@ Appl. Gen. Topol. 17, no. 2(2016), 159-172 doi:10.4995/agt.2016.4704 c© AGT, UPV, 2016 Digital fixed points, approximate fixed points, and universal functions Laurence Boxer a, Ozgur Ege b, Ismet Karaca c, Jonathan Lopez d, and Joel Louwsma e a Department of Computer and Information Sciences, Niagara University, NY 14109, USA; and Department of Computer Science and Engineering, SUNY at Buffalo, Buffalo, NY, USA. (boxer@niagara.edu) b Department of Mathematics, Celal Bayar University, Muradiye, Manisa 45140, Turkey. (ozgur.ege@cbu.edu.tr) c Department of Mathematics, Ege University, Bornova, Izmir 35100, Turkey. (ismet.karaca@ege.edu.tr) d Department of Mathematics, Canisius College, Buffalo, NY 14208, USA. (lopez11@canisius.edu) e Department of Mathematics, Niagara University, NY 14109, USA. (jlouwsma@niagara.edu) Abstract A. Rosenfeld [23] introduced the notion of a digitally continuous func- tion between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digi- tally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP). 2010 MSC: Primary 55M20; Secondary 55N35. Keywords: digital image; digitally continuous; digital topology; fixed point. 1. Introduction In digital topology, we study geometric and topological properties of digital images via tools adapted from geometric and algebraic topology. Prominent among these tools is a digital version of continuous functions. In the current Received 16 February 2016 – Accepted 17 July 2016 http://dx.doi.org/10.4995/agt.2016.4704 L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma paper, we study fixed points and approximate fixed points of digitally contin- uous functions. We present a number of original results and some corrections of previously published assertions. The paper is organized as follows. Section 2 reviews background material. In section 3, we show that a digital image X has the Fixed Point Property (FPP) if and only if X has a single point. In section 4 we introduce approximate fixed points and the Approximate Fixed Point Property (AFPP). We give examples of digital images that have, and that don’t have, this property. In section 5 we study universal functions on digital images and their relation to the AFPP. In section 6 we correct errors that appeared in earlier papers. Concluding remarks appear in section 7. 2. Preliminaries 2.1. General Properties. A fixed point of a function f : X → X is a point x ∈ X such that f(x) = x. For a finite set X, we denote by |X| the number of distinct members of X. Let N be the set of natural numbers and let Z denote the set of integers. Then Zn is the set of lattice points in Euclidean n−dimensional space. A digital image is a pair (X,κ), where ∅ 6= X ⊂ Zn for some positive integer n and κ is an adjacency relation on X. Technically, then, a digital image (X,κ) is an undirected graph whose vertex set is the set of members of X and whose edge set is the set of unordered pairs {x0,x1} ⊂ X such that x0 6= x1 and x0 and x1 are κ-adjacent. Adjacency relations commonly used for digital images include the following [22]. Two points p and q in Z2 are 8 −adjacent if they are distinct and differ by at most 1 in each coordinate; p and q in Z2 are 4 − adjacent if they are 8-adjacent and differ in exactly one coordinate. Two points p and q in Z3 are 26−adjacent if they are distinct and differ by at most 1 in each coordinate; they are 18−adjacent if they are 26-adjacent and differ in at most two coordinates; they are 6−adjacent if they are 18-adjacent and differ in exactly one coordinate. For k ∈ {4, 8, 6, 18, 26}, a k − neighbor of a lattice point p is a point that is k−adjacent to p. The adjacencies discussed above are generalized as