Coincidence Point Sets in Digital Topology
aa r X i v : . [ m a t h . GN ] S e p COINCIDENCE POINT SETS IN DIGITAL TOPOLOGY
MUHAMMAD SIRAJO ABDULLAHI, POOM KUMAM, AND JAMILU ABUBAKAR
Abstract.
In this article, we investigate some properties of the coincidencepoint set of digitally continuous maps. Following the Rosenfeld graphicalmodel which seems more combinatorial than topological, we expect to achieveresults that might not be analogous to the classical topological fixed pointtheory. We also introduce and study some topological invariants related tothe coincidence and common fixed point sets for continuous maps on a digitalimage. Moreover, we study how these coincidence point sets are affected byrigidity and deformation retraction. Lastly, we present briefly a concept ofdivergence degree of a point in a digital image. Introduction
Topology is a branch of Mathematics that studies the relationship betweenspaces, especially equivalence between them under continuous mappings. It pro-vides a lot of ease to many applications by reducing cost of computation throughproviding theoretical foundations and methods more efficient than the non topo-logical ones. Fixed point theory in particular, plays an important and fundamentalrole in numerous areas of mathematics including functional and mathematical anal-ysis, pure and applied topology, fuzzy theory etc. It has always provided us witha major theoretical tool in fields as widely as differential equations, topology, eco-nomics, game theory, dynamics, optimal control and functional analysis which leadsto various and important applications in mathematics and applied sciences.In metric spaces, this theory begins with the Banach fixed point theorem [5] (alsoknown as the Banach contraction principle), which guarantees the existence anduniqueness of a fixed point of a certain map f : X −→ X of a complete metric space X , it additionally provides a constructive method of finding such a fixed point ofthe map f . For this direction (see. [1, 2, 4, 31, 35]).Topologically, the tools of fixed point theory are: the Lefschetz number, fixedpoint index, Nielsen number and the topological degree (for root problems). Inclassical topology, the value of M F ( f ) (i.e. the minimum number of fixed pointsin the homotopy class of f ) is generally hard to compute. The Lefschetz number L ( f ) and the Nielsen number N ( f ) (homotopy invariant lower bound for M F ( f ))are often used to obtain M F ( f ), where the former is homological in nature andgives a very rough indication of homotopy invariant fixed point information, whilethe later is more sophisticated and geometrical in nature. [21]. Mathematics Subject Classification.
Primary: 47H10, 54E35; Secondary: 68U10.
Key words and phrases.
Digital topology, Coincidence point set, Common fixed point set,Digital continuous maps, Fixed points, Retractions.The first and third authors were supported by the “Petchra Pra Jom Klao Ph.D. ResearchScholarship from King Mongkut’s University of Technology Thonburi”.
On the other hand, digital topology deals with the questions of how and towhat extent that topological concepts can meaningfully and usefully be applied toa binary image [24]. It is mainly concerned with studying mathematical propertiesof n -dimentional digital images [33]. This study was initiated in the early 1970s byAzriel Rosenfeld [32] (see also [33]) and Mylopoulos and Pavlidis [30]. It has sinceprovided the theoretical foundations for important image processing operationssuch as object counting, image thinning, image segmentation, boundary detection,contour filling, computer graphics and mathematical morphology etc (see. [6, 18,27]).A digital curve can be described as a sequence of digital points, or equivalentlyas a path of vertices on a graph [13, 20]. In general, we can define a digital sur-face based on direct adjacency and indirect adjacency [16]. The concept of digitalsurfaces was proposed by Artzy et. al. [3], where they defined it as the face ofsome solid object. In 1981, Morgenthaler and Rosenfeld gave a different definitionof digital surfaces [29]. They stated that a digital surface locally splits a neighbor-hood into two disconnected components. They also gave some classification results,which later Kong and Roscoe [25] investigated further and concluded that mostof those do not exist in terms of real world examples. This motivated Chen andZhang [13] to give another definition mainly for (6,26)-surfaces, called parallel-movebased surfaces. They also obtained and proved the digital surface classification the-orem [13] (see also [16]). This inspired Chen and Rong [14] to calculate the genusand homology groups of 3-dimensional digital objects with the help of the classicalGauss-Bonnet Theorem and the Alexander Duality respectively.An n -dimensional manifold is a topological space where each point has a neigh-borhood that is homeomorphic to an n -dimensional Euclidean space. In 1993,Chen and Zhang proposed a simple extension of digital surfaces to define a digital n -manifold [15]. Melin [28] also studied digital n -manifolds