Two Dimensional Discrete Dynamics of Integral Value Transformations
Jayanta Kumar Das Sudhakar Sahoo Sk. Sarif Hassan Pabitra Pal Choudhury










Applied Statistics Unit, Indian Statistical Institute
 Kolkata-700108, India

dasjayantakumar89@gmail.com


Institute of Mathematics and Applications
 Bhubaneswar-751029,  India.

sudhakar.sahoo@gmail.com


Department of Mathematics, Pingla Thana Mahavidyalaya

Paschim Medinipur-721140,  India.

sarimif@gmail.com


Applied Statistics Unit, Indian Statistical Institute
 Kolkata-700108, India

pabitrapalchoudhury@gmail.com

Two Dimensional Discrete Dynamics of Integral Value Transformations
    Pabitra Pal Choudhury
    December 30, 2023
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(to be inserted by publisher)



A notion of dimension preservative map, Integral Value Transformations (IVTs) is defined over ℕ^k using the set of p-adic functions. Thereafter, two dimensional Integral Value Transformations (IVTs) is systematically analyzed over ℕ×ℕ using pair of two variable Boolean functions. The dynamics of IVTs over ℕ×ℕ=ℕ^2 is studied from algebraic perspective. It is seen that the dynamics of the IVTs solely depends on the dynamics (state transition diagram) of the pair of two variable Boolean functions. A set of sixteen Collatz-like IVTs are identified in two dimensions. Also, the dynamical system of IVTs having attractor with one, two, three and four cycles are studied. Additionally, some quantitative information of Integral Value Transformations (IVTs) in different bases and dimensions are also discussed. 







§ INTRODUCTION

Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line <cit.>. There are plenty of works carried out over last couple of decades in discrete dynamical systems as well as in arithmetic dynamics <cit.>. What is not very well studied yet is the discrete dynamics over spaces like natural numbers. In 2010, the notion of Integral Value Transformations (IVTs) which is a new paradigm in arithmetic dynamics <cit.> has been introduced. Later in 2014, a special class of one dimensional Integral Value Transformations (called Collatz-like) is studied in <cit.>. A couple of other works on IVTs and their applications in arithmetic logical circuit design and in Biology are also studied <cit.>. Applications of such transformations are noteworthy in the area of Cryptology and other field of discrete mathematics <cit.>.


In this present article, an attempt has been initiated to understand the discrete dynamics of Collatz-like IVTs including other two dimensional Integral Value Transformations. It is seen that the dynamics of the IVTs are depending on the state transition diagram of two dimensional IVTs of the pair of two variable Boolean functions <cit.>. There are four types of dynamics seen with regards to attractors with different cycle length(s) 1, 2, 3 and 4. All these dynamical systems end with attractors (global and/or local) with a maximum of four length cycles. In addition, from the algebraic aspects the dynamical system of IVTs are investigated. Discrete dynamics is itself very appealing from its application viewpoint. The present study of the dynamical systems over two dimensions can be as pathways for network designing converging to the desired attractor(s).       

The rest of paper is organized as follows. In Section 2, the definitions of IVT in higher dimensions and in particular in two dimensions over the set of natural numbers with zero are given. The various algebraic properties of the dynamical system of IVTs are described in this section. The Section 3 deals with the state transition diagram for the IVTs of various algebraic classes. Some important theorems including all possible type of attractors are presented in this section. The Section 4 deals with the dynamics of IVTs using their profound trajectories. The Section 4 also deals with the classifications of 256 possible IVTs in two dimensions based on their dynamics over ℕ×ℕ. The quantitative measure for the various classes in different bases and dimensions are also incorporated in this section. This Section 4 also includes the dynamics of some special class of IVTs. The Section 5 concludes the key findings of current article and future research endeavors.




