INTRODUCTION
Let G be a finite abelian group. We consider our study here when G = {0,1}, X_{G} = G^{Z2} be the space of doubly indexed sequences over G and it is subspace
The twodimensional shift of finite type α: X_{G} → X_{G} defined by:
of the space G^{Z2} of doubly indexed sequences over a finite abelian
compact group G. The space X_{G} was introduced by Ward
(1993).
To fined the Zeta function, it requires from us study the periodic points in X_{G }by α: X_{G} → X_{G}.
The group Z^{2} acts natural on the space X_{G} via left and upward shifts we show the periodic point data of X_{G} determine the group G.
This allows us to view X_{G} as a Z^{2}space for every subgroup U of Z^{2}, we define Uperiodic point to be those x ε X_{G} by the action of the subgroup U. This is natural generalization of the notion of periodic points in case of any ordinary Zaction. If U has finite index in Z^{2}, the number F_{U} of Uperiodic points x ε X_{G} is finite.
Compare this situation to the full shift, i.e., the shift action on the space X_{G}. If the index of U in Z^{2} is m, then obviously F_{U} = G^{m} for any group G.
It was also introduced by Lind (1996), Roettger
(2005) and AlRefaei and Noorani (2007) new way
to calculate periodic point numbers based on the study of linear algebra of
F_{2} the field and get the zeta function which generality by Lind
(1996).
Huffman code has been widely used in text, image and video compression. For
example, it is used to compress the result of quantization stage in JPEG (Hashemain,
1995). The simplest data structure used in the Huffman decoding is the Huffman
tree. Array data structure (Chen et al., 1999)
has been used to implement the corresponding complete ternary tree for the Huffman
tree.
Modeling and simulation of thermal dynamic system has been studied by Bhatti
et al. (2006). Whereas, Shabbak et al.
(2011) proposed multivariate control chart which is less sensitive to the
sustained shift in mean process. Another related work has been proposed by AbuShawiesh
et al. (2009) and Goegebeur et al. (2005).
Huffman code has been widely used in text, image and video compression. For
example, it is used to compress the result of quantization stage in JPEG (Hashemain,
1995). The simplest data structure used in the Huffman decoding is the Huffman
tree. Array data structure (Chen et al., 1999)
has been used to implement the corresponding complete ternary tree for the Huffman
tree.
Here our study will be about the compact group G = {0,1}. And so we observe that there are several of subgroups U of Z^{2} with finite indexes but in our work here we can classify the periodic points for subgroups U of Z^{2}.
CALCULATION FOR THE ZETA FUNCTION
Here, we will introduce the formula for the zeta function of left and upward shifts α according to the quality of the subgroups which are used in the following propositions:
Proposition 1: The zeta function of a 2dimensional shift of finite type via left and upward shifts α according to his subgroups U of Z^{2} in the formula (2) which defined as follows:
Will be given by ζ_{x} (Z) = 1/1Z
Proof: It was introduced by Ward (1993) and
Lind (1996) that for finite abilean (x_{s, I})
ε X_{G} is a Uperiodic point if and only if:
The number of periodic points for the special subgroups U as in (2) is exactly the No. of solutions to Eq. 3.
In this case at G = {0,1} we observe that for (x_{s, I}) ε X_{G}, x_{s,i} will be 0 or 1. And for appositive integer k = 0, 1, 2. We observe that 2^{k}1 is odd No. Then (2^{k}1) 0 = 0 and (2^{k}1). That is, for any subgroup U of Z^{2} in the formula (2), the zeroelement will be only periodic point, i.e., F_{U} = 1.
Hence the zeta function will define by:
ζ_{x} (Z) = exp (log (1Z)) = 1/1Z
Proposition 2: The zeta functions of a 2dimensional shift of finite type via left and upward shifts α according to the subgroups U of Z^{2} which defined as follows:
Will be given by:
Proof: It was introduced by Ward (1998) that
(x_{s, I}) ε X_{G} is an Uperiodic point if and:
The No. of periodic points for the special subgroups U as in (4) is exactly the No. of solutions to Eq. 5, 6 above.
When G = {0, 1}, we observe that for (x_{s, I}) ε X_{G} will be 0 or 1. Hence, for appositive integer m = 0, 1, 2… we observe
From definition of X_{G} , for (x_{s, I}) ε X_{G}
must satisfy the following equation :
We observe that the equation has four solutions:
Hence, if the System for Eq. 5, 6 has an
unique solution, then for any subgroup U of Z^{2} in the formula (4),
the zeroelement will be only periodic point, i.e., F_{U} = 1. And if
the system for Eq. 5, 6 has more than one
solution, then by the solutions for Eq. 7, for any subgroup
U of Z^{2} in the formula (4), F_{U} = 4.