using Khalimsky topo-logical approach. Finding the orientability of digital manifolds is very significant intopology, as it is used to determine if a manifold contains a Mobius band. The dig-ital Mobius band was first discovered by Lee and Rosenfeld [23]. Afterwards, Chen[13] designed an algorithm for determining whether a digital surface is orientableor not.Until late 1980’s, all works in digital topology were based on a graph-theoreticapproach rather than topological, in which binary images are made into graphs byimposing adjacency relations on Z n . For 2-dimensional binary images, the mostfrequently used adjacency relation is the (8,4) adjacency relation. The major prob-lem of the graph-based approach to digital topology is that of determining whatadjacency relations on Z n might reasonably be used. One would normally wantto use adjacency relations such that fundamental topological properties of Z n havenatural analogues for the graphs obtained from binary images. In [26] Kong et al. addressed this problem for Z and Z . See [24] and references therein, for moredetails.In [11], the authors examined some properties of the fixed point set of a digitallycontinuous function. They believed that digital setting requires new methods thatare not analogous to those of classical topological fixed point theory, and henceobtained results that often differ greatly from standard results in classical topology.They introduced some topological invariants related to fixed points for continuousself-maps on digital images, and study their properties. Their main contribution is OINCIDENCE POINT SETS IN DIGITAL TOPOLOGY 3 the fixed point spectrum F ( X ) of a digital image. i.e. the set of all numbers thatcan appear as the number of fixed points for some continuous self-map.Motivated by the work of Boxer and Staecker in [11], and the fact that coinci-dence theory has been greatly influenced by fixed point theory, in this manuscriptwe will investigate some properties of the coincidence point set of digitally con-tinuous maps. We also introduce and study some topological invariants related tocoincidence point sets and common fixed point sets for continuous maps on a digitalimage. Moreover, we study how these coincidence point sets are affected by rigidityand deformation retraction. Further, since the Rosenfeld graphical approach weintend to follow, seems more combinatorial than topological, we similarly expect tooften achieve results that were not necessarily analogous to the classical topologicalcoincidence point theory.The organization of the paper is as follows: Section 1 houses an introduction tothis research direction. In Section 2, we reviewed some basic and background ma-terial needed for this study. We introduce coincidence point spectrum and presentsome of its properties with some examples in Section 3. In Section 4, we introducecommon fixed point spectrum, highlight some of its properties and present some il-lustrative examples. Section 5 studies how retractions interact with the coincidenceand common fixed point spectra. In Section 6, we introduce and study the diver-gence degree obtained from the complement of the coincidence point set. Finally,in Section 7 we state our concluding remarks.2. Preliminaries
Let N and Z denote the sets of natural numbers and integers respectively. Let usalso denote by X the number of elements (i.e. the cardinality) of a set X. Fromnow on, we denote by id and c , the identity map (i.e. id ( x ) = x for all x ∈ X ) andthe constant map (i.e. c ( x ) = x for all x ∈ X with x ∈ X fixed) respectively.Traditionally, a digital image is a pair ( X, κ ) , where X ⊂ Z n for some n ∈ N and κ is an adjacency relation on X , which is symmetric and antireflexive. Therefore,we may view a digital image ( X, κ ) as a graph for which X is the vertex set and κ determines the edge set. Usually, X is finite and the adjacency relation reflectssome type of “closeness” of the adjacent points in Z n . When these usual conditionshold, one may consider the digital image as a model of a black and white realworld digital image in which the black points (i.e. foreground) are representedby the members of X and the white points (i.e. background) by the members ofcomplement of X (i.e. Z n \ X ) [11].We write x ↔ κ y to indicate that x and y are κ -adjacent or x ↔ y whenever κ isunderstood or it is unnecessary to mention. Further, we use the notation x ⇔ κ y, to indicate that x and y are κ -adjacent or are equal and use x ⇔ y whenever κ isunderstood.In this paper, we will use the following type of adjacency. For t ∈ N with1 ≤ t ≤ n , any 2 (two) points p = ( p , p , . . . , p n ) and q = ( q , q , . . . , q n ) in Z n (with p = q ) are said to be κ ( t, n ) or κ -adjacent if at most t of their coordinatesdiffers by ± , and all others coincide. Note that, the number of points adjacentto any element of Z n which we represent by the κ ( t, n )-adjacency relation on Z n isdetermined by the number t ∈ N and can be obtain by the following formula, which MUHAMMAD SIRAJO ABDULLAHI, POOM KUMAM, AND JAMILU ABUBAKAR
Figure 1.
