Stronger chaotic features of the complemented shift map on m-symbol space

Introduction

We describe the mathematical preliminaries needed for the subsequent chapters in Section 2. In Section 3, we found that (∑ 𝑚𝑚 , 𝜎𝜎 𝑐𝑐 ) has chaotic dependence on initial conditions, which means that 𝜎𝜎 𝑐𝑐 satisfies strong chaotic properties. We verify that 𝜎𝜎 𝑐𝑐 is totally transitive on ∑ 𝑚𝑚 and prove that 𝜎𝜎 and 𝜎𝜎 𝑐𝑐 are conjugate topologically. Finally, we also present an example that the complemented shift map 𝜎𝜎 𝑐𝑐: Ʃ m → Ʃ m is chaotic.

Materials and Methods

This section is devoted to introducing some important definitions, propositions, lemmas, theorems, which are necessary to prove the main result of the paper. In this paper, we work on m-symbol sequence space, which is

where 𝑑𝑑(≥ 2) ∈ ℕ, under the distance function

𝑚𝑚 𝑘𝑘 𝑘𝑘≥1

for 𝑥𝑥 = (𝑥𝑥 1 , 𝑥𝑥 2 , 𝑥𝑥 3 , … … ), 𝑦𝑦 = (𝑦𝑦 1 , 𝑦𝑦 2 , 𝑦𝑦 3 , … … ) ∈ ∑ 𝑚𝑚 and discuss the chaoticity of the complemented shift map. Many works have been done on this system for the particular case of 𝑑𝑑 = 2. Since ∑ 2 , our results are also valid in (∑ 𝑚𝑚 , 𝜎𝜎 𝑐𝑐 ). Further, it may be mentioned that our study falls in the more excellent domain of chaotic dynamics. Here we introduce the concept of Complemented shift map on ∑ 𝑚𝑚 . Topological conjugacy between maps is a very powerful tool in the discussion of dynamical systems. Topological conjugacy can be used to make predictions of the behaviour of a dynamical system up to comparing it with another dynamical system whose specific properties are known [5] . The Topological conjugacy feature has an essential role in studying the chaotic behaviour of a map. With the help of this feature, we can explore the chaotic significance by comparing one map with another map. Topological conjugacy has such importance as it can protect many topological dynamical properties.

Definition 2.1 (Complemented shift map)