Now we will take some special cases in this formula as follow:
• 
If a = 1, c = 1, b = 0 then from Eq. 5,
6 we get: 
x_{s, i+1} = x_{s, I} + x_{s+1,
I} 
x_{s,i} = x_{s+1, t} and x_{s,
I} = x_{s, I} +x_{s+1, t} this implies to 
x_{s,i} = 0, hence F_{u} = 1.
• 
If a = 2, b = 1, d = 1 then from Eq. 5 and
6, we get 
From Eq. 8 x_{s1, t} and from 9 x_{s, I}
this implies to F_{U} = 1
• 
If a = 3, b = 4, d = 1 then from Eq. 5 and
6, we get 
Then this equation has more then one solution therefore F_{U} = 4
We can continuous by the same way to calculate the periodic points for some special cases again as showed in Table 1.
Where, a, d>0, 0≤b <a and F_{U }is the periodic point.
Hence, we can define the zeta function by:
Table 1: 
Periodic points for some special cases 

Proposition 3: The zeta function of a 2dimensional shift of finite type via left and upward shifts α according to the subgroups U of Z^{2} which defined as follows:
will be given by:
Proof: We observe that the formula (10) is special case from the formula (4) by make b = 0. Hence, we can put b = 0 in the Eq. 6 to get that:
Now we will take some special cases in this formula as follow:
• 
We observe that if a = 1 and d is even No. then from Eq.
5 and 6 we get: 
x_{s, t} = x_{s+a, t} And 
This is implies to:
We observe that is
even number for 1≤ I < d Hence, from (12) we get:
• 
If a = 1 and d = 1 then from Eq. 5 and 6
we get: 
From Eq. 13 we observe that:
Hence from Eq. 14 we get:
• 
If a = 2 and d = 1 then from Eq. 5 and 6
we get: 
If d is even then from (1) we get that F_{U} = 1
If d is odd then from (15) we observe that:
Hence, from Eq. 16 we get that x_{s+(d1), t}, this implies to F_{U }= 1.
Then e we can define the zeta function by:
We can also compute F_{U} using some linear algebra over G (we can
consider G a field). A point x ε X_{G} that is Uinvariant must
have horizontal periodic a, so is determined by an:
element from the vector space F^{a}_{2}.
Let I_{a} be axa identity matrix. Let P_{a} be axa permutation matrix corresponding to the cyclic shift of elementary basis vectors F^{a}_{2}.
The condition of Uperiodicity of (x_{s,t}) translates to the condition:
Hence:
Then the zeta function in this case equal:
We will take some cases in this form
• 
If a = 1, c = 1, b = 0, then, 
P_{a} + I_{a} = 2 Then y = y ⇒ y = 0 because
2*1 = 0 ≠ 1 not allow so 2*0 = 0. Hence F_{U }= 1
• 
If a = 1, d = r, b = 0, then 
(Pa + 1a)^{r} = 2^{r}
2^{r} y = y ⇒ y = 0. Then F_{U }= 1.
• 
If a = 2, d = 1, b = 0 Then: 
Hence:
Let
Hence:
There fore y_{1} = y_{2},
Either y_{1} = 0, y_{2} ⇒ y_{1} y_{2}
= 0 is possible or y_{1} = 1 is impossible because y_{2} ⇒
y_{1} y_{2} = 0 but y_{1} y_{2} = y_{1}
is not zero. Hence F_{U }= 1.
• 
If a = 2, d = 1, b = 0. Then: 
Then y_{1} = y_{2} = 0. Hence F_{U }= 1.
Let a = 2 , d = 2, b = 0:
The we get that y_{1} = 0, y_{2} = 0 . Hence F_{U }=
1.
And when a = 2, d = r, b = 0, then F_{U }= 1. When a = 3, d = 1, b =
0 we get that:
Then:
So y_{1} = y_{2} = y_{3} = 0, therefore, F_{U }=
1.
Also similarly, when a = 3, d = 1, b = 1, we get that:
Let y_{1} = 0 ⇒ y_{3} = y_{2} then y_{3}
= 1 and y_{3} = 0, y_{2} = 0 and let:
So:
Hence, F_{U }= 4.
Also when a = 3, d = 1, b = 2, similarly, F_{U }= 1.
In the same way, we can get in all theses cases (a = 3, d = 2, b = 0), (a = 3, d = 2, b = 1) that F_{U }= 1 and in these cases (a = 3, d = 2, b = 2), (a = 3, d = 3, b = 0) that F_{U }= 1..
By using Mathematica, we will get this result:
CONCLUSIONS
The periodic point was calculated by using two methods namely, the index subgroup and the linear algebra of the field F_{2}. According to these two methods, the Zeta function for two dimensional shift of finite type by left and upward shifts α has been studied. For validation of the present results, three cases study have been presented to get the Zeta function.