Some Chaotic Results of Product on Zero Dimension Spaces

In this work , we study different chaotic properties of the product space on a onestep shift of a finite type, as well as other spaces. We prove that the product is Lyapunove ε –unstable if and only if at least one or is Lyapunove ε – unstable. Also, we show that and locally everywhere onto (l.e.o) if and only if is locally everywhere onto (l.e.o) .


Introduction
A discrete dynamical system is a way of describing the passage in discrete times of all points in a given space . A continuous map describes the rule of changes of each point. Therefore we can define a discrete dynamical system by a map . We analyze its dynamical behavior to determine if the system settles steady to equipoise, goes on in repeating cycles, or does something more complex. A chaotic dynamical system is an unpredictable system which can be found within complicated dynamical systems, as well as the almost trivial systems. Several efforts have been made to give the notion of chaos a mathematically precise meaning. However, chaos is not simple to define and has no universally concordant definition [1].
Until the end of the 1980s the subject of chaotic dynamics was limited mainly to research-oriented publications by Devaney, where his famous definition of chaos was (A map is said to be Devaney chaotic if it satisfies; is topologically transitive, the periodic points of is dense, and posses sensitive dependence on initial conditions or simply (SDIC)) [1,2].

ISSN: 0067-2904
Kamel and AL-Shara'a Iraqi Journal of Science, 2020, Vol. 61, No. 2, pp: 428-434 424 In another work [3], Dzul-kifli and Good showed that the set of points with prime period at least is dense for each if is Devaney chaotic on a compact metric space with no isolated points . In their article [4], Baloush and Dzul-kifli introduced six various one-step shifts of finite types, with totally different dynamic demeanor, and clearified the dynamics of each space Other authors [5] showed that the expression "Locally Everywhere Onto" implies many other chaos properties such as mixing, totally transitive ,and blending. Another study investigated how chaos conditions on maps hold over to their products [6].
Finally, Iftichar Talb [7] proved that some properties on a map were carried over to their product, and he also contracted the conditions of two maps to achieve that the product is chaotic. Let be a map with a metric . Given if for every neighborhood ( ) of , there exist ( ) and with ( ( ) ( )) , then the map is said Lyapunov -unstable at a point .

Definition 2.6. [5]
Let be a map that is said to be locally everywhere onto if for every open set of , there exists a positive integer such that ( ) .

On The Shift of Finite Type
The product space * + * * + * + + is a topological space. A convenient basis for * + is given by the cylinder sets , -* * + + Where and * + for each . when equipped with the metric defined as where is the minimal number such that since this gives the cylinder sets as open ball ( ) , -, [8]. An -block is a finite sequence of symbols of the length , i.e. . We now define a continuous map on* + , called the shift map as ( ) . Shift map deletes the first entry of the sequence in * + to produce the image of the sequence under . Shift space is a closed subset of full-2-shift which is invariant under [9] . We write instead of . The most widely studied shift spaces are called shifts of finite type, defined as follows

Definition 3.1 [9]
A shift space * + is a shift of finite type if there is a finite number of blocks over symbols 0 and 1 where the blocks do not occur in any element of . The blocks are said forbidden blocks in .
Because we only have four possible distinct blocks of length two, i.e. 00, 01,10 and 11, then we have 16 sets of forbidden blocks. For each * + * + is the one-step SFT with a set of forbidden blocks . Then , there exists that some of them are singletons, empty set or the whole * + , which have trivial dynamics and are not of our interest. There exists six distinct one-step shift of finite types over two symbols, and , with sets of forbidden blocks * + * + * + * + * + and * + respectively [1]. They are shown below through matrices and their own graph

Main Results
In this section , some results on * + are proved.
This means that is SDIC. Let is not SDIC , this means that given any there exists * + * + such that for a certain cylinder set , containing , the inequality ( ( ) ( )) ⁄ holds for every ́ * ́ ́ ́ ́ + and positive integer . Identically, there is * + such that for a certain cylinder set , containing , the inequality ( ( ) ( ) ) ⁄ holds for every ́ * ́ ́ ́ ́ + and positive integer .
so that it is not SDIC, which disagrees with the hypothesis.

Theorem 4.3
Let the one-step SFT and which has the sets of forbidden blocks * + * + , let * + be a map and let the product is topologically transitive on * +, then the map is topologically transitive on . Proof: Let , ́ ́ ́-, be non-empty cylinder sets in , so the sets and are cylinder sets in . As is transitive , there is a positive integer for this ( ) ( ) . then ( ) ( ) It follows that ( ( ) ) ( ( ) ) , so ( ) , thus is topologically transitive.

Theorem 4.4
Let the one-step SFT which has a set of forbidden blocks * + , the map is topologically mixing then is topologically mixing . Proof: Let be a topologically mixing map . Given , there exists cylinder sets such that and By the assumption that there exist and such that ( ) for and ( ) for . For * + we get (( ) ( )) ( ) ( ( ) ( )) ( ) ( ( ) ) ( ( ) ) Which means that is topologically mixing. ■ Theorem 4.5 Let the one-step SFT and which have the sets of forbidden blocks * + , * +, the map * + is weakly blending if and only if * + is weakly blending .

Proof:
If is weakly blending let and be non-empty cylinder sets in , then there are and non-empty cylinder of and there are and cylinder in such that and . Since is weakly blending , then such that  ) is a nonempty set then is strongly blending.

Some properties in the product Space
In this section , we prove the results on the product maps defined on any metric spaces .