, where the complement of 𝑠𝑠 𝑖𝑖 is 𝑠𝑠 𝑖𝑖 𝑐𝑐 . Here we see that the map shifts the first element of a point and then changes all others into its complement. [3] For any pair of non-empty open sets 𝑈𝑈, 𝑉𝑉 ⊂ 𝐴𝐴 ∃ 𝑘𝑘 ≥ 0 such that 𝑓𝑓 𝑘𝑘 (𝑈𝑈) ∩ 𝑉𝑉 ≠ 𝜙𝜙, then 𝑓𝑓: 𝐴𝐴 → 𝐴𝐴 is called transitive topologically, where 𝑓𝑓 is a continuous map and (𝐴𝐴, 𝑑𝑑) is a compact metric space. [1] A pair (𝑥𝑥, 𝑦𝑦) ∈ 𝐴𝐴 2 is called a Li-Yorke pair with modulus 𝛿𝛿 > 0 if lim 𝑛𝑛→∞ sup 𝑑𝑑(𝑓𝑓 𝑛𝑛 (𝑥𝑥), 𝑓𝑓 𝑛𝑛 (𝑦𝑦)) ≥ 𝛿𝛿 and lim 𝑛𝑛→∞ inf 𝑑𝑑�𝑓𝑓 𝑛𝑛 (𝑥𝑥), 𝑓𝑓 𝑛𝑛 (𝑦𝑦)� = 0 , where f is a continuous transformation on the compact metric space (𝐴𝐴, 𝑑𝑑). The set of all Li-Yorke pairs of modulus 𝛿𝛿 > 0 is denoted by 𝐿𝐿𝐿𝐿(𝑓𝑓, 𝛿𝛿). [1] A map 𝑓𝑓: 𝐴𝐴 → 𝐴𝐴 which is continuous has strong sensitive dependence on initial conditions if any 𝑥𝑥 ∈ 𝐴𝐴 and a non-empty open set 𝑈𝑈 of 𝐴𝐴 (not necessarily an open neighbourhood of 𝑥𝑥), there exist 𝑦𝑦 ∈ 𝑈𝑈 and 𝑛𝑛 ≥ 0 such that 𝑑𝑑�𝑓𝑓 𝑛𝑛 (𝑥𝑥), 𝑓𝑓 𝑛𝑛 (𝑦𝑦)� is maximum in 𝐴𝐴, where (𝐴𝐴, 𝑑𝑑) is a compact metric space. Definition 2.5 (Chaotic dependence on initial conditions) [8] For any 𝑥𝑥 ∈ 𝐴𝐴 and every neighbourhood 𝑁𝑁(𝑥𝑥) of 𝑥𝑥 there is a 𝑦𝑦 ∈ 𝑁𝑁(𝑥𝑥) such that the pair (𝑥𝑥, 𝑦𝑦) ∈ 𝐴𝐴 2 is Li-Yorke then a dynamical systems (𝐴𝐴, 𝑓𝑓) is called chaotic dependence on initial conditions. Definition 2.6 (Topologically mixing) [7] Given two non-empty open sets 𝑈𝑈, 𝑉𝑉 ⊂ 𝑋𝑋, ∃ 𝑛𝑛 0 such that 𝑓𝑓 𝑛𝑛 (𝑈𝑈) ∩ 𝑉𝑉 ≠ ∅ for all 𝑛𝑛 ≥ 𝑛𝑛 0 then f is topologically mixing where (𝑋𝑋, 𝑇𝑇) is a topological space. Definition 2.7 (Weakly topologically mixing) [7] For every non-empty pair 𝑈𝑈, 𝑉𝑉 ⊂ 𝐴𝐴, there exists a positive integer 𝑘𝑘 such that 𝑓𝑓 𝑘𝑘 (𝑈𝑈) ∩ 𝑉𝑉 ≠ ∅ then f is weakly topologically mixing, and 𝑓𝑓: 𝐴𝐴 → 𝐴𝐴 is a map on a topological space (𝐴𝐴, 𝑇𝑇).

Definition 2.2 (Topologically transitive)

Definition 2.3 (Li -Yorke Pair)

Definition 2.4 (Strong Sensitive Dependence on initial conditions)

Note:

From the above two definitions, it follows that topologically mixing property implies weakly topologically mixing property. Definition 2.8 (Generically 𝜹𝜹-chaotic) [2] If 𝐿𝐿𝐿𝐿(𝑓𝑓, 𝛿𝛿) is residual in 𝑋𝑋 2 then the continuous map 𝑓𝑓: 𝐴𝐴 → 𝐴𝐴 on a compact metric space 𝐴𝐴 is called generically 𝛿𝛿-chaotic. We also need the following Propositions, Lemma and Theorem. Proposition 2.1: Consider a continuous map 𝑓𝑓: 𝐴𝐴 → 𝐴𝐴 on a compact metric space A. If 𝑓𝑓 is topologically weak mixing, it is generically 𝛿𝛿-chaotic on A with 𝛿𝛿 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(𝐴𝐴). Proposition 2.2: If 𝑓𝑓: 𝐴𝐴 → 𝐴𝐴 has dense periodic points and is transitive, then 𝑓𝑓 has sensitive dependence on initial conditions [16] . Lemma 2.1: Let 𝑠𝑠, 𝑡𝑡 ∈ Ʃ 2 and 𝑠𝑠 𝑖𝑖 = 𝑡𝑡 𝑖𝑖 , 𝑑𝑑 = 0,1, … … … , 𝑑𝑑. Then 𝑑𝑑(𝑠𝑠, 𝑡𝑡) <

Main Results

We prove the chaotic characteristics of the complemented shift map in this section. Considering a particular property of the dynamical systems, which is the total transitivity. In this section, at first, we prove that 𝜎𝜎 𝑐𝑐 is continuous on ∑ 𝑚𝑚 . We give an example of a function (continuous) that satisfies the vital property of chaos that is topologically transitive but not chaotic. We are representing here a compatible example that all transitive maps (topologically) are not totally transitive and establish that 𝜎𝜎 and 𝜎𝜎 𝑐𝑐 are topologically conjugate. In the last part of this section, we solve a problem that is the dynamical systems (∑ 𝑚𝑚 , 𝜎𝜎 𝑐𝑐 ) is chaotic.

