 Advances in Pure Mathematics, 2013, 3, 292-296 http://dx.doi.org/10.4236/apm.2013.32041 Published Online March 2013 (http://www.scirp.org/journal/apm) Topological Dynamics in Tandem with Permutation Groups Isaac Kwame Dontwi, William Obeng-Denteh, Stephen K. Manu, Richard Nyarko Yeboah Department of Mathematics, College of Science, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Email: ikedontwi@hotmail.com, obengdentehw@yahoo.com, deoxy2005@yahoo.com, aricky2008@yahoo.com Received September 14, 2012; revised November 16, 2012; accepted December 3, 2012 ABSTRACT The purpose of this study was to delve into the aspects of abstract algebra that has a link with topological dynamics in terms of permutation and symmetric groups. This would aid users to appreciate the role it plays in the theory and appli-cation of topological dynamics. The usage of matlab programming to carry out the permutations was carried out. The study contributes to the literature by providing candid explanation and usage of data-based evidence documenting the extent to which topological dynamics operates. Keywords: Permutation Groups; Recurrence; Topological Dynamics; Matlab Code; Flows 1. Introduction Topological dynamics in  and  form an aspect of the theory of dynamical systems in  where qualitative , asymptotic  properties [6-9] of dynamical systems are taken into account from the viewpoint of general to-pology in . Reference  provides the first self-con- tained comprehensive exposition of the theory of dyna- mical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. Topological dynamics is defined as the study of asymp-totic [11-14] or long term properties of families of maps in  of topological spaces. Abstract topological dy-namics is usually developed in the context of flows as in . A flow ,XTT is a jointly continuous action as in  of the topological group  on the topological space X . This means that there is a continuous map  from ,,XTX tex xx tx with  and which clearly indicates that e is the identity of and ,s tsx,,Ttstx TX. A topological dynami-cal system is the central point in the study of topological dynamics. This constitutes a topological space, in tandem with a continuous transformation, a continuous flow, in other words a semigroup of continuous transformations of that space. This state of affairs brings the activity of permutation groups [21-23] and semigroups action  into the scenario. Recorded studies of permutations of the earliest peri-ods can be found in Sefer Yetsirah or Book of Creation which was written by an unknown Jewish author some-time before the eighth century . This was improved upon with the passage of time. 2. Problem Statement and the Way Forward The main problem that was considered was to identify the link between topological dynamics and permutation groups. The usage of permutation groups was used syn-onymously with Cartesian product. It became necessary to decipher the link based on the underlying factors enu-merated stemming from the laid down rules from credi-ble sources . A periodic point returns to itself every hour on the hour; but almost periodic point returns to a neighbourhood every hour within the hour. This is what has been established to lend credence to topological dy-namics in tandem with permutation groups. 3. Preliminary Notations and Definitions A topological transformation group, or transformation group is explained to be an ordered triple ,,πXTπ such that the following conditions are satisfied: 1) X is a topological space which is called the phase space: T is called the topological group known as the phase space and is a map of the cartesian product XTX, called the phase map;  into x2) for allex xXπ where e denotes the identity element of T, and the value of at the point ,ofxtXTx be represented by tπ; 3) is continuous. In analyzing a transformation group ,,πXT the phase map ,xtxt gives the values of two kinds of Copyright © 2013 SciRes. APM I. K. DONTWI ET AL. 293maps when one of the variables x, t is replaced by a con-stant. They are transition and motion in . Depending on additive groups of real numbers or integers a con-tinuous flow or discrete flow arises. These two kinds of flows are related in that a continuous flow determines many discrete flows by taking cyclic subgroups of R. Conversely, a discrete flow determines a continuous flow when the phase space X is extended to the Cartesian product of X and the closed interval. The crux of the work here cruises on recurrence. Sup-pose is a homeomorphism of X onto X, a point x of the phase space is said be almost periodic under  and is said to be almost periodic at x provided that if U is a neighbourhood of x, then there exists a relatively dense subset A of additive group of integers such that nxUnASD for all . A periodic point returns to itself every hour on the hour; but almost periodic point returns to a neighbourhood every hour within the hour. This is what has been established to lend credence to topological dy-namics in tandem with permutation groups. 