# Three Types of Distributional Chaos for a Sequence of Uniformly Convergent Continuous Maps

Authors: Risong Li; Tianxiu Lu; Jingmin Pi; Waseem Anwar
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Lethss=1∞ be a sequence of continuous maps on a compact metric space W which converges uniformly to a continuous map h on W. In this paper, some equivalence conditions or necessary conditions for the limit map h 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)).


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## Body

## 1. Introduction

In this paper, a topological dynamical system (shortly, TDS) is a pairW,h, where h:W⟶W is a continuous surjective map that acts on a compact metric space W with a metric e. And let ℤ+ 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 hss∈1,2,⋯ are continuous transitive functions on a metric space W,e, hs converge uniformly to a function h, and then, a few sufficient conditions for the limit function h 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 hs∘⋯∘h1a,a∈0,1 was studied; the author considered whether the simplicity (respectively, chaoticity) of h implies the simplicity (respectively, chaoticity) of h1,∞ (where h1,∞=hss=1∞ is a sequence of continuous interval maps converging uniformly to a continuous map h). More recently, [4] considered nonautonomous discrete dynamical systems h1,∞, which were given by surjective continuous map sequence hs:I⟶I converged uniformly to a map h:0,1⟶0,1. 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 W,h is sensitive if for any region E of the phase space W there are two points in E and some s∈0,1,⋯ satisfying that the sth iterate of the two points under the map h is significantly separated. The size of the set of all s∈ℤ+ 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 h is sensitive. In [7], we obtain an equivalence condition for that the uniform map h is topologically transitive (resp., syndetically transitive, topologically weak mixing, and topological mixing). Moreover, necessary conditions for the uniform map h 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 hss=1∞ on a compact metric space W converges uniformly to a continuous map h:W⟶W, we present an equivalence condition under which the uniform map h is syndetically sensitive or cofinitely sensitive or multisensitive or ergodically sensitive, and a sufficient condition under which the uniform map h is totally transitive or topologically weak mixing. In [18], we gave an equivalence condition under which the uniform limit map is F-transitive or weakly F-sensitive or F-sensitive or F1,F2-sensitive and a necessary condition for the uniform limit map to be weakly F-sensitive or F-sensitive or F1,F2-sensitive.In this paper, on a compact metric spaceW, for a continuous map sequence hnn=1∞ which converges uniformly to a continuous map h, we obtain equivalence conditions under which the uniform map h is distributional chaotic or distributional chaotic in a sequence or DC1 or DC2 or DC3 and necessary conditions for the uniform map h to be distributional chaotic or distributional chaotic in a sequence or DC1 or DC2 or DC3.In Section2, some concepts are recalled. Distributional chaotic properties are obtained and proved in Section 3.


## 2. Preliminaries

For any dynamical systemsW1,e1 and W2,e2, define e∞e1,e2=supa∈Wee1a,e2a, where W is a perfect metric space (see [8]).Letqjj=1∞ be a sequence of integers such that qj>0 and qj+1>qj for any j∈1,2,⋯. For a continuous map h:W⟶W of a metric space W with metric e, any a,b∈W and any t>0, let
(1)ϕabt,h,qj=liminfm⟶∞1m∑j=1mχ0,tehqja,ehqjb,ϕab∗t,h,qj=limsupm⟶∞1m∑j=1mχ0,tehqja,ehqjb,where χAy is 1 if y∈A and 0 otherwise. Let E⊂W and a,b∈E with a≠b. The set E is said to be a distributional chaotic set in a sequence, and a,b are said to be a pair of points displaying distributional chaos in a sequence if
(1)
ϕabp,h,qjj=1∞=0, for some p>0(2)
ϕab∗t,h,qjj=1∞=1, for any t>0A continuous maph:W⟶W 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 h:W⟶W is said to be distributional chaotic if it is distributional chaotic in the sequence of positive integers.Letϕabt,h=ϕabt,h,1,2,⋯ and ϕab∗t,h=ϕxy∗t,h,1,2,⋯ for any t≥0. For a continuous map h on a metric space W, if there is an uncountable set E⊂W, for any a,b∈E with a≠b, ϕab∗t,h≡1 for every t>0 and ϕabt0,h=0 for some t0>0, or ϕab∗t,h≡1 for every t>0 and ϕxyt0,h<ϕab∗t0,h for some t0>0, or ϕabt,h<ϕab∗t,h for every t>0, then we say that h is DC1, DC2, or DC3, respectively (see [19]).