follows. Let u,n be positive integers, 1 ≤ u ≤ n. Distinct points p,q ∈ Zn are called cu-adjacent if there are at most u distinct coordinates j for which |pj − qj| = 1, and for all other coordinates j, pj = qj. The notation cu represents the number of points q ∈ Zn that are adjacent to a given point p ∈ Zn in this sense. Thus the values mentioned above: if n = 1 we have c1 = 2; if n = 2 we have c1 = 4 and c2 = 8; if n = 3 we have c1 = 6, c2 = 18, and c3 = 26. Yet more general adjacency relations are discussed in [19]. Let κ be an adjacency relation defined on Zn. A digital image X ⊂ Zn is κ − connected [19] if and only if for every pair of points {x,y} ⊂ X, x 6= y, there exists a set {x0,x1, . . . ,xc} ⊂ X such that x = x0, xc = y, and xi and xi+1 are κ−neighbors, i ∈{0, 1, . . . ,c− 1}. A κ-component of X is a maximal κ-connected subset of X. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 160 Digital fixed and approximate fixed points Often, we must assume some adjacency relation for the white pixels in Zn, i.e., the pixels of Zn\X (the pixels that belong to X are sometimes referred to as black pixels). In this paper, we are not concerned with adjacencies between white pixels. Definition 2.1 ([3]). Let a,b ∈ Z, a < b. A digital interval is a set of the form [a,b]Z = {z ∈ Z | a ≤ z ≤ b} in which 2−adjacency is assumed. Definition 2.2 ([4]; see also [23]). Let X ⊂ Zn0 , Y ⊂ Zn1 . Let f : X → Y be a function. Let κi be an adjacency relation defined on Z ni , i ∈ {0, 1}. We say f is (κ0,κ1)−continuous if for every κ0−connected subset A of X, f(A) is a κ1−connected subset of Y . See also [11, 12], where similar notions are referred to as immersions, grad- ually varied operators, and gradually varied mappings. If a and b are members of a digital image (X,κ), we write a ↔κ b, or a ↔ b when κ is understood, to indicate that either a = b or a and b are κ-adjacent. We say a function satisfying Definition 2.2 is digitally continuous. This definition implies the following. Proposition 2.3 ([4]; see also [23]). Let X and Y be digital images. Then the function f : X → Y is (κ0,κ1)-continuous if and only if for every {x0,x1}⊂ X such that x0 and x1 are κ0−adjacent, f(x0) ↔κ1 f(x1). For example, if κ is an adjacency relation on a digital image Y , then f : [a,b]Z → Y is (2,κ)−continuous if and only if for every {c,c + 1} ⊂ [a,b]Z, f(c) ↔κ f(c + 1). We have the following. Proposition 2.4 ([4]). Composition preserves digital continuity, i.e., if f : X → Y and g : Y → Z are, respectively, (κ0,κ1)−continuous and (κ1,κ2)−continuous functions, then the composite function g ◦f : X → Z is (κ0,κ2)−continuous. We say digital images (X,κ) and (Y,λ) are (κ,λ) − isomorphic (called (κ,λ) − homeomorphic in [3, 5]) if there is a bijection h : X → Y that is (κ,λ)-continuous, such that the function h−1 : Y → X is (λ,κ)-continuous. Classical notions of topology [2] yielded the concept of digital retraction in [3]. Let (X,κ) be a digital image and let A be a nonempty subset of X. A retraction of X onto A is a (κ,κ)-continuous function r : X → A such that r(a) = a for all a ∈ A. A digital simple closed curve is a digital image X = {xi}m−1i=0 , with m ≥ 4, such that the points of X are labeled circularly, i.e., xi and xj are adjacent if and only if j = (i− 1) (mod m) or j = (i + 1) (mod m). 2.2. Digital homotopy. A homotopy between continuous functions may be thought of as a continuous deformation of one of the functions into the other over a time period. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 161 L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma Definition 2.5 ([4]; see also [21]). Let X and Y be digital images. Let f,g : X → Y be (κ,κ′)-continuous functions. Suppose there is a positive integer m and a function F : X × [0,m]Z → Y such that • for all x ∈ X, F(x, 0) = f(x) and F(x,m) = g(x); • for all x ∈ X, the induced function Fx : [0,m]Z → Y defined by Fx(t) = F(x,t) for all t ∈ [0,m]Z is (2,κ′)−continuous. • for all t ∈ [0,m]Z, the induced function Ft : X → Y defined by Ft(x) = F(x,t) for all x ∈ X is (κ,κ′)−continuous. Then F is a digital (κ,κ′)−homotopy between f and g, and f and g are digitally (κ,κ′)−homotopic in Y . When the adjacency relations κ and κ′ are understood in context, we say f and g are digitally homotopic to abbreviate “digitally (κ,κ′)−homotopic in Y .” Definition 2.6. A digital image (X,κ) is κ-contractible [21, 3] if its identity map is (κ,κ)-homotopic to a constant function p for some p ∈ X. When κ is understood, we speak of contractibility for short. 2.3. Digital simplicial homology. Our presentation of digital simplicial ho- mology is taken from that of [16]. A set of m + 1 distinct mutually adjacent points is an m-simplex. Definition 2.7. If αq is the number of (κ,q)-simplices in X and m = max{q ∈ N∗ |αq 6= 0}, then m is the dimension of (X,κ), denoted dim(X,κ) or dim(X), and the Euler characteristic of (X,κ), χ(X,κ), is defined by χ(X,κ) = m∑ q=0 (−1)qαq. 2 For q ∈ N∗, the group of q-chains of (X,κ), denoted Cκq (X), is the free Abelian group with basis being the set of q-simplices of X. Let δq : C κ q (X) → Cκq−1(X) defined by δq(< p0,p1, . . . ,pq >) = { ∑q i=0(−1) i < p0,p1, . . . , p̂i, . . . ,pq > if 0 ≤ q ≤ dim(X); 0 if q > dim(X), where p̂i means that pi is omitted from the vertices of the simplex considered. Then δq is a homomorphism, and we have δq−1 ◦ δq = 0 [1]. This gives rise to the following groups [9]. • Zκq (X) = Kerδq, the group of digital simplicial q-cycles of X. • Bκq (X) = Imδq+1, the group of digital simplicial q-boundaries of X. • The quotient group Hκq (X) = Zκq (X) /Bκq (X), the q-th digital simpli- cial homology group of X. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 162 Digital fixed and approximate fixed points We have the following. Theorem 2.8 ([9]). Let (X,κ) be a directed digital simplicial complex of di- mension m. • Hq(X) is a finitely generated abelian group for every q ≥ 0. • Hq(X) is a trivial group for all q > m. • Hm(X) is a free abelian group, possibly {0}. 3. Fixed point property We say a digital image (X,κ) has the fixed point property (FPP) if every (κ,κ)-continuous function f : X → X has a fixed point. Some properties of digital images with the FPP were studied in [14]. However, the following shows that for digital images with cu-adjacencies, the FPP is not very interesting. Theorem 3.1. Let (X,κ) be a digital image. Then (X,κ) has the FPP if and only if |X| = 1. Proof. Clearly, if |X| = 1 then (X,κ) has the FPP. Now suppose |X| > 1. If (X,κ) has more than 1 κ-component, then there is a (κ,κ) continuous map f : X → X such that for all x ∈ X, x and f(x) are in different κ-components of X. Such a map does not have a fixed point. Therefore, we may assume X is κ-connected. Since |X| > 1, there are distinct κ-adjacent points x0,x1 ∈ X. Consider the map f : X → X given by f(x) = { x0 if x 6= x0; x1 if x = x0. Consider a pair y0,y1 of κ-adjacent members of X. • If one of these points, say, y0, coincides with x0, we have f(y0) = f(x0) = x1 and, since y1 6= x0, f(y1) = x0, so f(y0) and f(y1) are κ-adjacent. • If both y0 and y1 are distinct from x0, then f(y0) = x1 = f(y1) Therefore, f is (κ,κ)-continuous. Clearly, f does not have a fixed point. There- fore, (X,κ) does not have the FPP. � 4. Approximate fixed points Given a digital image (X,κ) and a (κ,κ)-continuous function f : X → X, we say p ∈ X is an approximate fixed point of f if either f(p) = p, or p and f(p) are κ-adjacent. We say a digital image (X,κ) has the approximate fixed point property (AFPP) if every (κ,κ)-continuous function f : X → X has an approximate fixed point. Theorem 4.1 ([23]). Let I = Πni=1 [ai,bi]Z. Then (I,cn) has the AFPP. Theorems 3.1 and 4.1 show that it is worthwhile to consider the AFPP, rather than the FPP, for digital images. We have the following. Theorem 4.2. Suppose (X,κ) has the AFPP and there is a (κ,λ)-isomorphism h : X → Y . Then (Y,λ) has the AFPP. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 163 L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma Proof. Let f : Y → Y be (λ,λ)-continuous. By Proposition 2.4, the function g = h−1 ◦f ◦h : X → X is (κ,κ) continuous, so our hypothesis implies there exists p ∈ X such that p ↔κ g(p). Then h(p) ↔λ h◦g(p) = h◦h−1 ◦f ◦h(p) = f(h(p)), so h(p) is an approximate fixed point of f. � Proposition 4.3. A digital simple closed curve of 4 or more points does not have the AFPP. Proof. Let S = ({si}m−1i=0 ,κ) with si and sj κ-adjacent if and only if i = (j + 1) modm or i = (j − 1) modm. Then the function f : S → S defined by f(si) = s(i+2) modm is (κ,κ)-continuous, and, for each i, si and f(si) are neither equal nor κ-adjacent. � Next, we show retractions preserve the AFPP. Theorem 4.4. Let (X,κ) be a digital image, and let Y ⊂ X be a (κ,κ)-retract of X. If (X,κ) has the AFPP, then (Y,κ) has the AFPP. Proof. Let r : X → Y be a (κ,κ) retraction. Let f : Y → Y be a (κ,κ)- continuous function. Let i : Y → X be the inclusion map. By Proposition 2.4, g = i ◦ f ◦ r : X → X is (κ,κ)-continuous. Therefore, g has an approximate fixed point x0 ∈ X. Let x1 = g(x0) ∈ Y . By choice of x0, it follows that x0 ↔ x1. Then x1 = g(x0) ↔ g(x1) = i◦f ◦r(x1) = i◦f(x1) = f(x1). Thus x1 is an approximate fixed point of f. � Following a classical construction of topology, the wedge of two digital images (A,κ) and (B,λ), denoted A ∧ B, is defined [17] as the union of the digital images (A′,µ) and (B′,µ), where • A′ ∩B′ has a single point, p; • If a ∈ A′ and b ∈ B′ are µ-adjacent, then either a = p or b = p; • (A′,µ) and (A,κ) are isomorphic; and • (B′,µ) and (B,λ) are isomorphic. In practice, we often have κ = λ = µ, A = A′, B = B′. We have the following. Theorem 4.5. Let A and B be digital images. Then (A∧B,κ) has the AFPP if and only if both (A,κ) and (B,κ) have the AFPP. Proof. Let A∩B = {p}. Let pA,pB : A∧B → A∧B be the functions pA(x) = { x if x ∈ A; p if x ∈ B. pB(x) = { p if x ∈ A; x if x ∈ B. It is easily seen that both of these functions are well defined and (κ,κ)-continuous. Also, let iA : A → A∧B and iB : B → A∧B be the inclusion functions, which are clearly (κ,κ)-continuous. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 164 Digital fixed and approximate fixed points Suppose (A,κ) and (B,κ) have the AFPP. Let f : A∧B → A∧B be (κ,κ)- continuous. We must show that there exists a point of A ∧ B that is equal or κ-adjacent to its image under f. If f(p) = p, then we have realized that goal. Otherwise, without loss of generality, f(p) ∈ A\{p}. By Proposition 2.4, h = pA ◦ f ◦ iA : A → A is (κ,κ)-continuous. Since A has the AFPP, there exists a ∈ A such that (4.1) h(a) ↔κ a. If f(a) ∈ B, then p = pA ◦f(a) = pA ◦f ◦ iA(a) = h(a). It follows from statement (4.1) that (4.2) p ↔κ a. If f(a) 6= p then f(a) ∈ B \ {p} and f(p) ∈ A \ {p}, so f(a) and f(p) are distinct, non-adjacent points. This is a contradiction of statement (4.2), since f is continuous. Therefore, we have f(a) ∈ A. Then f(a) = pA ◦f(a) = pA ◦f ◦ iA(a) = h(a). It follows from statement (4.1) that f(a) ↔κ a. Since f