A 2-dimensional digital image with a 4-adjacency relation.appears in [18]: κ := κ ( t, n ) = t X i =1 i C ni , where C ni = n !( n − i )! i ! . Following the graph theoretic approach of studying n -dimensional digital images,we will use the notions of κ -adjacency relations on Z n and a digital κ -neighborhoodas have been extensively used in the literature. More precisely, using the κ -adjacency relations as defined above, we say that a digital κ -neighborhood of apoint p in Z n is the set defined and denoted as [33]: N κ ( p ) := { q | q ⇔ κ p } . Also, the following notation is often use to denote a kind of neighborhood, the socalled deleted digital κ -neighborhood of a point p in Z n [27]. N ∗ κ ( p ) := N κ ( p ) \ { p } . For a, b ∈ Z with a (cid:12) b, the set [ a, b ] Z = { n ∈ Z | a ≤ n ≤ b } with 2-adjacencyrelation is called a “digital interval” [7]. We say that two subsets ( A, κ ) and (
B, κ )of (
X, κ ) are “ κ -adjacent” to each other if A ∩ B = ∅ and there are points a ∈ A and b ∈ B such that a and b are κ -adjacent to each other. A set X ⊂ Z n is called“ κ -connected” if it is not a union of two disjoint non-empty sets that are not κ -adjacent to each other [19]. For a digital image ( X, κ ) the “ κ -component” of x ∈ X is defined to be the largest κ -connected subset of ( X, κ ) containing the point x . Definition 2.1. [34] Let (
X, κ ) and ( Y, κ ) be digital images. A function f : X −→ Y is ( κ , κ )-continuous, if for every κ -connected subset A of X, f ( A ) is a κ -connected subset of Y. The function f is called digitally continuous whenever κ and κ are understood.If ( X, κ ) = ( Y, κ ) (i.e. X = Y and κ = κ = κ ) we say that a function is κ -continuous to abbreviate ( κ, κ )-continuous. Theorem 2.2. [8]
A function f : X −→ Y between digital images ( X, κ ) and ( Y, κ ) is ( κ , κ ) -continuous if and only if for every x, y ∈ X, f ( x ) ⇔ κ f ( y ) whenever x ↔ κ y . Theorem 2.3. [8]
Let f : X −→ Y and g : Y −→ Z be continuous functionsbetween digital images ( X, κ ) , ( Y, κ ) and ( Z, κ ) . Then g ◦ f : ( X, κ ) −→ ( Z, κ ) is continuous. Definition 2.4. [22] A digital κ -path in a digital image ( X, κ ) is a (2 , κ )-continuousfunction γ : [0 , m ] Z −→ X . Further, γ is called a digital κ -loop if γ (0) = γ ( m ) , andthe point p = γ (0) is the base point of the loop γ. Moreover, γ is called a trivialloop if γ is a constant function. OINCIDENCE POINT SETS IN DIGITAL TOPOLOGY 5
Figure 2.
A 2-dimensional digital image with an 8-adjacency relation.For a digital image (
X, κ ) , we define C ( X, κ ) = { f : X −→ X | f is κ -continuous } . Recall that, a topological space X has the fixed point property (FPP, for short)if every continuous function f : X −→ X has a fixed point. A similar definitionhas appeared in digital topology as follows: Definition 2.5. [34] A digital image (
X, κ ) has the fixed point property (FPP) ifevery κ -continuous f : X −→ X has a fixed point.However, this property turns out to be very trivial, since the only digital imagewith the fixed point property (FPP) is a single point as was established in [9] asfollows: Theorem 2.6.