## 3. Main Results

In all theorems in this section,W,e is a metric space. hn be a sequence of continuous maps which converges uniformly to h on W. And it is always assumed that limn⟶∞e∞hnn,hn=0.Theorem 1.
Letqjj=1∞ be a strictly infinitely increasing sequence of positive integers. Then, E is a distributional chaotic set in the sequence qjj=1∞ for h if and only if the following conditions are satisfied:
(1)
For any givena,b∈E with a≠b, ϕabp,hnn,qjj=1∞=0 for some p>0, where(2)ϕabt,hnn,qjj=1∞=liminfn⟶∞1n♯1,2,⋯,n∩m∈ℕ:ehqmqma,hqmqmb<t,for any t>0.
(2)
For any givena,b∈E with a≠b, ϕab∗t,hnn,qjj=1∞=1 for any t>0, where(3)ϕab∗t,fnn,qjj=1∞=limsupn⟶∞1n♯1,2,⋯,n∩m∈ℕ:ehqmqma,hqmqmb<t,for any t>0, where ♯· is the cardinal number of a set.Proof (necessity).
Assume thatD is a distributional chaotic set in the sequence qjj=1∞ for h.
On the one hand, for any givena,b∈D:a≠b, ϕab∗1/3t,h,qjj=1∞=1 for any t>0. By the assumption that
(4)limn⟶∞e∞hnn,hn=0,for any given t>0, there is a positive integer nt such that
(5)e∞hmm,hm<13t,for any m≥nt. Then, we have
(6)ehmma,hmmb≤ehmma,hma+ehma,hmb+ehmmb,hmb<13t+13t+13t<t,for any m≥nt. This shows that ϕab∗t,hnn,qjj=1∞=1 for any t>0. Hence, condition (1) holds.
On the other hand, for any givena,b∈D:a≠b, ϕab3p,h,qjj=1∞=0 for some p>0. This implies that
(7)limsupn⟶∞1n♯j:ehqja,hqjb≥3p,1≤j≤n=1.
By (4), for the above p>0, there is an integer np>0 such that
(8)ehnna,hnnb−ehna,hnb≤ehnna,hna+ehnnb,hnb<p+p=2p,for any a,b∈W and any integer n≥np. Therefore,
(9)ehna,hnb<ehnna,hnnb+2p,for any integer n≥np. That is,
(10)ehnna,hnnb>ehna,hnb−2p>p,for any integer n≥np. This implies that
(11)limsupn⟶∞1n♯j:ehqjqja,hqjqjb≥3p,1≤j≤n=1.
That is,ϕabp,hnn,qjj=1∞=0 for the above p>0. So, condition (2) is true.
(Sufficiency) suppose that conditions (1) and (2) are true. From the above argument, we know that(12)ehma,hmb≤ehmma,hma+ehmma,hmmb+ehmmb,hmb<13t+13t+13t<t,for any m≥nt, any a,b∈D:a≠b, and any t>0.
Sinceϕab∗1/3t,hnn,qjj=1∞=1 for any t>0 and for any a,b∈D:a≠b, ϕab∗t,h,qjj=1∞=1 for any t>0 and any a,b∈D:a≠b. By hypothesis, for any given a,b∈D:a≠b, ϕab3p,hnn,qjj=1∞=0 for some p>0. This means that
(13)limsupn⟶∞1n♯j:ehqjqja,hqjqjb≥3p,1≤j≤n=1.
By (8),
(14)ehnna,hnnb<ehna,hnb+2p,for any integer n≥np. That is,
(15)ehna,hnb>ehnna,hnnb−2p>p,for any integer n≥np. This implies that
(16)limsupn⟶∞1n♯j:ehqja,hqjb≥3p,1≤j≤n=1,which means that ϕabp,h,qjj=1∞=0 for the above p>0. So, D is a distributional chaotic set in the sequence qjj=1∞ for h.
Thus, the entire proof is completed.Theorem 2.
E is a distributional chaotic set for h if and only if the following conditions are true:
(1)
