# The Chaotic Properties of Increasing Gap Shifts

Authors: Nor Syahmina Kamarudin; Malouh Baloush; Syahida Che Dzul-Kifli
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

It is well known that locally everywhere onto, totally transitive, and topologically mixing are equivalent on shift of finite type. It turns out that this relation does not hold true on shift of infinite type. We introduce the increasing gap shift and determine its chaotic properties. The increasing gap shift and the sigma star shift serve as counterexamples to show the relation between the three chaos notions on shift of infinite type.


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## Body

## 1. Introduction

The main ingredients for Devaney chaos is topologically transitive, dense periodic points, and sensitive dependence on initial conditions as originally defined by Devaney [1]. Later, it turns out that the third condition is redundant and therefore being excluded from the definition afterwards [2]. Even though other definitions of chaos with different senses were introduced, Devaney chaos has received more attention among the others. See Guirao et al. [3] for different versions of chaos definitions. On top of that there are more chaos notions that have been brought up to expose more characterizations of dynamical systems that exhibit randomness behavior. The incomplete list of the chaos notions is totally transitive, topologically mixing (or mixing), blending, locally everywhere onto (shortly l.e.o and also known as topologically exact), specification property, strong dense periodicity property and expansive. Following the introduction of many chaos notions, researchers started to find connections between those notions on variety topological spaces. Thomson [4] formed a hierarchy of three chaos notions on the interval, circle, torus and sphere. Li and Ye [5] had listed some recent development of chaos theory in topological dynamics by focusing on some versions of chaos with their relationships. See Fan et al. [6], Crannell [7], Alseda et al. [8], Denker et al. [9], Syahida and Good [10], and Malouh and Syahida [11] for relations on graph maps, shift spaces, closed interval, etc.It is well known that having the property of transitivity is sufficient enough for a system on the interval [12] and, on the infinite shift space, it is obvious that transitivity implies dense periodic points and therefore implies SDIC [2]. This is not true if we replace the property of transitivity with the other ingredient of Devaney chaos, dense periodicity property. It turns out, however, that strengthening this dense periodicity property yields different results. On shift of finite type, the implication is true [13] but not the case on the unit interval [10]. The study of chaotic behavior on shift of finite type has been done extensively in various approaches. However, analogues to results on shift of finite type are not much explored on shift of infinite type even though the main gaps between these two spaces are finiteness of its number of forbidden blocks. It is well known that totally transitive, topologically mixing, l.e.o., and specification property are equivalent on shift of finite type (see Crannell [7] and Fan et al. [6]). These equivalences are not necessarily true on general. At least Thomson [4] has provided an example on the interval with topologically mixing and totally transitive properties but without locally everywhere onto, while Ruette [14] has given an example that there is an interval map which is topologically mixing but it is not locally everywhere onto. The sigma star (see [7]) is another counterexample on shift of infinite type to demonstrate this phenomenon. In this work, we will consider a certain example of shift of infinite type and discuss its dynamical properties in order to see the differences between relations among these chaos notions on these two types of shift spaces.In the other perspective, Baker and Ghenciu [15] study the dynamical properties of certain S-gap shift and introduce two shift spaces: boundedly supermultiplicative (BSM) shifts and balanced shifts. Dawoud and Somaye [16] also look at topological dynamics of S-gap shift in terms of almost finite type, eventually constant, sofic etc.