§ DEFINITIONS & BASIC ALGEBRAIC STRUCTURES IN 𝔹_2 AND 𝕋



 §.§ Definitions and basic algebraic structures


 
Integral Value Transformation in the base p and dimension k (IVT^p, k_i_1,i_2,⋯ ,i_k) over ℕ^k (ℕ is set of all natural numbers including zero) is defined as 

    IVT^p,k_i_1,i_2,⋯ ,i_k(m_1,m_2,⋯, m_k)=(IVT^p,k_i_1 (m_1,m_2,⋯, m_k), IVT^p,k_i_2 (m_1,m_2,⋯, m_k),⋯,
     IVT^p,k_i_k (m_1,m_2,⋯, m_k))

	
Here (m_1,m_2,⋯, m_k) is an element of ℕ^k. The function IVT^p,k_i is defined using f_i in <cit.>. f_i is a p-adic function. 
 



 
Integral Value Transformation in two dimension (k=2) with base (p=2): (IVT^2,2_i,j) over ℕ×ℕ is defined as 
	
    IVT^2,2_i,j(m,n)=(IVT^2,2_i (m,n), IVT^2,2_j (m,n))

	
Here (m,n) is an element of ℕ×ℕ and the function f_i or f_j is the 2-variable Boolean functions and pair-wise in combining they are represented as f_i,j which is formally defined in the Definition 3.


 



The map f_i,j is defined on the set 𝔹_2={00,01,10,11} as 

    f_i,j(x,y)=(f_i(x,y), f_j(x,y))


Here (x,y) belongs to 𝔹_2.
In the rest of article, we shall use IVT_i,j instead of IVT^2,2_i,j for notational simplicity.


Here we define a discrete dynamical system over ℕ×ℕ as 
    x_n+1=IVT_i,j(x_n)
 where x_n ∈ℕ×ℕ.  


	For every x_0 ∈ℕ×ℕ, if trajectory of the discrete dynamical system x_n+1=IVT_i,j(x_n) eventually converges to a point (0,0) ∈ℕ×ℕ, then we call such dynamical system x_n+1=IVT_i,j(x_n) as Collatz-like IVT. 


Let 𝕊 be the set of all possible two variable Boolean functions. Clearly, there are exactly 16 such functions. 


Let 𝕋 be the set of all possible IVT_i,j defined over the set ℕ×ℕ. Clearly, there are exactly 256 such IVTs. Our aim is to comprehend the dynamics of these maps. 

The following properties and remarks in two dimensional integral value transformation from algebraic perspective are given.



  * Properties: 


  * The space (𝔹_2, +_2, *_2) forms a vector space over 𝔽_2={0, 1}. Here +_2 and *_2 are modulo 2 addition and scalar multiplication. The dimension of the vector space is 2.


  * The basis of the vector space (𝔹_2, +_2, *_2) consists of two elements (0,1) and (1,0). The dimension of the vector space is 2.



  * The space (𝕊, ⊕_2, ⊗_2) forms a vector space over 𝔽_2={0, 1}. Here ⊕_2, and ⊗_2 are modulo 2 point wise addition and scalar point-wise multiplication. 


  * The basis of the vector space (𝕊, ⊕_2, ⊗_2) consists of 4 elements f_1, f_2, f_4, f_8. The dimension of the vector space is 4.


  * The space (𝕋, ⊕_2, ⊗_2) forms a vector space over 𝔽_2={0, 1}. Here ⊕_2, and ⊗_2 are modulo 2 function addition and scalar function multiplication.


  * The basis of the vector space (𝕋, ⊕_2, ⊗_2) consists of 16 elements IVT_1,1, IVT_1,2, IVT_1,4, IVT_1,8, IVT_2,1, IVT_2,2, IVT_2,4, IVT_2,8, IVT_4,1, IVT_4,2, IVT_4,4, IVT_4,8, IVT_8,1, IVT_8,2, IVT_8,4, IVT_8,8. The dimension of the vector space is 16.


  * There are 4 linear Boolean functions viz. f_0, f_6, f_10 and f_12 in 𝕊. 