For anya,b∈E:a≠b, ϕabp,hnn=0 for some p>0, where(17)ϕabt,hnn=liminfn⟶∞1n♯1,2,⋯,n∩m∈ℕ:ehmma,hmmb<t,for any t>0.
(2)
For anya,b∈E:a≠b, ϕab∗t,hnn=1 for any t>0, where(18)ϕab∗t,hnn=limsupn⟶∞1n♯1,2,⋯,n∩m∈ℕ:ehmma,hmmb<t,for any t>0.Proof.
Letqjj=1∞=1,2,⋯, and then, by Theorem 1, the conclusion holds.Theorem 3.
h is DC1 if and only if there are two points a,b∈W:a≠b such that the following conditions are satisfied:
(1)
ϕabp,hnn=0 for some p>0(2)
ϕab∗t,hnn=1 for any t>0Proof.
By Theorem2, the conclusion is obvious. So it is omitted.Theorem 4.
h is DC2 if and only if there are two points a,b∈W:a≠b such that the following conditions are satisfied:
(1)
ϕabp,hnn<1 for all p>0 in an interval(2)
ϕab∗t,hnn=1 for any t>0Proof (necessity).
Assume thath is DC2. By the similar proof as Theorem 1, condition (2) holds.
LetD is a distributional chaotic set of DC2 of h. Then, for any two points a and b in D, ϕab3p,h<1 for some interval c,d⊂0,1 and any p∈c,d, which means that
(19)limsupn⟶∞1n♯j:ehja,hjb≥3p,1≤j≤n>0,for any p∈c,d. By (4), for the above p∈c,d, there is an integer np>0 such that (8) holds for any a,b∈W and any integer n≥np. Therefore,
(20)ehnna,hnnb>efna,hnb−2p>p,for any integer n≥np. This implies that
(21)limsupn⟶∞1n♯j:ehjja,hjjb≥3p,1≤j≤n>0.
That is,ϕabp,hnn<1 for any p∈c,d. Hence, condition (1) holds.
(Sufficiency) assume that conditions (1) and (2) hold. By the above discussion, we get that (12) holds for any m≥nt and any t>0.
Sinceϕab∗1/3t,hnn=1 for any t>0 and for the above a and b, then ϕab∗t,h=1 for any t>0 and for these two points a and b. By hypothesis, we obtain that for the above two points a and b, ϕab3p,hnn<1 for some interval c,d⊂0,1 and any p∈c,d. This means that
(22)limsupn⟶∞1n♯j:ehjja,hjjb≥3p,1≤j≤n>0,for any p∈c,d. By (8),
(23)ehna,hnb>ehnna,hnny−2p>p,for any integer n≥np. This implies that
(24)limsupn⟶∞1n♯j:ehja,hjb≥3p,1≤j≤n>0.
That is,ϕabp,h<1 for any p∈c,d. By the definition, h is DC2.
Thus, the entire proof is ended.Theorem 5.
Ifh is DC3, then there are a,b∈W:a≠b and some interval c,d⊂0,1 such that for any p∈1/3c,1/3d,
(25)ϕabp,hnn≤ϕab3p,h<ϕab∗3p,h≤ϕab∗9p,hnn.Proof.
Suppose thath is DC3. By the definition, there is a,b∈W:a≠b such that ϕabt,h<ϕab∗t,h for some interval c,d⊂0,1 and any t∈c,d. By the assumption that
(26)limn⟶∞e∞hnn,hn=0,there is some positive integer nt such that
(27)e∞hmm,hm<t,for any m≥nt. Then, we have
(28)ehmma,hmmb≤ehmma,hma+ehma,hmb+ehmmb,hmb<t+t+t=3t,for any m≥nt. This implies that ϕab∗3t,hnn≥ϕab∗t,h. Clearly, for the above two points a and b and any p>0,
(29)1−ϕab3p,h=limsupn⟶∞1n♯j:ehja,hjb≥3p,1≤j≤n.
By (4), for the above p>0, there is an integer np>0 such that (8) holds for any a,b∈W and any integer n≥np. Therefore,