## 2. Preliminary

To define shift space and its subspaces (shift of finite type and shift of infinite type), we letAn=0,1,2,…,n-1 be the alphabet set of n symbols. Consider the full n-shift; ∑n=x=xkk∈N0∣xk∈An as the collection of all sequences over symbols in An. The shift map σ on ∑n is defined as σx0x1x2x3…=x1x2x3…. We call =w0w1…wl-1, where wi∈An for every i=0,1,2,…,l-1 as a block of length l. For a block w, we denote Cw as a collection of any sequence in ∑n which started with the block w, i.e., Cw=x=x0x1x2x3…∈∑n∣x0x1x2…xl-1=w0w1w2…wl-1. The set Cw is called a cylinder set. The metric on ∑n is ds,t=1/2j, where j is the first entry of s and t in ∑n such that sj≠tj. The family of cylinder sets Cw where w is any allowed block form a basis for a topology on full n-shift, ∑n induced by a metric d.A closed subspaceX of ∑n is called a shift space if it is invariant under the map σ. We may see that, for every shift space, there is a set F of forbidden blocks where X is a collection of any sequences in ∑n which do not contain any block in F. We may then denote X as XF. If the set F is finite, then XF is shift of finite type and vice versa. Therefore, shift of finite type and its complement, shift of infinite type, are just distinguished based on the finiteness of the number of forbidden blocks but we found that they have different chaotic properties. Crannell [7] introduced the sigma star as the following set: ∑∗=x0x1x2x3…∈∑3∣xi=0,xj=2⇒i-j≠2p,forallp=0,1,2,…, and it is a shift of infinite type. It is shown that ∑∗,σ is totally transitive and strongly blending but not topologically mixing.A dynamical systemXF,σ is said to be topologically transitive if, for any pair of open sets U and V, there exists an integer n≥1, such that σnU∩V≠∅ and it is totally transitive if σn is topologically transitive for all positive integers n≥1. Mixing is when, for every pair of nonempty open subsets U,V⊂XF there exists an integer N∈N such that σn(U)⋂V≠∅, for all n>N and locally everywhere onto is when, for every open subset U⊂XF, there exists a positive integer n such that σnU=XF. σ is said to be strongly blending if for every pair of nonempty open subsets U,V⊂XF there exists an integer n>0 such that σn(U)∩σnV contains a nonempty open subset. A strong dense periodicity property is when, for all n∈N, the set of periodic points of prime period n is dense in the whole system.It is stated earlier that totally transitive, topologically mixing, l.e.o., and specification property are equivalent on shift of finite type. However, the sigma star is a counterexample on shift of infinite type to cross out mixing from the list of equivalence chaos notions. In the sequel section, the increasing gap shift is introduced and its chaotic dynamical properties are explored to show that totally transitive and l.e.o. is also not equivalence on infinite type shifts.