Theorem 3.1:

The dynamical systems (∑ 𝑚𝑚 , 𝜎𝜎 𝑐𝑐 ) is continuous on the symbol space ∑ 𝑚𝑚 .

Proof: We pick n so large that

Theorem 3.2:

The complemented shift map 𝜎𝜎 𝑐𝑐: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 is topologically transitive.

Proof: To establish that the complemented shift map 𝜎𝜎 𝑐𝑐 is topologically transitive, we have to show that for any two non-empty open sets P and

and 𝐵𝐵(𝑣𝑣, 𝑟𝑟) ⊆ 𝑄𝑄. We choose 𝑛𝑛 ∈ ℕ such that

which agrees with u up to the n th term. Therefore by the Proximity theorem, we have that 𝑑𝑑(𝑢𝑢, 𝑤𝑤) ≤ 1 𝑚𝑚 𝑛𝑛 < 𝑟𝑟 ⇒ 𝑤𝑤 ∈ 𝐵𝐵(𝑢𝑢, 𝑟𝑟) ⊆ 𝑃𝑃 and consequently it follows that 𝜎𝜎 𝑐𝑐𝑛𝑛 (𝑤𝑤) ∈ 𝜎𝜎 𝑐𝑐𝑛𝑛 (𝑃𝑃).

So it follows that 𝜎𝜎 𝑐𝑐𝑛𝑛 (𝑃𝑃) ∩ 𝑄𝑄 ≠ ∅ and hence 𝜎𝜎 𝑐𝑐: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 is topologically transitive. We know that the chaotic maps are Li-Yorke sensitive and all topologically transitive [2] . Here we establish an example of a continuous function which is Li-Yorke sensitive maps or topologically transitive but not chaotic in the sense of B. S. Du.

Example 3.1: Consider a function 𝐽𝐽(𝑥𝑥):

The above function 𝐽𝐽(𝑥𝑥) is a continuous function. We can simultaneously prove that 𝐽𝐽(𝑥𝑥) is Li-Yorke sensitive and topologically transitive [2] . However, in the sense of B. S. Du it is not chaotic because -11

21

(the period two-point) and the interval [0,1] are jumping alternatively [2] and never get close to each other. If 𝑟𝑟 = min{𝑟𝑟 1 , 𝑟𝑟 2 } then 𝐵𝐵(𝑢𝑢, 𝑟𝑟) ⊆ 𝑃𝑃 and 𝐵𝐵(𝑣𝑣, 𝑟𝑟) ⊆ 𝑄𝑄 and choose 𝑘𝑘 ∈ 𝑁𝑁 such that 1 𝑚𝑚 𝑘𝑘 < 𝑟𝑟. We then set up a sequence {𝑤𝑤 𝑛𝑛 } of points in ∑ 𝑚𝑚 + with the help of k, u and v such that

Here, every 𝑤𝑤 𝑖𝑖 , 𝑑𝑑 ≥ 2 is constructed by using the finite word obtained by taking first (i-1) consecutive symbols of a fixed sequence 𝑑𝑑 = (𝑑𝑑 1 , 𝑑𝑑 2 , 𝑑𝑑 3 , … … 𝑑𝑑 𝑖𝑖-1 , . . . ) ∈ ∑ 𝑚𝑚 chosen arbitrarily. More precisely, the first k letters of 𝑤𝑤 𝑖𝑖 , for each 𝑑𝑑 ≥ 2, is the finite word 𝑢𝑢 𝜎𝜎 𝑐𝑐(𝑘𝑘+𝑖𝑖-1) (𝑤𝑤 𝑖𝑖 ) ∈ 𝜎𝜎 𝑐𝑐(𝑘𝑘+𝑖𝑖-1) (𝐵𝐵(𝑢𝑢, 𝑟𝑟))⊆ 𝜎𝜎 𝑐𝑐(𝑘𝑘+𝑖𝑖-1) (𝑃𝑃) for all 𝑑𝑑 ∈ 𝑁𝑁.