A digital image ( X, κ ) has the FPP if and only if X = 1 . Definition 2.7. [7] A function f : X −→ Y between digital images ( X, κ ) and( Y, κ ) is called an isomorphism if f is a digitally continuous bijection such that f − is digitally continuous. Definition 2.8. [8] Let (
X, κ ) and ( Y, κ ) be digital images. Suppose that f, g : X −→ Y are ( κ , κ )-continuous functions, there is a positive integer m and afunction H : X × [0 , m ] Z −→ Y such that:1. For all x ∈ X, H ( x,
0) = f ( x ) and H ( x, m ) = g ( x );2. For all x ∈ X, the induced function H x : [0 , m ] Z −→ Y defined by H x ( t ) = H ( x, t ) , for all t ∈ [0 , m ] Z is ( c , κ )-continuous. That is, H x ( t ) is a κ -path in Y ;3. For all t ∈ [0 , m ] Z , the induced function H t : X −→ Y defined by H t ( x ) = H ( x, t ) , for all x ∈ X is ( κ , κ )-continuous.Then H is a digital homotopy (or κ -homotopy) between f and g . Thus, the func-tions f and g are said to be digitally homotopic (or κ -homotopic) and denoted by f ≃ g. Note that if m = 1 , then f and g are said to be κ -homotopic in one step. Definition 2.9. [22] A continuous function f : X −→ Y is called digitally null-homotopic in Y if f is digitally homotopic to a constant function c . Moreover,a digital image ( X, κ ) is said to be digitally contractible (or κ -contractible) if itsidentity map id is digitally nullhomotopic. Definition 2.10. [11, 17] A function f : X −→ Y is called rigid if no continuousmap is homotopic to f except f itself. Moreover, when the identity map id : X −→ X is rigid, we say that X is rigid. MUHAMMAD SIRAJO ABDULLAHI, POOM KUMAM, AND JAMILU ABUBAKAR Coincidence Point Spectrum
In [10], the authors gave a brief treatment of homotopy-invariant fixed pointtheory. Following suit, we will now give a more general view of their treatment byextending it to a more general concept namely; the homotopy-invariant coincidencepoint theory. Let us begin, by respectively defining the quantities
M F ( f ) and XF ( f ) as the minimum number and maximum number of fixed points among allmaps homotopic to f .For a self-map f : X −→ X, we always have0 ≤ M F ( f ) ≤ XF ( f ) ≤ X. Any one of the above inequalities can be strict or equality depending on thesituation or conditions at hand.
Definition 3.1. [11] Let f : X −→ X be a mapping on X . Then(i) The homotopy fixed point spectrum of f is defined as: S ( f ) = { g ) | g ≃ f } ⊆ { , , . . . , X } ;(ii) The fixed point spectrum of X is defined as: F ( X ) = { f ) | f : X −→ X is continuous } . For simplicity, in the sequel we shall be using “continuous” instead of “ κ -continuous”.Let’s also denote a digital image ( X, κ ) as simply X, since we will not be referencingthe adjacency relation explicitly, and we will often refer to “digital images” as simply“images”. Moreover, from now on, we will consider the functions f , f : X −→ Y to be continuous maps between connected digital images X and Y , unless statedotherwise.Now, let’s consider the set C ( f , f ) , which we call the “coincidence point set”of the maps f and f .C ( f , f ) := { x ∈ X | f ( x ) = f ( x ) } . Whenever we deform f and f , the size and shape of C ( f , f ) may varygreatly. However, in topological coincidence theory we are not interested in anysuch inessential changes. We rather tend to capture only those features whichremain unchanged by arbitrary homotopies.In this paper, we are more concerned about the size of the set C ( f , f ), and onepossible tool to measure the set C ( f , f ) is “the minimum number of coincidencepoints” (i.e. M C ( f , f )) which we define as: M C ( f , f ) := min { C ( g , g ) | g ≃ f and g ≃ f } . Theorem 3.2.