## 3. The Increasing Gap Shift

The increasing gap shift is an example from a larger class of subshift known in literature asS-gap shifts ΣS. To define S-gap shift, fix an increasing subset S in N0=N∪0. If S is finite, define ΣS to be the set of all binary sequences for which 1’s occur infinitely often and the number of 0’s between successive occurrences of 1 is an integer in S. When S is infinite, we need to allow points that begin or end with an infinite string of 0’s. The increasing gap shift ΣI is an S-gap shift with S=I=an∣a0=1,an=an-1+nforeveryn∈N0⊂N0. Since it has infinitely many forbidden blocks, it is shift of infinite type. Chaos properties of ΣI will be explored. Hence we have the following result.Theorem 1.
Σ I , σ is topologically mixing.Proof.
LetCw and Cv be basic open subsets of Σ(I) for allowable blocks w and v with length l and k, respectively. We consider cases for w and v. For the first case, let w=w0w1…wl-p1-2100…00︸p1 and v=00…00︸p21vp2+1…vk-1, where p1 and p2 are both in I. Choose N=l+1. Let n>N such that n=N+r=l+1+r for some r∈N. For any y∈Cv, we may take x=w11…11︸r+1y∈Cw such that σl+r+1x=σnx=y∈Cv. Hence, for every n>l+1, σn(Cw)∩Cv≠∅. Therefore, ΣI,σ is mixing for this case. There are 5 other cases to be considered. In each case, we choose different value of N such that, for every n=N+r>N and for some chosen y∈Cv, there exists x∈Cw such that σnx=y∈Cv. The points x and y are chosen differently in each case. All cases for w and v, chosen value of N, points x and y are given in Table 1. Note that p1 and p2 are any elements in I whereas c1 and c2 are not in I. For any c1∉I, let g1∈N such that c1+g1∈I. The same goes to c2 and g2.
Note that since0∉I, the case for w ending with 1 is one of the cases w=w0w1⋯wl-c1-2100⋯00︸c1 for any c1∉I. The same goes for the case when v started with 1; it is one of the case for v=00⋯00︸c21vc2+1⋯vk-1 for any c2∉I. Therefore, ΣI,σ is topologically mixing.Table 1
Cases for blocksw and v.
Blockw of length l Blockv of length k N y ∈ C v x ∈ C w w 0 w 1 ⋯ w l - p 1 - 2 00 ⋯ 00 ︸ p 2 1 v p 2 + 1 ⋯ v k - 1 l + 1 Any point w 11 ⋯ 11 ︸ r + 1 y 1 00 ⋯ 00 ︸ p 1 w 0 w 1 ⋯ w l - p 1 - 2 00 ⋯ 00 ︸ c 2 1 v c 2 + 1 ⋯ v k - 1 l + g 2 + 1 Any point w 11 ⋯ 11 ︸ r + 1 00 ⋯ 00 ︸ g 2 y 1 00 ⋯ 00 ︸ p 1 w 0 w 1 ⋯ w l - p 1 - 2 00 ⋯ 00 ︸ k l 000 ¯ w 11 … 11 ︸ r y 1 00 ⋯ 00 ︸ p 1 w 0 w 1 ⋯ w l - c 1 - 2 00 ⋯ 00 ︸ p 2 1 v p 2 + 1 ⋯ v k - 1 l + g 1 + 1 Any point w 00 ⋯ 00 ︸ g 1 11 ⋯ 11 ︸ r + 1 y 1 00 ⋯ 00 ︸ c 1 w 0 w 1 ⋯ w l - c 1 - 2 00 ⋯ 00 ︸ c 2 1 v c 2 + 1 ⋯ v k - 1 l + g 1 + g 2 + 1 Any point w 00 ⋯ 00 ︸ g 1 11 ⋯ 11 ︸ r + 1 1 00 ⋯ 00 ︸ c 1 00 ⋯ 00 ︸ g 2 y w 0 w 1 ⋯ w l - c 1 - 2 00 ⋯ 00 ︸ k l 000 ¯ w 11 … 11 ︸ r y 1 00 ⋯ 00 ︸ c 1The next result verifies thatΣI,σ is also Devaney chaotic.Theorem 2.
LetPn⊂ΣI be a collection of periodic points of increasing gap shift with prime period at least n. Then Pn is dense in ΣI for any n∈N.Proof.
LetCw be a basic open subset of ΣI for an allowed block w=w0w1…wl-1 in ΣI and n∈N. We will consider the possibilities for the block w, where the first case is w=00…00︸p11wp1+1…wl-p2-2100…00︸p2 for two elements p1 and p2 in I. For this case, take an element x in Cw such that x=00…00︸p11wp1+1…wl-p2-2100…00︸p211⋯11︸n-=w11⋯11︸n- is a periodic point of prime period greater than n. Therefore, Cw∩Pn≠∅. The other cases and its chosen periodic points with prime period at least n are presented in Table 2. Note that p1 and p2 are any elements in I whereas c1 and c2 are not in I. For any c1∉I, let g1∈N such that c1+g1∈I. The same goes to c2 and g2.
Since0∉I, the cases for w that ended or started with 1 are one of the cases in the table. Therefore, the table completes the proof.Table 2
Cases for blockw.