Also 𝜎𝜎 𝑐𝑐(𝑘𝑘+𝑖𝑖-1) (𝑤𝑤 𝑖𝑖 ) = (𝑣𝑣 1 , 𝑣𝑣 2 , 𝑣𝑣 3 … … ) ∈ 𝑉𝑉, 𝜎𝜎 𝑐𝑐(𝑘𝑘+𝑖𝑖-1) (𝑤𝑤 𝑖𝑖 ) ∈ 𝜎𝜎 𝑐𝑐(𝑘𝑘+𝑖𝑖-1) (𝑃𝑃) imply that 𝜎𝜎 𝑐𝑐(𝑘𝑘+𝑖𝑖-1) (𝑃𝑃) ∩ 𝑄𝑄 ≠ ∅, for all 𝑑𝑑 ≥ 2.

Therefore,

𝜎𝜎 𝑐𝑐𝑛𝑛 (𝑃𝑃) ∩ 𝑄𝑄 ≠ ∅, for all 𝑛𝑛 ≥ 𝑘𝑘.

Hence, the complemented shift map is topologically mixing.

Theorem 3.4:

The complemented shift map 𝜎𝜎 𝑐𝑐 on ∑ 𝑚𝑚 is generically 𝛿𝛿 -chaotic with 𝛿𝛿 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(∑ 𝑚𝑚 ) = 1.

Proof:

We have proved that 𝜎𝜎 𝑐𝑐 on ∑ 𝑚𝑚 is topologically mixing. Since a topologically mixing map which is continuous on a compact metric space is also topologically weak mixing, so 𝜎𝜎 𝑐𝑐 is topologically weak mixing. Also, using Proposition 2.1, we observe that a continuous topologically weak mixing map on a compact metric space 𝐴𝐴 is generically 𝛿𝛿 -chaotic on 𝑋𝑋 with 𝛿𝛿 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(𝐴𝐴). So it follows that 𝜎𝜎 𝑐𝑐 on ∑ 𝑚𝑚 is generically 𝛿𝛿 -chaotic with 𝛿𝛿 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(∑ 𝑚𝑚 ) = 1. Further, if we guess that 𝑡𝑡 1 𝑡𝑡 2 … … 𝑡𝑡 𝑝𝑝 are 𝑝𝑝 finite sequences then 𝑡𝑡 1 𝑡𝑡 2 … … 𝑡𝑡 𝑝𝑝 can be defined in a like manner as above.

), and so on. Note that for any even integer 𝑘𝑘 > 0, 𝑊𝑊(𝑝𝑝, 2𝑛𝑛 + 𝑘𝑘) is a finite string of length (2𝑛𝑛 + 𝑘𝑘).

(iii) We take 𝑡𝑡 ∈ ∑ 𝑚𝑚 such that, 𝑡𝑡 = (𝑝𝑝 0 𝑝𝑝 1 … … … . 𝑝𝑝 𝑛𝑛 𝑊𝑊(𝑝𝑝, 2𝑛𝑛 + 2)𝑊𝑊(𝑝𝑝, 2𝑛𝑛 + 4)𝑊𝑊(𝑝𝑝, 2𝑛𝑛 + 6) … … … … … ).

Using those four notations and Lemma 2.1 as above, we now prove the theorem. By making 𝑝𝑝 and 𝑡𝑡 agree up to

Now consider the following two cases, which are favourable to prove this theorem.

Case I:

We consider 𝑛𝑛 is an odd integer.

Note that 𝑡𝑡 formation of infinitely many finite sequences of the type 𝑊𝑊(𝑝𝑝, 2𝑛𝑛 + 𝑘𝑘). We get, Hence,

Case II: Here 𝑛𝑛 is an even integer.

In this case, also 𝑡𝑡 formation of infinitely many finite sequences of the type 𝑊𝑊(𝑝𝑝, 2𝑛𝑛 + 𝑘𝑘).