  * There are 16 linear maps in 𝕋 and they are IVT_0,0, IVT_0,6, IVT_0,10, IVT_0,12, IVT_6,0, IVT_6,6, IVT_6,10, IVT_6,12, IVT_10,0, IVT_10,6, IVT_10,10, IVT_10,12, IVT_12,0, IVT_12,6, IVT_12,10 and IVT_12,12. 


  * Sum and Composition to two linear maps in 𝕋 is linear.


  * There are 24 bijective f_i,j in 𝕊. The twelve of the bijective functions are f_3,5, f_3,6, f_3,9, f_3,10, f_5,6, f_5,9, f_5,12, f_6,10, f_6,12, f_9,10, f_9,12, and f_10,12.


  * There are only 6 isomorphisms (linear and bijective) exist in 𝕋. They are f_6,10, f_6,12,, f_10,6, f_10,12, f_12,6, and f_12,10.      




  * Remarks:


  * IVT_i,j∈𝕋 is a basis element of 𝕋 if and only if f_i and f_j are also basis elements in 𝕊.


  * IVT_i,j∈𝕋 is linear if and only if f_i and f_j are also linear.


  * If f_i,j is linear then f_j,i is also linear.


  * None of the basis function in 𝕋 is linear.


  * If f_i,j is bijective then f_j,i is also bijective.





 §.§ IVTs through STDs

In graph theory, a directed graph G is defined as G(V,E,F); where F is a function such that every edge in E an oder pair of vertices in V <cit.>. In our context, we have designed the state transition diagram (STD) of f_i,j as directed graph. 

Illustration: For an example, consider the domain of 2-variable Boolean functions which have four unique possibilities i.e. 00, 01, 10 and 11 for its constituent variables as shown in Table 1. 




From the truth table representation of 2-variable Boolean function f_6,13, we find that (0,0) → (0,1), (0,1)→ (1,0), (1,0)→ (1,1) and (1,1)→ (0,1) are the required transitions. Similarly, the transitions are shown for the function f_0,12. Based on these four transitions a directed graph called STD (G=(V,E)) can be drawn where the vertices or nodes in the graph are four possible bit pairs i.e. V= 𝔹_2 and the directed edges (E) are the transitions or mapping from the set 𝔹_2 to 𝔹_2. So, STD identifies the local transformations for all possible combinations of domain set to the rules, given the truth table of one or more rules of n-variable. The STDs of function f_6,13 and f_0,12 are shown in Fig. 1 and it is noted that there will be always four nodes each node with an out degree edge/transition, but the in degree edge for a particular node can vary from zero to maximum of four. From Fig. 1 (f_6,13), the in-degree of node (0,1) is 2, (1,1) is 1, (1,0) is 1, and (0,0) is 0. Total in-degrees are the summand of all in-degrees of four nodes that must be equal to 4. 



The STDs of linear functions are given in Fig. 2. The STDs of only linear IVT_i,j are given, we omit other STDs of linear f_j,i. The STDs of basis functions are given in Fig. 3. The STDs of only basis IVT_i,j are given, we omit other STDs of basis f_j,i. The STDs of bijective functions are given in Fig. 4. The STDs of only bijective IVT_i,j are given, we skip STDs of other bijective f_j,i. The STDs of isomorphisms are given in Fig. 5. The STDs of only IVT_i,j which are isomorphic are given, we skip other STDs of isomorphism f_j,i.












§ DYNAMICS OF IVT_I,J USING STATE TRANSITION DIAGRAM OF F_I,J



 §.§ Dynamics of STDs when p=2, k=2

For i, j ∈{0, 1,…,15} in two dimensions, there are 2^4× 2^4=256 functions f_i,j. From the STDs of those 256 functions f_i,j, their dynamics are pretty clear. There are specifically two kinds of dynamics possible; one is attracting to a single attractor and other kind is periodic attractor of period (pr), 2 ≤ pr ≤ 4. For examples, the single attractor function is f_0,12 and period 3 attractor function is f_6,13 (Fig. 1). 