Let
X, Y be isomorphic digital images and f , f : X −→ X becontinuous mappings. Then there are continuous mappings g , g : Y −→ Y suchthat C ( f , f ) = C ( g , g ) . Proof.
Let Φ : X −→ Y be an isomorphism and A = C ( f , f ) . Since Φ is one-to-one, A ) = A. Let g , g : Y −→ Y be defined by g i = Φ ◦ f i ◦ Φ − , for i = 1 , . OINCIDENCE POINT SETS IN DIGITAL TOPOLOGY 7
Now, for an arbitrary y ∈ Φ( A ) , let x = Φ − ( y ) . Then g ( y ) = Φ ◦ f ◦ Φ − ( y )= Φ ◦ f ( x )= Φ ◦ f ( x )= Φ ◦ f ◦ Φ − ( y )= g ( y ) . (3.1)Let B = C ( g , g ), then it follows thatΦ( A ) ⊆ B, hence A ≤ B. Similarly, let y ∈ B (arbitrary) and x = Φ − ( y ) . Then f ( y ) = Φ − ◦ g ◦ Φ( y )= Φ − ◦ g ( x )= Φ − ◦ g ( x )= Φ − ◦ g ◦ Φ( y )= f ( y ) . (3.2)It follows that Φ − ( B ) ⊆ A. Therefore B ≤ A. Thus C ( f , f ) = C ( g , g )as required. (cid:3) In the next few paragraphs, we will recall some classical topological notions.Notably, the following theorem proves that any change in the coincidence set C ( f, g )that may be effected by deforming both f and g can also be effected by deformingjust f. However, this property might not necessarily be true in digital topologicalsetting as we will discuss later.
Theorem 3.3. [12]
Let f, g : X −→ Y be mappings of a topological space X intoa topological manifold Y, and suppose that f ′ and g ′ are homotopic to f and g respectively. Then there is a map f ′′ homotopic to f , such that C ( f ′′ , g ) = C ( f ′ , g ′ ) . The following result is a consequence of Theorem 3.3.
Corollary 3.4. [12] If we deform only one of the two maps f, g in Theorem 3.3 bya homotopy while leaving the other fixed. Then
M C ∗ ( f, g ) = M C ( f, g ) . MUHAMMAD SIRAJO ABDULLAHI, POOM KUMAM, AND JAMILU ABUBAKAR
For example. Let f be a self map of X. Then
M C ( f, id) = min { g ) | g ≃ f } = M F ( f ) . i.e. the minimum number of coincidence points coincide with the classical minimumnumber of fixed points which plays a central role in the classical topological fixedpoint theory.The statement in Theorem 3.3 only holds for continuous maps on manifolds (orslightly more general spaces than that). However, it does not hold even for continu-ous maps on polyhedra. One of the major limitations of Nielsen coincidence theoryis that there is no way of dealing with homotopy-invariant coincidence countingwhere only one map varies by homotopy. When the space is a manifold it’s nomore a problem because of Brooks result (Theorem 3.3), but even when the spaceis a polyhedron there is really no way to proceed.So, we believe that it would be interesting to investigate whether or not Theorem3.3 and Corollary 3.4 holds in the setting of digital spaces. A partial answer to thisproblem is given in Proposition 3.11.Now, for some maps f , f : X −→ Y, we may define the following set HCS ( f , f ) , which we call the “homotopy coincidence point spectrum” of the functions f and f as follows: HCS ( f , f ) = { C ( g , g ) | g ≃ f and g ≃ f } ⊆ { , , . . . , X } . Remark . (i) M C ( f , f ) = min HCS ( f , f );(ii) Moreover, both M C ( f , f ) and HCS ( f , f ) are homotopy invariants forany continuous functions f and f .Now, we may also consider the “coincidence point spectrum” of X , which wedefine and denote as: CS ( X ) = { C ( f , f ) | f , f : X −→ Y are continuous } . The following immediately follows as a consequences of Theorem 3.2 above.
Corollary 3.6.