4. Methods A permutation group is a group of permutations on a finite set of positive integers. Permutation groups are therefore all operations for groups which can be applied to them. Here permutation multiplications are carried out on permutations of the main set. 3 is seen as a group 3 of symmetries of an equilateral triangle in which two copies of it with vertices 1, 2, and 3 can be placed with one covering the other with vertices on top of verti-ces in . The subscripted Greek letters i and i are for rotations and mirror images in bisectors of angles. The second one is the 4 which is the group of sym-metries of the square. The subscripted Greek letters iiD, and i are for rotations, mirror images in per-pendiculars bisectors of sides, and for diagonal flips re-spectively. Matlab was then used to develop codes for the permutations. 5. Results and Discussion Two important examples would be explored to unearth the surprising revelations in the processes. See Figure 1 for the operations of the matlab code. Considering the group 3with elements where S3! 61, 2,3.A2312331212321300 001 102 2123123 123123123 123123 123123123 231231123 123123123 312312 The permutations are listed with names using Greek let-ters assigned to them viz.; 3S0112123 123,,123 231123 123,,132 321 Now, The detailed process of the permutations can be found in . The multiplication table for is depicted in Table 1. The next is to form the dihedral-group 4 of permu- tations to the ways that two copies of a square with ver- tices 1-4 can be placed with one covering the other with vertices on top of vertices. is the group of symme- tries of the square. D4D012312121234 1234,,1234 23411234 1234,3412 41231234 1234,,2143 43211234 1234,3214 1432   0000110221234 12341234 12341234 12341234 12341234 23411234 23411234 12341234 34121234 3412   SThe following representations are used; Now The multiplication table for is shown in Table 2. 4 Theorem: (Cayley’s Theorem) every group is isor- morphic to a group of permutations. An important addi- ion is that in group theory, Cayley’s theorem which was t Copyright © 2013 SciRes. APM I. K. DONTWI ET AL. Copyright © 2013 SciRes. APM 294 Figure 1. Pou1 has been depic ted. Table 1. The multiplication table for 3S. 0 1 2 1 2 3 0 0 1 2 1 2 3 1 1 2 0 3 1 2 2 2 0 1 2 3 1 1 1 2 3 0 1 2 2 2 3 1 2 0 1 3 3 1 2 1 2 0 STable 2. The multiplication table for 4. 0 1 2 3 1 2 1 2 0 0 1 2 3 1 2 1 2 1 1 2 3 0 1 2 2 1 2 2 3 0 1 2 1 2 1 3 3 0 1 2 2 1 1 2 1 1 2 2 1 0 2 3 1 2 2 1 1 2 2 0 1 3 1 1 1 2 2 1 3 0 2 2 2 2 1 1 3 1 2 0 I. K. DONTWI ET AL. 295 named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G in . In fact this can be explained as an instance of the group action of G on the elements of G. A permutation of a set G is any objective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and this denoted by Sym(G). Cayley’s theorem outlines all groups on the same ped-estal, by taking into account any group which might be made up of infinite groups such as (R,+) as a permutation group of some underlying set. This establishes the fact that theorems which holds for permutation groups are counted to hold for groups in general. The group being used here satisfies the axioms for topological group. In topological dynamics it involves topological groups and their operations [2,27]. 6. Using Matlab to Compute the Results above Instead of Manual Approach Algorithm In general this work has been trying to establish a link between topological dynamics and permutation groups which has been likened to Cartesian product. Matlab has been used to generate the permutations explained above. See Figure 1 for an example of the usage of the matlab code. The code could be extended to cover Sn. 7. Conclusions The crux of the work here cruised on recurrence in . A periodic point returns to itself every hour on the hour; but almost periodic point returns to a neighbourhood every hour within the hour. The recurrence was depicted in the permutations that were done. All the parameters in the form of the Greek symbols recurred in the table pro- vided. When this trend is continued for all time the re- currence nature would be mimicked along the same trend. This is what has been established to lend credence to topological dynamics in tandem with permutation groups. It is worthy to note that from all the tables each col- umn gives a permutation of the group set. It is obvious that at least every finite group is isomorphic to a sub- group of the group. Isomorphism is about one-to-one correspondence satisfying a particular relation. This is the way topological dynamics apply permutation in find-ing the symmetries of objects and their rotations as well and matlab has been useful over here. REFERENCES  E. Glasner and B. Weiss, “On the Interplay between Measurable and Topological Dynamics,” Most, Vol. 1, 2004, pp. 1-47.  A. Katok and B. 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