So we get, 𝐿𝐿𝑡𝑡 𝑑𝑑𝑛𝑛𝑓𝑓 𝑛𝑛→∞ 𝑑𝑑(𝜎𝜎 𝑐𝑐𝑛𝑛 (𝑝𝑝), 𝜎𝜎 𝑐𝑐𝑛𝑛 (𝑡𝑡))

By virtue of (3.5), it is proved that the pair (𝑝𝑝, 𝑡𝑡) is Li-Yorke. Hence the dynamical system (∑ 𝑚𝑚 , 𝜎𝜎 𝑐𝑐 ) has chaotic dependence on initial conditions. he following example establishes that any map which is continuous has strong sensitive dependence on initial conditions, then it has sensitive dependence on initial conditions, but the converse is not always true. Here 𝑀𝑀 is a continuous map, and also it can be quickly showed that the function has sensitive dependence on initial conditions.

We observe that the maximum distance between any two points of [-1,1] is equal to 2. We consider the point - and the open interval 𝑈𝑈 = (0,1). Then there exists no point 𝑦𝑦 ∈ 𝑈𝑈 such that 𝑑𝑑�𝑀𝑀 𝑛𝑛 (𝑥𝑥), 𝑀𝑀 𝑛𝑛 (𝑦𝑦)� = 2, for any 𝑛𝑛 ≥ 0. Hence 𝑀𝑀(𝑥𝑥) does not strong sensitive dependence on initial conditions [2] . Let 𝑠𝑠 = (𝑠𝑠 0 𝑠𝑠 1 𝑠𝑠 2 … … ) and 𝑡𝑡 = (𝑡𝑡 0 𝑡𝑡 1 𝑡𝑡 2 … … ) be any points of ∑ 𝑚𝑚 , choose 𝜖𝜖 1 > 0 be arbitrary and an even integer n so large that

⇒ 𝑠𝑠 𝑖𝑖 = 𝑡𝑡 𝑖𝑖 for 𝑑𝑑 = 0,1,2, … … . , 𝑛𝑛 using Lemma 2.1

Which proves that on the symbol space∑ 𝑚𝑚 , our assuming map 𝑔𝑔: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 is continuous. Similarly, we can prove that the inverse of 𝑔𝑔 is also continuous. Next, we prove that the map 𝑔𝑔: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 is bijective. Let 𝑔𝑔(𝑠𝑠) = 𝑔𝑔(𝑡𝑡), then we get (𝑠𝑠 0 𝑠𝑠

Hence 𝜎𝜎 ০ 𝑔𝑔(𝑠𝑠) = 𝑔𝑔 ০ 𝜎𝜎 𝑐𝑐 (𝑠𝑠). So 𝜎𝜎 𝑐𝑐: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 and 𝜎𝜎: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 are conjugate topologically.

The study of transitive maps on compact spaces is widely motivated. It is essential to know if there exists some iterate that is not transitive for a transitive map. The following example gives that all topologically transitive maps are not totally transitive.

Example 3.3: Consider 𝐺𝐺(𝑥𝑥) from [0,1] onto itself is defined by

, 0 ≤ 𝑥𝑥 ≤ Since G is transitive, it has an orbit that is dense in [0,1] and hence, the interval is the unique invariant compact with a non-empty interior. It can be spontaneously shown that the map G is topologically transitive on [0,1], and we can see that the subintervals �0,

Solution:

In the significance of Li-Yorke, the complemented shift map 𝜎𝜎 𝑐𝑐: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 is chaotic because, according to the definition of Devaney, it is chaotic, and we know that Devaney chaos implicates Li-Yorke. By the scrambled set S consisting of points,⎾ 𝛼𝛼 where ⎾ 𝛼𝛼 is defined by, ⎾ 𝛼𝛼 = 𝛼𝛼 0 01𝛼𝛼 0 𝛼𝛼 0 𝛼𝛼 1 𝛼𝛼 1 0011𝛼𝛼 0 𝛼𝛼 0 𝛼𝛼 0 𝛼𝛼 1 𝛼𝛼 1 𝛼𝛼 1 𝛼𝛼 2 𝛼𝛼 2 𝛼𝛼 2 000111 … … … …, ∀ 𝛼𝛼 = (𝛼𝛼 0 𝛼𝛼 1 𝛼𝛼 2 … … … ) in ∑ 𝑚𝑚 , we can prove it directly. So in the symbol ∑ 𝑚𝑚 , 𝜎𝜎 𝑐𝑐 is a strong chaotic map. As the shift map is frequently used to model the chaoticity of a dynamical system, we can now use the complemented shift map in the position of the shift map to ideal the chaoticity of a dynamical system. Hence for chaotic dynamical systems, we accomplish that 𝜎𝜎 𝑐𝑐 is a new model.

