above , there is an integer such that
for any , any integer , and any integer . Therefore,
for any integer and any integer . That is,
for any integer . This implies that
for any integer . That is, for the above and any integer .

Thus, the proof is finished.

Theorem 8. *If the integer in Lemma 6 is independent of and is DC1, then there is a distributional chaotic set for which is uncountable such that the following hold:
*(1)*For any and any , there is an integer such that for any integer *(2)*For any , there are some and some integer such that for any integer *

*Proof. *Since DC1 is the special case of distributional chaotic in a sequence, then it is obvious.

Theorem 9. *If the integer in Lemma 6 is independent of and is DC2, then there are two points with such that the following are true:
*(1)*For any , there is an integer such that for any integer *(2)*There are some interval such that for any , there is some integer such that for any integer *

*Proof. *Suppose that is DC2.
(1)The proof is similar to Theorem 7. So it is omitted(2)By the definition, for any given , for some interval and any . This means that for any ,By hypothesis and the above argument, for any given , there is an integer such that (36) holds for any , any integer , and any integer . Therefore,
for any integer and any integer . This implies that
for any integer . That is, for any and any integer .

Thus, the proof is complete.

Theorem 10. *If the integer in Lemma 6 is independent of and is DC3, then there are and some interval , for any , there is an integer such that
for any .*

*Proof. *Suppose that is DC3. Then, by the definition, there are two points such that for some interval and any . By hypothesis and Lemma 6, for any given , there is , which is independent of , such that for any ,
for any integer and any . So,
for any integer , any integer , and any . This implies that
for any and any . By the definition, for any and the above two points and ,
By hypothesis and the above argument, for the above , there is an integer such that (36) holds for any , any integer , and any integer . Therefore,
for any integer and any integer . This implies that
for any integer and any . That is, for the above and any integer .

Thus, the proof is ended.

*Remark 11. *There are some other future studies; for example, the following conclusion is true or not.

Under the notations and the assumptions of Theorem 10, there exists an integer such that is DC3 for any .

#### Data Availability

The data used to support the findings of this study are currently under embargo. Requests for data, 6 months after publication of this article, will be considered by the corresponding author.

#### Conflicts of Interest

The authors declare no conflict of interest.

#### Authors’ Contributions

Risong Li and Tianxiu Lu were responsible for conceptualization; Risong Li, Tianxiu Lu, and Jingmin Pi were responsible for validation; Tianxiu Lu and Waseem Anwar were responsible for formal analysis; Jingmin Pi was responsible for investigation; Risong Li was responsible for writing original draft preparation; Tianxiu Lu and Waseem Anwar were responsible for writing, review, and editing; Tianxiu Lu was responsible for supervision; Risong Li and Tianxiu Lu were responsible for funding acquisition.

#### Acknowledgments

Many thanks are due to experts. This work was supported by the Project of Department of Science and Technology of Sichuan Province (No. 2021ZYD0005), the Ministry of Education, Science and Technology Development Center (No. 2020QT13), the Opening Project of Key Laboratory of Sichuan Province (Nos. 2018RZJ03, 2018QZJ03, and 2020WZJ01), and the Scientific Research Project of Sichuan University of Science and Engineering (No. 2020RC24).