The dynamical system x_n+1=IVT_i,j(x_n) is Collatz-like if and only if STD of f^k_i,j((x,y))=(0,0) for all (x,y) ∈𝔹_2, for some k ∈ℕ. Here f^k_i,j denotes the k-composition of the function defined as f^k_i,j=f_i,j(f^k-1_i,j). 
 



There are exactly 16 Collatz-like x_n+1=IVT_i,j(x_n) in 𝕋.



As per definition of Collatz-like function, we are looking for functions f_i,j such that for some k, 
    f^k_i,j((x,y))=(0,0)
 for all (x,y) ∈𝔹_2. We will check and count through the possible variety of STDs structure of f_i,j.

Suppose f_i,j has the required property and let G_f_i,j be the directed graph (STD) with four vertices- a is (0,0), b is (0,1), c is (1,0), d is (1,1) and an edge (x,y) → (m,n) whenever f_i,j(x,y) = (m,n).  Each vertex of G_f_i,j has out degree 1. We make the following claims, all of which are simple.

1. G_f_i,j must have at least one cycle, say C.

2. G_f_i,j must be connected, and it has only one component.

3. The cycle C is the only cycle in G_f_i,j, since every vertex must be in the same component as (0,0).  

4. The cycle C is a sink: for each (x,y), f^k_i,j(x,y) is in C for all sufficiently large k and (0,0) ∈ C.

5. If (0,0)  f^k_i,j((0,0)), then f^k_i,j(a) f^k_i,j(f((0,0))) for every k. And no other vertex is in C.

	
So we have (0,0) = f_i,j((0,0)). Therefore, we could have only 4 topologies (STDs) for G_f_i,j as shown below. The vertex a is fixed and by interchanging b, c and d in different levels/positions we can easily enumerate how many elements that have f^k_i,j((x,y)) = (0,0).




 
Topology:      
[
> = stealth, 
shorten > = .4pt, 
auto,
node distance = 1cm, 
semithick 
]	
every state=[
draw = black,
thick,
fill = white,
minimum size = 1mm
]
[state] (1) a;
[state] (2) [below of=1] c;
[state] (3) [left of=2] b;
[state] (4) [right of=2] d;

[->] (1) edge [loop above] node  (1);
[->] (2) edge node  (1);
[->] (3) edge node  (1);
[->] (4) edge node  (1);

   
[
> = stealth, 
shorten > = .5pt, 
auto,
node distance = 1cm, 
semithick 
]	
every state=[
draw = black,
thick,
fill = white,
minimum size = 2mm
]
[state] (1) a;
[state] (2) [below left of=1] b;
[state] (3) [below right of=1] c;
[state] (4) [below of=2] d;

[->] (1) edge [loop above] node  (1);
[->] (2) edge node  (1);
[->] (3) edge node  (1);
[->] (4) edge node  (2);
 
    
[
> = stealth, 
shorten > = .5pt, 
auto,
node distance = 1cm, 
semithick 
]	
every state=[
draw = black,
thick,
fill = white,
minimum size = 2mm
]
[state] (1) a;
[state] (2) [below of=1] b;
[state] (3) [below left of=2] c;
[state] (4) [below right of=2] d;

[->] (1) edge [loop above] node  (1);
[->] (2) edge node  (1);
[->] (3) edge node  (2);
[->] (4) edge node  (2);
 
   
[
> = stealth, 
shorten > = .3pt, 
auto,
node distance = 1cm, 
semithick 
]	
every state=[
draw = black,
thick,
fill = white,
minimum size = 1mm
]
[state] (1) a;
[state] (2) [below of=1] b;
[state] (3) [below of=2] c;
[state] (4) [below of=3] d;

[->] (1) edge [loop above] node  (1);
[->] (2) edge node  (1);
[->] (3) edge node  (2);
[->] (4) edge node  (3);
 
 Number of functions:     1     6     3     6 
 Functions (f_i,j:(i,j)):     (0,0)      (0,4), (0,8), (2,0), (2,2), (4,4), (8,0)     (0,12), (6,6), (10,0)     (2,6), (2,8), (4,12), (6,4), (8,4), (10,2) 
 


Hence, the total number of Collatz-like functions in two dimensions is 16.