(30)ehnna,hnnb>ehna,hnb−2p>p,for any integer n≥np. This implies that
(31)1−ϕabp,hnn=limsupn⟶∞1n♯j:ehjja,hjjy≥p,1≤j≤n≥1−ϕab3p,h.
That is,ϕabp,hnn≤ϕab3p,h for any p>0. Consequently, for any p∈1/3a,1/3b,
(32)ϕabp,hnn≤ϕxy3p,h<ϕab∗3p,h≤ϕab∗9p,hnn.
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 anyε>0 and any integer l>0, there is an integer n0>0 (possibly depending on l) satisfying that for any integer n>n0, ehnla,hla<ε for any a∈W.Theorem 7.
Letqjj=1∞ be a strictly infinitely increasing sequence of positive integers. If the integer n0 in Lemma 6 is independent of l and h is distributional chaotic in a sequence qjj=1∞, then there is an uncountable distributional chaotic set E⊂W in the sequence qjj=1∞ for h such that the following conclusions are true:
(1)
For anya,b∈E:a≠b and any t>0, there is an integer Nt such that ϕab∗t,hn,qj=1 for any integer n>Nt(2)
For anya,b∈E:a≠b, there are some p>0 and some integer np such that ϕabp,hn,qj=0 for any integer n>npProof.
Suppose thath is distributional chaotic in a sequence qjj=1∞.
(1)
By the definition, there is an uncountable distributional chaotic setD⊂W in the sequence qjj=1∞ for h such that for any given a,b∈D:a≠b, ϕab∗1/3t,h,qjj=1∞=1 for any t>0. By hypothesis and Lemma 6, for any given t>0, there is Nt, which is independent of l, such that for any n>Nt,(33)ehnla,hla<13t,for any integer l≥1 and any a∈W. Therefore,
(34)ehnja,hnjb≤ehja,hjb+ehnja,hja+ehnjb,hjb<3·13t=t,
for any integern>Nt, any integer j>0, and any a,b∈D. Consequently, ϕab∗1/3t,h,qjj=1∞=1 implies that ϕab∗t,hn,qjj=1∞=1 for any n>Nt and any t>0.
(2)
Sinceh is distributional chaotic in a sequence qjj=1∞, then for any given a,b∈W:a≠b, ϕab3p,h,qjj=1∞=0 for some p>0. This means that(35)limsupn⟶∞1n♯j:ehqja,hqjb≥3p,1≤j≤n=1.
By hypothesis and the above argument, for the abovep>0, there is an integer np>0 such that
(36)ehnja,hnjb−ehja,hjb≤ehnna,hna+ehnnb,hnb<p+p=2p,for any a,b∈W, any integer n≥np, and any integer j>0. Therefore,
(37)ehja,hjb<ehnja,hnjb+2p,for any integer n≥np and any integer j>0. That is,
(38)ehnja,hnjb>ehja,hjb−2p>p,for any integer n≥np. This implies that
(39)limsupm⟶∞1m♯j:ehnqja,hnqjb≥3p,1≤j≤m=1,for any integer n≥np. That is, ϕabp,hn,qjj=1∞=0 for the above p>0 and any integer n≥np.
Thus, the proof is finished.Theorem 8.
If the integern0 in Lemma 6 is independent of l and h is DC1, then there is a distributional chaotic set E⊂W for h which is uncountable such that the following hold:
(1)
For anya,b∈E:a≠b and any t>0, there is an integer Nt such that ϕab∗t,hn=1 for any integer n>Nt(2)
