The Chaotic Properties of Increasing Gap Shifts

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.


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 2 International Journal of Mathematics and Mathematical Sciences 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 -gap shift and introduce two shift spaces: boundedly supermultiplicative (BSM) shifts and balanced shifts.Dawoud and Somaye [16] also look at topological dynamics of -gap shift in terms of almost finite type, eventually constant, sofic etc.

Preliminary
To define shift space and its subspaces (shift of finite type and shift of infinite type), we let   = {0, 1, 2, . . .,  − 1} be the alphabet set of  symbols.Consider the full -shift; ∑  = {x = {  } ∈N 0 |   ∈   } as the collection of all sequences over symbols in   .The shift map  on ∑  is defined as ( 0  1  2  3 . ..) = ( 1  2  3 . ..).We call =  0  1 . . . −1 , where   ∈   for every  = 0, 1, 2, . . .,  − 1 as a block of length .For a block , we denote   as a collection of any sequence in ∑  which started with the block , i.e., where  is the first entry of s and t in ∑  such that   ̸ =   .The family of cylinder sets   where  is any allowed block form a basis for a topology on full -shift, ∑  induced by a metric .
A closed subspace  of ∑  is called a shift space if it is invariant under the map .We may see that, for every shift space, there is a set  of forbidden blocks where  is a collection of any sequences in ∑  which do not contain any block in .We may then denote  as   .If the set  is finite, then   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: , for all  = 0, 1, 2, . ..}, and it is a shift of infinite type.It is shown that (∑ * , ) is totally transitive and strongly blending but not topologically mixing.
A dynamical system (  , ) is said to be topologically transitive if, for any pair of open sets  and , there exists an integer  ≥ 1, such that   () ∩  ̸ = 0 and it is totally transitive if   is topologically transitive for all positive integers  ≥ 1. Mixing is when, for every pair of nonempty open subsets ,  ⊂   there exists an integer  ∈ N such that   () ⋂  ̸ = 0, for all  >  and locally everywhere onto is when, for every open subset  ⊂   , there exists a positive integer  such that   () =   . is said to be strongly blending if for every pair of nonempty open subsets ,  ⊂   there exists an integer  > 0 such that   () ∩   () contains a nonempty open subset.A strong dense periodicity property is when, for all  ∈ N, the set of periodic points of prime period  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.

The Increasing Gap Shift
The increasing gap shift is an example from a larger class of subshift known in literature as -gap shifts Σ().To define -gap shift, fix an increasing subset  in N 0 = N ∪ {0}.If  is finite, define Σ() 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 .When  is infinite, we need to allow points that begin or end with an infinite string of 0's.The increasing gap shift Σ() is an -gap Since it has infinitely many forbidden blocks, it is shift of infinite type.Chaos properties of Σ() will be explored.Hence we have the following result.

Theorem 1. (Σ(𝐼), 𝜎) is topologically mixing.
Proof.Let   and  V be basic open subsets of Σ() for allowable blocks  and V with length  and , respectively.We consider cases for  and V.For the first case, let  =  0  1 . . . − 1 −2 100 . . .00 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟  1 and V = 00...00 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ where  1 and  2 are both in .Choose  =  + 1.Let  >  such that  =  +  =  + 1 +  for some  ∈ N.For any y ∈  V , we may take x = 11 . . .11 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ +1 y ∈   such that  ++1 (x) =   (x) = y ∈  V .Hence, for every  >  + 1,   (  ) ∩  V ̸ = 0. Therefore, (Σ(), ) is mixing for this case.There are 5 other cases to be considered.In each case, we choose different value of  such that, for every  = + >  and for some chosen y ∈  V , there exists x ∈   such that   (x) = y ∈  V .The points x and y are chosen differently in each case.All cases for  and V, chosen value of , points x and y are given in Table 1.Note that  1 and  2 are any elements in  whereas  1 and  2 are not in .For any  1 ∉ , let  1 ∈ N such that  1 +  1 ∈ .The same goes to  2 and  2 .
Note that since 0 ∉ , the case for  ending with 1 is one of the cases for any  1 ∉ .The same goes for the case when V started with 1; it is one of the case for V = 00 ⋅ ⋅ ⋅ 00 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (Σ(), ) is topologically mixing.
Theorem 2. Let   ⊂ Σ() be a collection of periodic points of increasing gap shi with prime period at least .en   is dense in Σ() for any  ∈ N.
Proof.Let   be a basic open subset of Σ() for an allowed block  =  0  1 . . . −1 in Σ() and  ∈ N. We will consider the possibilities for the block , where the first case is  = 00 . . .00 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ for two elements  1 and  2 in .For this case, take an element x in   such that  Since 0 ∉ , the cases for  that ended or started with 1 are one of the cases in the table.Therefore, the table completes the proof.
The increasing gap shift is strongly blending as stated by the following theorem.

Theorem 3. (Σ(𝐼), 𝜎) is strongly blending.
Proof.Let   and  V be two basic open subsets of Σ() for two allowable blocks  =  0  1 . . . −1 and with lengths  and , respectively.We consider all possibilities of  and V by looking at its possibilities of 1 to be suffix of the blocks.
The first case is whenever 1 and V1 are both allowed.Let  ∈ N such that  > ,  >  and  =  +  1 ,  =  +  2 for some  1 ,  2 ∈ N. Let us choose  to be an allowed block in Σ() such that 1 is also allowed and let z ∈   .Then, take x = 11 . . .11 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ The second case is whenever 1 is allowed but not V1.Without loss of generality, we assume that

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.
such that  +  ∈

Table 1 :
Cases for blocks  and V.
1  1 +1 . ..− 2 −2 100 ...00 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ The other cases and its chosen periodic points with prime period at least  are presented in Table2.Note that  1 and  2 are any elements in  whereas  1 and  2 are not in .For any  1 ∉ , let  1 ∈ N such that  1 +  1 ∈ .The same goes to  2 and  2 .
is a periodic point of prime period greater than .Therefore,   ∩   ̸ = 0.

Table 2 :
Cases for block .