It is noted that the attractor (0,0) is a global attractor for the Collatz-like IVTs, that is for every initial value taken from ℕ×ℕ, the dynamical system takes all the trajectories to (0,0).   



	The dynamical system x_n+1=IVT_i,j(x_n) possesses multiple attractor(s) of the form (2^s-1,0), s∈ℕ if and only if f^k_i,j((x,y))=(1,0), for all (x,y) ∈𝔹_2, for some k ∈ℕ. 
 



	There are countably infinite number of local attractors of the form (2^s-1,0), s∈ℕ. 



	For any x_0 ∈ℕ×ℕ, the dynamical system eventually proceeds to a point of the form (2^s-1,0), according to definition of the dynamical system map IVT_i,j. It is obvious that the set of all possible points of the form (2^s-1,0), is a subset of the ℕ×ℕ. Since the set ℕ×ℕ is countable and so the subset is countable. Hence the theorem is followed.   




	The dynamical system x_n+1=IVT_i,j(x_n) possesses multiple attractor of the form (0, 2^s-1) if and only if f^k_i,j((x,y))=(0,1), for all (x,y) ∈𝔹_2, s∈ℕ for some k ∈ℕ.
 


	There are countably infinite number of local attractors of the form (0,2^s-1), s∈ℕ for the dynamical system x_n+1=IVT_i,j(x_n). 




	The dynamical system x_n+1=IVT_i,j(x_n) possesses multiple attractors of the form (2^s-1,2^t-1), s, t∈ℕ if and only if f^k_i,j=(1,1) for some k ∈ℕ.
 


	There are countably infinite number of local attractors of the form (2^s-1,2^t-1) s, t∈ℕ for the dynamical system x_n+1=IVT_i,j(x_n). 



It is noted that there are STDs of some function f_i,j such that there are periodic attractor of period p ≥ 2. 
 In such cases, the dynamical system x_n+1=IVT_i,j(x_n) will possess to multiple attractors of form (u,2^t-1), (2^t-1,u), (0,u), (u,0) and  (u,v). There are few dynamical systems which are sensitive to the initial conditions. In the following section, we shall classify all such types of dynamical systems based on their dynamical behavior.



 §.§ STD dynamics when p≥ 2, k≥3

In p-adic and k-variable functions (or k-dimension), we have p^p^k number of functions f_i where i∈{0,1,⋯,2^k-1}. Depending on k value, we select the number of functions in k dimensions and in combining we represent as f_i_1,i_2,⋯ ,i_k and length of each function is p^k.


In general, for any i, j ∈{0, 1,…2^k-1} in k dimensions, there are p^p^k× p^p^k=p^2p^k functions f_i_1,i_2,⋯ ,i_k. In that case, we could have the attractor of period (pr), 1 ≤ pr ≤ p^k. The number of functions is exponentially increasing with the increment of p and k. In principle, the discrete dynamics of such IVTs can also be obtained using STDs as we did it in two dimensions (p=2,k=2). The vertex of the STD in p and k dimension is (a_1,a_2,⋯ a_k); where a_i∈{0,1,⋯ p^k}. For an example, the STDs of two Collatz-like functions f_128,72,160 (p=2,k=3) and f_2187,12690 (p=3,k=2) and a STD of function f_41,19230 (p=3,k=2) with attractors 5 and 2 is shown in Fig. 6.






The number of global attractors for different p and k value is shown in Table 2.



Based on the computational evidence, the total number of Collatz-like IVTs in the base p with k dimension is conjectured.