Let X and Y be isomorphic digital images. Then CS ( X ) = CS ( Y ) . To avoid confusion, when we allow only one of the two maps to be deformed by ahomotopy while keeping the other map fixed, we let
M C ∗ ( f , f ) , HCS ∗ ( f , f ) and CS ∗ ( X ) to denote M C ( f , f ) , HCS ( f , f ) and CS ( X ) respectively. For instance,we have M C ∗ ( f , f ) := min { C ( g , f ) | g ≃ f and f is fixed } . Theorem 3.7.
Suppose that X is a rigid digital image. Let id and c be the identityand constant mappings respectively. Then HCS ( id, c ) = S ( c ) and M C ( id, c ) = M F ( c ) . Proof.
The results follows immediately from Corollary 3.4 and the fact that X isrigid. Example 3.8.
Let X be a rigid digital image and f : X −→ X be a continuousmapping. Then HCS ( f, id ) = S ( f ) . OINCIDENCE POINT SETS IN DIGITAL TOPOLOGY 9
12 53 4
Figure 3.
The digital image C . Example 3.9.
Let X be a connected digital image, f : X −→ X be a continuousmapping and the constant mapping c be rigid. Then HCS ( f, c ) = { } . Remark . In Example 3.9 above, we realise that the assumption that c is rigidis very strong and therefore forced X to be a single point, which makes the examplea little bit not too interesting. Proposition 3.11. If X is a rigid image then Corollary 3.4 holds. Example 3.12.
Let C be the cycle of 5 points, id be the identity map, and c bea constant map. Then C ( id, c ) = 1 . If we change id by homotopy to some other map say f , we will always have C ( f, c ) = 1 since f must be a rotation [11]. Therefore, the spectrum of coin-cidences when we change only the first map by homotopy is just the set { } . i.e. HCS ∗ ( id, c ) = { } .However, if we are allowed to change both maps by homotopy, then we canchange c to some other map say g which has 0 , , { , , , } . i.e. HCS ( id, c ) = { , , , } . Moreover, in this particular example, when we interchange the position of thetwo mappings, we have
HCS ∗ ( c, id ) = HCS ( c, id ) = { , , , } .In fact, any cycle of 5 or more points can hold a similar result to this example.In other words, there’s nothing special about the cycle of 5 points specifically. Wechoose the cycle of 5 points, id and c to emphasize that HCS ∗ ( f, g ) can be differentfrom HCS ( f, g ) for any continuous maps f and g . Remark . (i) Example 3.12 above shows that Theorem 3.3 is false in thesetting of digital spaces;(ii) The equality HCS ∗ ( f , f ) = HCS ∗ ( f , f ) need not necessarily always betrue for any mappings f and f .4. Common Fixed Point Spectrum
In this section, we present the concept of common fixed point set, some relatedinvariants and results were also discuss. Let f , f : X −→ X, we define the“common fixed point set” of f and f as: CF ( f , f ) := { x ∈ X | f ( x ) = f ( x ) = x } . Theorem 4.1.
Let
X, Y be isomorphic digital images and f , f : X −→ X becontinuous mappings. Then there are continuous mappings g , g : Y −→ Y suchthat CF ( f , f ) = CF ( g , g ) . Proof.
The result follows from similar argument to the proof of Theorem 3.2. (cid:3)
Similar to the assertions in the previous section, we define the “minimum numberof common fixed points” of f and f as: M CF ( f , f ) := min { CF ( g , g ) | g ≃ f and g ≃ f } . Moreover, for some maps f , f : X −→ X, we may consider the following set HF S ( f , f ) , which we call the “homotopy common fixed point spectrum” of f and f : HF S ( f , f ) = { CF ( g , g ) | g ≃ f and g ≃ f } ⊆ { , , . . . , X } . We may also consider the following “common fixed point spectrum” of X definedas: CF S ( X ) = { CF ( f , f ) | f , f : X −→ X are continuous } . The following immediately follows as a consequences of Corrollary 4.1 above.
Corollary 4.2.