For anya,b∈E:a≠b, there are some p>0 and some integer np such that ϕabp,hn=0 for any integer n>npProof.
Since DC1 is the special case of distributional chaotic in a sequence, then it is obvious.Theorem 9.
If the integern0 in Lemma 6 is independent of l and h is DC2, then there are two points a,b∈W with a≠b such that the following are true:
(1)
For anyt>0, there is an integer Nt such that ϕab∗t,hn=1 for any integer n>Nt(2)
There are some intervalc,d⊂0,1 such that for any p>0, there is some integer np such that ϕabp,hn<1 for any integer n>npProof.
Suppose thath is DC2.
(1)
The proof is similar to Theorem7. So it is omitted(2)
By the definition, for any givena,b∈D:a≠b, ϕab3p,h<1 for some interval c,d⊂0,1 and any p∈c,d. This means that for any p∈c,d,(40)limsupn⟶∞1n♯j:ehja,hjb≥3p,1≤j≤n>0.
By hypothesis and the above argument, for any givenp∈c,d, there is an integer np>0 such that (36) holds for any a,b∈W, any integer n≥np, and any integer j>0. Therefore,
(41)ehnja,hnjb>ehja,hjb−2p>p,for any integer n≥np and any integer j>0. This implies that
(42)limsupm⟶∞1m♯j:ehnja,hnjb≥3p,1≤j≤m>0,for any integer n≥np. That is, ϕabp,hn<1 for any p∈c,d and any integer n≥np.
Thus, the proof is complete.Theorem 10.
If the integern0 in Lemma 6 is independent of l and h is DC3, then there are a,b∈W:a≠b and some interval c,d⊂0,1, for any p∈1/3c,1/3d, there is an integer Np>0 such that
(43)ϕabp,hn≤ϕab3p,h<ϕab∗3p,h≤ϕab∗9p,hn,for any n>Np.Proof.
Suppose thath is DC3. Then, by the definition, there are two points a,b∈W:a≠b such that ϕab1/3t,h<ϕab∗1/3t,h for some interval c,d⊂0,1 and any t∈c,d. By hypothesis and Lemma 6, for any given t>0, there is Nt, which is independent of l, such that for any n>Nt,
(44)ehnla,hla<13t,
for any integerl≥1 and any a∈W. So,
(45)ehnja,hnjb≤ehja,hjb+ehnja,hja+ehnjb,hjb<3·13t=t,for any integer n>Nt, any integer j>0, and any a,b∈D. This implies that
(46)ϕab∗t,hn≥ϕab∗13t,h,for any n>Nt and any t∈c,d. By the definition, for any p>0 and the above two points a and b,
(47)1−ϕab3p,h=limsupn⟶∞1n♯j:ehja,hjb≥3p,1≤j≤n.
By hypothesis and the above argument, for the abovep>0, there is an integer np>0 such that (36) holds for any a,b∈W, any integer n≥np, and any integer j>0. Therefore,
(48)ehnja,hnjb>ehja,hjb−2p>p,for any integer n≥np and any integer j>0. This implies that
(49)1−ϕabp,hn=limsupm⟶∞1m♯j:ehnja,hnjb≥p,1≤j≤m≥limsupn⟶∞1n♯j:ehja,hjb≥3p,1≤j≤n=1−ϕab3p,h,for any integer n≥np and any p>0. That is, ϕabp,hn≤ϕab3p,h for the above p>0 and any integer n≥np.
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 Theorem10, there exists an integer N>0 such that hn is DC3 for any n≥N.



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