The total number of Collatz-like IVTs in the base p with k dimension is p^k(p^k-2).






If l_k defines the path length from a particular point to the global attractor in the dynamics of IVT^p,k_i,j, then 1≤ l_k≤ (p^k-1).




§ DYNAMICS OF X_N+1=IVT_I,J(X_N)

Before getting into the detailed classifications of dynamics, a vivid example of a dynamical system x_n+1=IVT_i,j(x_n), x_n ∈ℕ×ℕ is adumbrated with different initial positive natural integers and their trajectories as shown in the Fig. 7. For an example, IVT_13,3(0,2)=(1,3) i.e. (0,2)→(1,3). The trajectories (path) can be changed with the changes of p value even if the taken function is fixed.





 §.§ Classifications of IVT_i,j based on their dynamics when p=2, k=2

In this section, we have classified the 256 of IVT_i,j into four disjoint classes (Class I-IV) according to the attractors of four different cycle length(s). For every class, several subclasses are identified according to the dynamic behavior of IVT_i,j over ℕ×ℕ. It can be seen that there are many (x,y) components in the dynamics of  IVT_i,j  where (x,y)∈ℕ×ℕ. For the simplicity of graph visualization, we have taken certain limit for both x and y i.e. x,y∈{0,1⋯ 7}. For some of the IVTs, all the components are connected (e.g. all Collatz-like IVTs) and other cases there could be of different variety.  For understanding the complete dynamics over the set of positive natural numbers in base p and dimension k, x,y∈{0,1,⋯, p^k-1} that covering all possible transitions in their p-adic representations. Here, p^k-1 is the minimum value that can be extended to desire limit.



  §.§.§ Class-I: Dynamics of IVT_i,j: attractor with a fixed point

A total of 125 IVT_i,j are classified into the Class-I as shown in Table 3. It is noted that each of the IVTs in the Class-I is having attractor with one length cycle (singleton attractor). There are 16 IVTs which are Collatz-like and rest are non-Collatz-like. For the non-Collatz-like dynamical systems, there are countable many local attractors with one cycle.



A dynamical system  IVT_10,0 having an attractor with one length cycle is given in the Fig. 8.



There are other 16 out of 125 which are sensitive to the initial conditions that is their trajectories are depending on initial values. Six possible IVT_i,j are taken and for all the cases five integer pairs trajectories are shown in Fig. 9. It is noted that each of these initial condition (taken pair) provides sensitive dynamical system ending with different cyclic attractors of length one. 






  §.§.§ Class-II: Dynamics of IVT_i,j: attractor with two length cycles


A total of 93 IVT_i,j are classified into the Class-II as shown in Table 4. All these IVTs in the Class-II are having attractors with two length cycles. Some of these dynamical systems are having global attractor of two length cycles. There are 42 out of 93 which are sensitive to the initial conditions that is their trajectories are depending on initial values. Thus, each of these initial conditions provides sensitive dynamical system ending with two length cycle attractor. 



A dynamical system  IVT_4,5 having an attractor with two length cycle (global attractor) is given in the Fig. 10. 





  §.§.§ Class-III: Dynamics of IVT_i,j: attractor with three length cycles


A total of 32 IVT_i,j are classified into the Class-III as shown in Table 5. All these IVTs in the Class-III are having attractor with three length cycles. There are some dynamical systems with global three length cycle attractors. There are 8 out of 32 which are sensitive to the initial conditions that is their trajectories are depending on initial values. Each of these initial conditions provides sensitive dynamical systems ending three length cycle attractor.



A dynamical system  IVT_9,1 having an attractor with three length cycle is shown in the Fig. 11.





  §.§.§ Class-IV: Dynamics of IVT_i,j: attractor with four length cycles


A total of 6 IVT_i,j are classified into the Class-IV as shown in Table 6. All these IVTs in the Class-IV are having attractors with four length cycles. These dynamical systems do possess global attractors with four length cycles. 