Let X and Y be isomorphic digital images. Then CF S ( X ) = CF S ( Y ) . Remark . (i) If f = f = f , then CF ( f , f ) = Fix( f );(ii) It is easy to see that F ( X ) is always a subset of CF S ( X ).By Remark 4.3, we obtain the following two corollaries. Corollary 4.4. [11] Let
X, Y be isomorphic digital images and f : X −→ X becontinuous mapping. Then there exists a continuous mapping g : Y −→ Y suchthat f ) = g ) . Corollary 4.5. [11] Let X and Y be isomorphic digital images. Then F ( X ) = F ( Y ) . Question 4.6. If f = f = f , do we always have(i) M CF ( f , f ) = M F ( f )?(ii) HF S ( f , f ) = S ( f )?(iii) CF S ( X ) = F ( X )?In response to Questions 4.6, we consider X ⊂ Z to be a digital image of unitcube of 8 points with 6-adjacency as shown in Figure 4. For any continuous mapping f : X −→ X , S ( f ) = { , , , , , , , } = F ( X ) since X is contractible [11]. Fur-ther, since HF S ( c, c ) = { , , , , , , , } and CF S ( X ) = { , , , , , , , } , wehave HF S ( c, c ) = S ( c ) = F ( X ) = CF S ( X ) . This further implies that
HF S ( f, f ) = CF S ( X ) for any continuous mapping f : X −→ X. Conjecture 4.7.
Let X be a contractible image and f : X −→ X be a continuousmapping. Then, HF S ( f, f ) = S ( f ) and CF S ( X ) = F ( X ). OINCIDENCE POINT SETS IN DIGITAL TOPOLOGY 11 x x x x x x x x Figure 4.
A contractible 3-dimensional digital image with a 6-adjacency relation. 5.
Retracts of X In this section, we study how retractions interact with the coincidence and com-mon fixed point spectra. To begin with, it is natural to ask whether or not, whenever A is a subset of an image X , we will have CS ( A ) ⊆ CS ( X ). The answer is negativeas shown by the following example. Example 5.1.
Let X be the digital image in Fig. 4. If A = X \{ x } then A ⊂ X and CS ( A ) = { , , , , , , , } 6⊆ { , , , , , , , } = CS ( X ) . However, if A ⊂ X is a retract of X , then we will have an affirmative answer asshown by Theorem 5.3 below. Definition 5.2. [7] Let A be a subset of a digital image X. A continuous function r : X −→ A is called a retraction, and A is a retract of X, if r ( a ) = a for all a ∈ A. Moreover, r is called a κ -deformation retraction, and A is a κ -deformation retractof X, if r satisfies i ◦ r ≃ κ id, where i : A −→ X is the inclusion map. Theorem 5.3.
Let A be a retract of an image X. Then CS ( A ) ⊆ CS ( X ) . Proof.
Let f , f : A −→ A be continuous functions and r : X −→ A be a retractionmapping. Now, we define the functions g , g : X −→ X as g = i ◦ f ◦ r and g = i ◦ f ◦ r, where i : A −→ X is the inclusion map. So, from Theorem 2.3, thefunctions g and g are continuous. Therefore, we have g ( x ) = f ( x ) if and only if x ∈ A and similarly g ( x ) = f ( x ) if and only if x ∈ A. Thus C ( f , f ) = C ( g , g ) , hence the assertion follows immediately since f and f are arbitrarily chosen. (cid:3) Theorem 5.4.
Let A be a retract of an image X. Then
CF S ( A ) ⊆ CF S ( X ) . Proof.
The assertions follows from a similar argument to the proof of Theorem 5.3. (cid:3)
Corollary 5.5. [11] Let A be a retract of an image X. Then F ( A ) ⊆ F ( X ) . Divergence Degree
In this section, we introduce the notion of divergence degree of a point x in animage X , which give us an estimate of “non-coincident indicator of the point x ”. Throughout this section, f and f are self maps on X . We begin with presentingan important definition we use to define the degree at which two given functionsdiffer at a point x . This we call the complement of the coincidence point set, whichwe denote by C ( f , f ) and define as: C ( f , f ) := { x ∈ X | f ( x ) = f ( x ) } . Whenever f ( x ) = f ( x ), we say that f and f does not meet at point x in X. Definition 6.1.
Let (
X, κ ) be a digital image with
X > x ∈ X. Thenthe “non-coincident indicator of x ” which we call the “Divergence Degree of x ” isdefine as: D ( x ) := min { C ( f , f ) | f ( x ) = f ( x ) and f , f are continuous } . Theorem 6.2.
