d-action Induced by Shift Map on 1-Step Shift of Finite Type over Two Symbols and k-type Transitive

The dynamics of a multidimensional dynamical system may sometimes be inherited from the dynamics of its classical dynamical system. In a multidimensional case, we introduce a new map called a d  -action on space X induced by a continuous map : f X X → as : d f T X X × →  such that ( ) ( , ) ( ) r n f T n x f x = , where d n∈ , x X ∈ and : d r →   is a map of the form 1 1 2 2 ( ) ... d d r n h n h n h n = + + + . We then look at how topological transitivity of f effects the behaviour of k -type transitivity of the d  -action, f T . To verify this, we look specifically at spaces called 1 -step shifts of finite type over two symbols which are equipped with a map called the shift map, σ . We apply some topological theories to prove the d  -action on 1 -step shifts of finite type over two symbols induced by the shift map, Tσ is k -type transitive for all { } 1,2,..., 2 k ∈ whenever σ is topologically transitive. We found a counterexample which shows that not all maps Tσ are k -type transitive for all { } 1,2,..., 2 k ∈ . However, we have also found some sufficient conditions for k -type transitivity for all { } 1,2,..., 2 k ∈ . In conclusions, the map Tσ on 1 -step shifts of finite type over two symbols induced by the shift map is k -type transitive for all { } 1,2,..., 2 k ∈ whenever either the shift map is topologically transitive or satisfies the sufficient conditions. This study helps to develop the study of k -chaotic behaviours of d  -action on the multidimensional dynamical system, contributions, and its application towards symbolic dynamics.

. In conclusions, the map T σ on 1 -step shifts of finite type over two symbols induced by the shift map is k -type transitive for all { }

Introduction
The study of d  -action has become the current interest among researchers which involves the observation of a multidimensional dynamical system. Compared to the classical dynamical system study (or d  -action), the behavior of d  -action is much more complex and almost impossible to be described. However, there are many discussions that have an approach study on the chaotic behavior of the d  -action.
The existence of d  -action on X originally came from group action as the general form. Instead, the main focus of group action was replaced with d  in the studies.
There are many finding results which focus on group actions and also some other special kind of group actions. For instance, Barzanouni et al. [1] had studied group actions on metric space with expansive properties. They found several relations of expansivity in various cases such as between subgroup actions and covering maps. The study had also introduced orbit expansivity to characterize the expansive action. While Wang and Zhang [2] had focused on group actions for which the group was countable and discrete. They defined the notions of local weak mixing and Li-Yorke chaos for this kind of action to show the relation between them. Next, they had also studied the topological entropy of actions of an infinite countable amenable group and actions of an infinite countable discrete sofic group on a shift of finite type.
Then, Cairns et al. [3] had also studied on group actions and defined six notions of dynamical transitivity and mixing in the context of group actions. Interestingly, they highlighted some relations between those six notions in which they are inherited by subgroups, by taking products and when passing to the induced action on hyperspace. In addition, their discussions were also focus on semi-conjugacy and actions of abelian groups.
There are some studies which directed to the chaos of semigroup actions. Wang et al. [4] studied the action of a semigroup and abelian monoid on Polish space. They learnt about the sensitive and syndetic transitive of the system. Then, they had results on which the system was chaotic by depending on chaos in the sense of Li-Yorke and Devaney. In [5], they also studied the action of semigroup and gave a focus on periodicity and transitivity. They also learnt some chaotic changes like sensitivity to initial conditions and equicontinuous.
Our major interest to study about d  -action is mainly because of some findings from Shah and Das [6,7,8] who studied and introduced the notions of k -type transitive and some other k -type chaos notions such as k -type periodic point, k -type sensitive dependence on initial conditions, k -type Devaney chaotic and k -type mixing. All of the k -type chaos notions are defined mostly related to the chaos notions in the classical dynamical system and therefore we may see some familiarities through the definitions. The k -type chaos notions do help the research to study the behavior of d  -action on a space X with easier understanding and a better way of structure. The study of Shah and Das in [6] was to focus on relationships between k -type Devaney chaotic of d  -action and its induced d  -action. Furthermore, they also highlight some relations especially involving k -type transitive, dense k -type periodic point, k -type sensitive dependence on the initial condition, k -type weak mixing and k -type mixing. While in [7], their major focus was on a relationship of k -type chaos notions within conjugacy, uniform conjugacy, and product spaces. They also mention the redundancy of k -type sensitive dependence on the initial condition for k -type Devaney chaotic as similar to the finding of Banks et al. in [9] for Devaney chaos on infinite metric spaces in the study of a classical dynamical system.
In [8], Shah and Das changed their focus to the notion of k -type collective sensitive and studied the relation of the new notion between uniform conjugacy and finite product.
They also find the relation between k -type sensitive dependence on initial conditions and k -type collective sensitive for induced d  -action. Besides that, Kim and Li [10] had introduced and studied the notion of k -type limit set and k -type non-wandering set of 2  -action. Their major purpose was to generalize the spectral decomposition theorem for k -type non-wandering points of a 2  -action. On the other hand, Lima [11] had an interest of study on d  -action which is ergodic. The research had also tried to find the connection between ergodic property and positive topological entropy within There are also some interests of study on d  -action for symbolic dynamical systems. Many studies are interested in looking at shifts of finite type and therefore they had defined d  -action on the shift of finite type. In [12], the research had learnt about the phenomenon of transition from the classical shifts of finite type to the multidimensional shifts of finite type. The discussion is mainly about an algebraic structure called Wang Tiling which appears in certain multidimensional shifts. While the study in [13] was interested in the entropy value of multidimensional shifts of finite type. Next, Boyle and Schraudner [14] had also extended the result by finding d  shifts of finite type with positive topological entropy but cannot factor topologically onto the d  Bernoulli shift on N symbols. However, Pavlov [15] had a different approach which studied on d  -shift spaces. Its main purpose was to give conditions which guarantee a d  -shift space to be nonsofic.
In this paper, we may introduce a new concept of d  -action called as a d  -action on X induced by a continuous map : f X X → . Then, we may focus on a specific kind of shift of finite type which is 1 -step shift of finite type over two symbols. Our main purpose is to relate the transitivity of the shift map σ to the k -type transitivity of d  -action induced by the shift map on 1 -step shift of finite type over two symbols.