A dynamical system  IVT_9,5 having an attractor with four length cycle is shown in the Fig. 12.





 §.§ Classifications of IVT^p,k_i,j based on their dynamics when p≥2, k≥ 3

In case of p=2, k=2, the cycle length of an attractor observed in the dynamics of IVT_i,j does not exceed 4 which is the maximum cycle length of an STD of f_i,j. It is quite impossible to observe the dynamics of individual IVT^p,k_i,j when p and k are sufficiently large. From the dynamics of STDs of some f^p,k_i,j, one can classify the corresponding IVT^p,k_i,j based on their dynamics ends up with cycle length exclusively 1,2,…, p^k (Fig. 6) into p^k number of classes. The cardinality of different classes based on dynamics of IVT^p,k_i,j having attractors of different cycle lengths (upto p^k) including an exclusive outer class are presented for different p and k in Table 7. The exclusive outer class contains IVT^p,k_i,j such that the length of the cycle of the attractors exceed the maximum cycle length (p^k) of the STD of f^p,k_i,j. Such a discrete dynamical systems is called here as chaotic dynamical system. 
 
An example of chaotic dynamical system of IVT^3,2_41,19230 using the function f_41,19230 (p=3, k=2) is shown in Fig. 13, whose STD is given in Fig. 6 (C). From the STD of f_41,19230, we could see two types of cycles (2 and 5) is present. But, the dynamical system shows an additional cycle of length 10 (l.c.m of 2 and 5). The reason can be easily seen from the ternary representation (here, p=3) of any pair (x,y)∈ℕ×ℕ. Take a pair (5,6) from the cycle of length 10 from Fig. 13. The ternary representation of (5,6) is (12,20). The pair wise component are 12 (first component from left) and 20 (second component from left). These two components are from two different cycles 2 and 5 respectively from STD of f_41,19230 (Fig. 6(C)). All the STD components are distinct. So, while calculating the IVT one by one iteration, the components 12 and 20 will come back/repeat to the same initial stage after  l.c.m (5,2) iterations or cycle of length l.c.m (5,2). This is true for arbitrary pair in any base p and also true if we increase the dimension k.







Number of bijective functions (f^p,k_i_1,i_2,⋯ i_k) in p-adic, k dimension is (p^k-1)!.





 §.§ Dynamics of two dimensional IVT_i,j from algebraic perspective


Here we go with classification of two dimensional IVT_i,j from their algebraic perspectives such as basis, linearity, bijective and isomorphism which are shown in Table 8. 



It is to be noted that the dynamics of the IVT_i,j is not always same as the dynamics of IVT_j,i although they belong to same algebraic class as previously pointed out. 




§ CONCLUSIONS AND FUTURE ENDEAVORS

In this manuscript, notion of dimension preservative Integral Value Transformations (IVTs) is defined over ℕ^k using the set of p-adic functions. The main focus is given in order to understand the two dimensional integral value transformation (p=2 and k=2, which are computationally feasible) and studied the dynamics of all possible pair functions in four different kinds of attractors. Among them, a special class of IVTs (Collatz-like) is computationally identified along with some others global and local dynamical systems. In addition, from the algebraic properties of IVT_i,j, the dynamics are critically observed. From the current analysis, it is evident that as p and k go high, the total numbers of Collatz like IVTs are increasing very fast. Consequently, the dynamical behavior of these IVTs is computationally difficult to simulate and apprehend. But, some of their dynamics can be vivid from our current analysis in two dimensions and some other cases. In this concern, we have shown some quantitative informations for other two special cases of IVTs (p=2,k=3 and p=3, k=2). Further, the analysis also clearly evident that as p and k go high, we come across with some chaotic dynamical behavior. In future endeavor, one can explore with some higher p and k considering the limitation of simulation.




 §.§ Acknowledgment

We thank to anonymous reviewers for their constructive suggestions and valuable directions that help to improve the article quality.

ws-ijbc

