International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 2936560, 5 pages

https://doi.org/10.1155/2019/2936560

## The Chaotic Properties of Increasing Gap Shifts

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Correspondence should be addressed to Syahida Che Dzul-Kifli; ym.ude.mku@adihays

Received 30 July 2018; Revised 7 November 2018; Accepted 28 January 2019; Published 27 February 2019

Academic Editor: Frédéric Mynard

Copyright © 2019 Nor Syahmina Kamarudin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.