Conclusion

In this work, we have established that some chaos-related properties of the complemented shift map and proved that 𝜎𝜎 𝑐𝑐 on ∑ 𝑚𝑚 is Devaney as well as Auslander-York Chaotic. It is generally 𝛿𝛿-chaotic with 𝛿𝛿 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(∑ 𝑚𝑚 ) = 1. We have shown that an example that is topologically transitive but not totally transitive. In this paper, we have proved that 𝜎𝜎 𝑐𝑐 is conjugate topologically to 𝜎𝜎. Finally, we proved that 𝜎𝜎 𝑐𝑐: ∑ 𝑚𝑚 → ∑ 𝑚𝑚 is chaotic.

Acknowledgement

AcknowledgementWe would like to thank James A. Yorke, University of Maryland, College Park, Maryland, for checking the manuscript and gave suggestions to improve the manuscripts.

References

1. Li TY, Yorke JA. Period three implies chaos, Amer. Math. Monthly 1975;82:985-992.
2. Auslander J, Yorke JA. Interval maps factors of maps and chaos. Tohoku Mathematical Journal 1980;32:177-188.
3. Devaney RL. An introduction to chaotic dynamical systems, 2nd edition, New York: Addison -Wesley, Redwood City, CA, 1989.
4. Block LS, Copple WA. Dynamics in One Dimension, Springer Lecture Notes, 1513, Springer Verlag, Berlin 1992, 5.
5. Biswas HR. Chaotic features of the generalized shift map and the complemented shift map, Barishal University Journal Part 2017;4(1):185-202,
6. Biswas HR, Md. Shahidul Islam. Chaotic features of the forward Shift Map on the Generalized m-Symbol Space, Journal of Applied Mathematics and Computation 2020;4(3):104-112
7. Robinson C. Dynamical systems: Stability, symbolic dynamics and chaos, 2nd Edition, CRC Press, Boca Raton, FL 1999.
8. Blanchard F, Glasner E, Kolyada S, Maass A. Li -Yorke pairs, J. Rei. Ange. Math 2000;547:51-68.
9. Bhaumik I, Choudhury BS. A new model of Chaotic Dynamics-the Complemented Shift Map, Journal of Physical Science, 2010;14:133-143
10. Bhaumik I, Choudhury BS. Totally Transitive Maps-A Short Note, Gen. Math. Notes, 2011;3(2):80-83.
11. Bhaumik I, Choudhury BS. Topologically conjugate maps and ω-chaos in symbol space, Int. J. Appl. Math 2010;23(2):309321
12. Biswas HR, Monirul M Islam. Shift Map and Cantor Set of Logistic Function. IOSR Journal of Mathematics 2020;16(3):0108
13. Ju H, Shao H, Choe Y, Shi Y. Conditions for maps to be topologically conjugate or semi-conjugate to subshifts of finite type and criteria of chaos. Dynamical Systems, International Journal 2016; 31(4),
14. Du BS. A dense orbit almost implies sensitivity to initial conditions, Bull. Inst. Math. Acad. Sinica 1998;26:85-94.
15. Du BS. On the invariance of Li-Yoke chaos of interval maps, J. Diff. Equ. Appl. 2005;11:823-826
16. Banks J, Brooks J, Cairns G, Stacey P. On Devaney's definition of chaos, Amer. Math. Monthly 1992;(99):332-334
17. Glasner E, Weiss B. Sensitive dependence on initial conditions, lop science, nonlinearity 1993;6(6):1067-1075
18. Ruette S. Chaos for continuous interval maps, www.math.u-psud.fr/ruette/, December 2003, 15
19. Parry W. Symbolic dynamics and transformation of the unit interval, Tran. Amer. Math. Soc 1966;122(2):368-378
20. Akin E, Kolyada S, Li-Yorke sensitivity, Nonlinearity 2003;16:421-433.
21. Dutta TK, Burhagohain A. Some properties of the Chaotic Shift map on the Generalized m-symbol space and topological conjugacy. Int. J. Math. Res 2015;7(2):173-190
22. Snoha L. Dense chaos, Comment. Math. Univ. Carolin 1992;33(4):747-752.