Blockw of length l Periodic pointx∈Cw w = 00 ⋯ 00 ︸ p 1 1 w p 1 + 1 ⋯ w l - p 2 - 2 1 00 ⋯ 00 ︸ p 2 x = w 11 ⋯ 11 ︸ n - w = 00 ⋯ 00 ︸ p 1 1 w p 1 + 1 ⋯ w l - c 2 - 2 1 00 ⋯ 00 ︸ c 2 x = w 00 ⋯ 00 ︸ g 2 11 ⋯ 11 ︸ n - w = 00 ⋯ 00 ︸ c 1 1 w c 1 + 1 ⋯ w l - p 2 - 2 1 00 ⋯ 00 ︸ p 2 x = w 11 ⋯ 11 ︸ n 00 ⋯ 00 ︸ g 1 - w = 00 ⋯ 00 ︸ c 1 1 w c 1 + 1 ⋯ w l - c 2 - 2 1 00 ⋯ 00 ︸ c 2 x = w 00 ⋯ 00 ︸ g 2 11 ⋯ 11 ︸ n 00 ⋯ 00 ︸ g 1 - w = 00 ⋯ 00 ︸ l, where l∈I x = w 11 ⋯ 11 ︸ n - w = 00 ⋯ 00 ︸ l, where l∉I x = w 00 ⋯ 00 ︸ g 11 ⋯ 11 ︸ n -, where g∈N such that l+g∈ISinceΣI does not have any isolated point, then Theorems 1 and 2 prove that ΣI,σ is Devaney chaotic.The increasing gap shift is strongly blending as stated by the following theorem.Theorem 3.
Σ I , σ is strongly blending.Proof.
LetCw and Cv be two basic open subsets of ΣI for two allowable blocks w=w0w1…wl-1 and v=v0v1…vk-1 with lengths l and k, respectively. We consider all possibilities of w and v by looking at its possibilities of 1 to be suffix of the blocks.
The first case is wheneverw1 and v1 are both allowed. Let n∈N such that n>l, n>k and n=l+r1,n=k+r2 for some r1,r2∈N. Let us choose u to be an allowed block in ΣI such that 1u is also allowed and let z∈Cu. Then, take x=w11…11︸r1z∈Cw and y=v11…11︸r2z∈Cv. Then, σl+r1x=σnx=z and σk+r2y=σny=z. Thus, z∈σn(Cw)∩σn(Cv). Therefore, Cu⊆σn(Cw)∩σn(Cv).
The second case is wheneverw1 is allowed but not v1. Without loss of generality, we assume that v=v0v1…vk-c2-2100…00︸c2, where c2∉I and c2+g2∈I for some integer g2. Let v′=v00…00︸g2. Now, w1 and v′1 are allowed. By the first case, there exists an integer n>0 such that σn(Cw)∩σn(Cv′) contains a nonempty open subset. Then, σn(Cw)∩σn(Cv) must also contain the nonempty open subset.
The last case is neitherw1 and v1 are allowed. Without loss of generality, let w=w0w1…wl-c1-2100…00︸c1 and v=v0v1…vk-c2-2100…00︸c2, where c1,c2∉I and c1+g1∈I, c2+g2∈I for some integer g1 and g2. Let w′=w00…00︸g1 and v′=v00…00︸g2 such that w′1 and v′1 are both allowed. By the first case, there exists an integer n>0 such that σn(Cw′)∩σn(Cv′) contains a nonempty open subset. Then, σn(Cw)∩σn(Cv) must also contain the nonempty open subset.
Therefore,ΣI,σ is strongly blending.However,ΣI is not l.e.o. as given in the following.Theorem 4.
Σ I , σ is not locally everywhere onto.Proof.
On the contrary, suppose thatΣI,σ is l.e.o. Then, for every basic open subset Cu⊂ΣI, there exists an integer k∈N such that σkCu=ΣI. Let Cu=C1 and σnC1=ΣI for some n∈N. Let m>n+1. Since am+1=am+m+1, then, from the way we define I, m+1 is the smallest integer r such that am+r∈I. Now let s∈Σ(I) such that s=00…00︸c1sc+1sc+2sc+3… where c=am+1. Since σnC1=ΣI, then there exists t∈C1 such that σnt=s. Then, t=1t1t2…tn-100…00︸c1sc+1sc+2sc+3…=1t1t2…tn-1s. Since t∈ΣI, then there exists an integer l≤n-1 such that t=1t1t2⋯100…00︸l00…00︸c1sc+1sc+2sc+3… and l+c∈I. So l+c=aj for some j∈N0. So l+am+1=aj. Since m+1 is the smallest integer r such that am+1+r∈I, then l≥m. But m>n+1 which implies l>n+1, and therefore it contradicts to l≤n-1. Therefore, ΣI,σ is not l.e.o.


## 4. Conclusion

Topologically transitive is one of the two components of Devaney chaos. In this paper we look at three chaos notions which are stronger than transitivity: totally transitive, topologically mixing, and l.e.o. It is generally known that totally transitive, topologically mixing, and locally everywhere onto are equivalent on shift of finite type. It has been discussed earlier that the sigma star is a shift of infinite type which satisfies totally transitive but not topologically mixing. We have presented another example of shift of infinite type, the increasing gap shift which has topologically mixing property but not locally everywhere onto. Therefore, these two counterexamples show that these three chaos notions are not equivalence on shift of infinite type. In addition, we have also shown that the increasing gap shift is strongly blending. The shift of finite type with forbidden blocks 00 and 10 is also strongly blending but not l.e.o. [17]. Therefore, strongly blending and l.e.o. are not equivalence on all shift spaces.Specification property is another chaos notion which is equivalent to l.e.o., mixing, and totally transitive on shift of finite type. However, as far as we are concerned this relation on shift infinite type is still unknown.



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