d  -action and Preliminary Definitions
Let 0 d > . We let ( , ) X ρ be a topological dynamical system and a d  -action on a space X was defined in most of the past studies as a continuous map , for all , d n m ∈  and for all x X ∈ .
In addition, : is a homeomorphism on X [6]. In the classical system, it is said a given continuous map : is topologically transitive if for every pair of open sets U and V of X , there exists an integer 0 n > such that ( ) For a d  -action, we must let Next, let us introduce d  -action on a space X induced by a continuous map f on X into itself in the following definition.

Let
:  shifts the same amount to the right [16]. A shift space is a closed, shift-invariant subset of 2 Σ . Equivalently, let F be any set of blocks (or later will be called as set of forbidden blocks), the set F X X = of sequences that do not contain any element of F is a shift space. If the set F is finite, then it is called as a shift of finite type. A shift of finite type is an M -step if the set of the forbidden block F contains all blocks which have length 1 M + . Therefore, 1 -step shift of finite type over two symbols is a shift space in which its forbidden block, F contains blocks of length 2 .

Shift Map on 1 -Step Shifts of Finite Type over Two Symbols
In this subsection, we will discuss the shift map on 1 -step shifts of finite type over two symbols. With only two symbols, we have four possible different blocks of length two i.e. 00, 01, 10, and 11, then we have 16 sets of forbidden blocks. , , X X X and 9 X . Then, Therefore, 2 ( )

 -action Induced by Shift Map on 1 -Step Shifts of Finite Type over Two Symbols
Our main objective is to study the behavior of d  -action induced by shift map on 1 -step shifts of finite type over two symbols. Firstly, a d  -action induced by shift map on 1 -step shifts of finite type over two symbols is given by the following definition.