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.
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 -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.
2. Preliminary
To define shift space and its subspaces (shift of finite type and shift of infinite type), we let be the alphabet set of symbols. Consider the full -shift; as the collection of all sequences over symbols in . The shift map on is defined as We call , where for every as a block of length . For a block , we denote as a collection of any sequence in which started with the block , i.e., . The set is called a cylinder set. The metric on is , where is the first entry of and 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: , 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 , such that and it is totally transitive if is topologically transitive for all positive integers . Mixing is when, for every pair of nonempty open subsets there exists an integer such that , 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 such that contains a nonempty open subset. A strong dense periodicity property is when, for all , 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.
3. The Increasing Gap Shift
The increasing gap shift is an example from a larger class of subshift known in literature as -gap shifts . To define -gap shift, fix an increasing subset in . If is finite, define to be the set of all binary sequences for which 1’s occur infinitely often and the number of ’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 ’s. The increasing gap shift is an -gap shift with Since it has infinitely many forbidden blocks, it is shift of infinite type. Chaos properties of will be explored. Hence we have the following result.
Theorem 1. is topologically mixing.
Proof. Let and be basic open subsets of for allowable blocks and with length and , respectively. We consider cases for and For the first case, let and , where and are both in Choose . Let such that for some For any , we may take such that . Hence, for every , . Therefore, 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 , there exists such that . The points and are chosen differently in each case. All cases for and , chosen value of , points and are given in Table 1. Note that and are any elements in whereas and are not in . For any , let such that The same goes to and .
Note that since , the case for ending with 1 is one of the cases for any . The same goes for the case when started with 1; it is one of the case for for any . Therefore, is topologically mixing.
The next result verifies that is also Devaney chaotic.
Theorem 2. Let be a collection of periodic points of increasing gap shift with prime period at least Then is dense in for any .
Proof. Let be a basic open subset of for an allowed block in and . We will consider the possibilities for the block , where the first case is for two elements and in . For this case, take an element in such that is a periodic point of prime period greater than . Therefore, . The other cases and its chosen periodic points with prime period at least are presented in Table 2. Note that and are any elements in whereas and are not in . For any , let such that The same goes to and .
Since , the cases for that ended or started with 1 are one of the cases in the table. Therefore, the table completes the proof.
Since does not have any isolated point, then Theorems 1 and 2 prove that is Devaney chaotic.
The increasing gap shift is strongly blending as stated by the following theorem.
Theorem 3. is strongly blending.
Proof. Let and be two basic open subsets of for two allowable blocks and with lengths and , respectively. We consider all possibilities of and by looking at its possibilities of 1 to be suffix of the blocks.
The first case is whenever 1 and are both allowed. Let such that , and for some Let us choose to be an allowed block in such that is also allowed and let . Then, take and Then, and . Thus, . Therefore, .
The second case is whenever is allowed but not Without loss of generality, we assume that , where and for some integer Let . Now, and are allowed. By the first case, there exists an integer such that contains a nonempty open subset. Then, must also contain the nonempty open subset.
The last case is neither and are allowed. Without loss of generality, let and , where and , for some integer and Let and such that and are both allowed. By the first case, there exists an integer such that contains a nonempty open subset. Then, must also contain the nonempty open subset.
Therefore, is strongly blending.
However, is not l.e.o. as given in the following.
Theorem 4. is not locally everywhere onto.
Proof. On the contrary, suppose that is l.e.o. Then, for every basic open subset , there exists an integer such that Let and for some Let . Since , then, from the way we define , is the smallest integer such that . Now let such that where . Since , then there exists such that . Then, Since , then there exists an integer such that and So for some So Since is the smallest integer such that , then But which implies , and therefore it contradicts to Therefore, 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.
Data Availability
There is no data used in this article.
Conflicts of Interest
The authors of this manuscript declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to thank Universiti Kebangsaan Malaysia and Center for Research and Instrumentation (CRIM) for the financial funding through FRGS/1/2017/STG06/UKM/02/2.