Let X be a connected digital image with n = X > . Then n − ∈ CS ( X ) if and only if there is some x ∈ X with D ( x ) = 1 . Proof.
It is not too difficult to see that n − ∈ CS ( X ) if and only if there exist f , f ∈ C ( X, κ ) with exactly n − x ∈ X notcoincident by f and f has D ( x ) = 1 . Hence proving the result. (cid:3)
Example 6.3.
Let X be the digital image in Fig. 2. Let f, p v and p h be selfmaps on X representing; 180 ◦ rotation of X , vertical flip of X and horizontal flipof X respectively, then f, p v and p h are continuous. Let g , g , g : X −→ X bemappings define as: g (7) = 5 , g (11) = 10 , g (18) = 16 , g ( x ) = x for x ∈ X \{ , , } ,g (1) = 3 , g (8) = 9 , g (12) = 14 , g ( x ) = x for x ∈ X \{ , , } and g (1) = 3 , g (7) = 5 , g (8) = 9 , g (11) = 10 , g (12) = 14 , g (18) = 16 , g ( x ) = x for x ∈ X \{ , , , , , } . Then g , g and g are all continuous.Let h : X −→ X be the mappings that maps the top bar into the bottom barand fixes all other points and h : X −→ X be the mappings that maps the bottombar into the top bar and fixes all other points. These are all the possible non trivial(different from id and c ) continuous functions on X providing different coincidencepoint sets.So, after some computations we obtain D ( x ) = 3 for x ∈ { , , , , , } ,D ( x ) = 14 for x ∈ { , , , , , , , } and D ( x ) = 17 for x ∈ { , , , } . OINCIDENCE POINT SETS IN DIGITAL TOPOLOGY 13 Conclusion
In this article, we introduced, studied and investigated some properties of thecoincidence point set of digitally continuous maps. Following the Rosenfeld graph-ical model which seems more combinatorial than topological, we achieved resultsthat are not analogous to the classical topological fixed point theory, for instance,in classical coincidence theory the only interesting homotopy invariant count ofthe number of coincidence points is
M C ( f , f ). Whereas, here we introduced HCS ( f , f ) which is not studied in the classical coincidence theory. We also intro-duced and studied some topological invariants related to coincidence and commonfixed point sets for continuous maps on a digital image. Moreover, we studied howthese coincidence point sets are affected by rigidity and deformation retraction.Also, we briefly introduced the concept of divergence degree of a point in a digitalimage and illustrated by example that D(x) can assume different values for differentchoice of point. Lastly, we are optimistic that these properties will be applicable inimage processing and its related disciplines in the nearest future.8. Acknowledgement
The authors acknowledge the financial support provided by the Center of Excel-lence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science,KMUTT. The first and the third authors were supported by “the Petchra Pra JomKlao Ph.D. Research Scholarship” from ‘King Mongkut’s University of TechnologyThonburi” (Grant No. 35/2017 and 38/2018 respectively). Finally, the authorswould like to thank Assoc. Prof. Peter Christoper Staecker for his careful readingand his valuable suggestions to the improvement of this paper, especially his ideaof Example 3.12.
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OINCIDENCE POINT SETS IN DIGITAL TOPOLOGY 15
Current address , (MS Abdullahi, P. Kumam and J. Abubakar):
KMUTTFixed Point ResearchLaboratory , KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed PointLaboratory, Department of Mathematics, Faculty of Science, King Mongkut’s University of Tech-nology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140,Thailand
E-mail address : [email protected] [M.S. Abdullahi] E-mail address : [email protected] [P. Kumam] E-mail address : [email protected] [J. Abubakar] Current address : Center of Excellence in
Theoretical and Computational Science (TaCS-CoE) ,Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
E-mail address : [email protected] [P. Kumam] (MS Abdullahi and J. Abubakar) Department of Mathematics, Faculty of Science, Us-manu Danfodiyo University, Sokoto, Nigeria
E-mail address : [email protected] [M.S. Abdullahi] E-mail address : [email protected] [J. Abubakar][email protected] [J. Abubakar]