was arbitrarily selected, it follows that (A∧B,κ) has the AFPP. Conversely, suppose (A∧B,κ) has the AFPP. Since the maps pA and pB are (κ,κ)-retractions of (A ∧ B,κ) onto (A,κ) and (B,κ), respectively, it follows from Theorem 4.4 that (A,κ) and (B,κ) have the AFPP. � 5. Universal functions and the AFPP In this section, we define the notion of a universal function and study its relation to the AFPP. Definition 5.1. Let (X,κ) and (Y,λ) be digital images. A (κ,λ)-continuous function f : X → Y is universal for (X,Y ) if given a (κ,λ)-continuous function g : X → Y , there exists x ∈ X such that f(x) ↔λ g(x). The notion of a dominating set in graph theory corresponds to the notion of a dense set in a topological space. Definition 5.2 ([10]). Let (X,κ) be a nonempty digital image. Let Y be a nonempty subset of X. We say Y is κ-dominating in X if for every x ∈ X there exists y ∈ Y such that x ↔κ y. Theorem 5.3. Let (X,κ) and (Y,λ) be digital images. Let f : X → Y be a universal function for (X,Y ). Then f(X) is λ-dominating in Y . Proof. Let y ∈ Y and consider the constant function cy : X → Y defined by cy(x) = y for all x ∈ X. This function is clearly (κ,λ) continuous. Since f is universal, there exists xy ∈ X such that f(xy) is either equal to or λ-adjacent to y. Since y was arbitrarily chosen, the assertion follows. � c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 165 L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma Proposition 5.4. Let X be a κ-connected digital image of m points. Let (Y,λ) be a digital interval or a digital simple closed curve of n points, with n > m+ 2. Then there is no universal function from X to Y . Proof. Let f : X → Y be a (κ,λ) continuous function. Then f(X) is a λ- connected subset of Y , and |f(X)| ≤ m < n. We show that Y \f(X) has a component with at least 2 points, one of which is not λ-adjacent to any member of f(X). • If Y is a digital interval [a,a + n−1]Z, then, since f(X) is a connected subset of Y , f(X) = [u,v]Z. Consider the following possibilities. – v ≤ a + n − 3. Then the endpoint a + n − 1 of Y \ f(X) is not adjacent to any point of f(X). – v > a + n− 3. Therefore, v ≥ a + n− 2. Then u = v −|f(X)| + 1 ≥ a + n− 2 −|f(X)| + 1 ≥ a + n− (m + 2) + 1 > a + 1. I.e., u ≥ a + 2, so the point a of Y \f(X) is not adjacent to any point of f(X). • If Y is a digital simple closed curve, we may assume Y = {yj}n−1j=0 , where ya and yb are adjacent if and only if a = (b + 1) mod n or a = (b− 1) mod n. Since f(X) is connected, we may assume without loss of generality that f(X) = {yj}rj=0 where 0 ≤ r < m < n−2. Then yr+2 is a point of Y \f(X) that is not adjacent to any point of f(X). Thus, f(X) is not λ-dominating in Y . The assertion follows from Theorem 5.3. � Proposition 5.5. Let (X,κ) be a digital image. Then (X,κ) has the AFPP if and only if the identity function 1X is universal for (X,X). Proof. The function 1X is universal if and only if for every (κ,κ)-continuous f : X → X, there exists x ∈ X such that f(x) ↔κ 1X(x) = x, which is true if and only if (X,κ) has the AFPP. � Theorem 5.6. Let (X,κ) and (Y,λ) be digital images and let U ⊂ X. If the restriction function f|U : (U,κ) → (Y,λ) is a universal function for (U,Y ), then f is a universal function for (X,Y ). Proof. Let h : X → Y be (κ,λ)-continuous. Since f|U is universal, there exists u ∈ U ⊂ X such that h(u) = h|U (u) ↔λ f|U (u) = f(u). Hence f is universal for (X,Y ). � Theorem 5.7. Let (W,κ), (X,λ), and (Y,µ) be digital images. Let f : W → X be (κ,λ)-continuous and let g : X → Y be (λ,µ)-continuous. If g◦f is universal, then g is also universal. Proof. Let h : X → Y be (λ,µ)-continuous. Since g◦f is universal, there exists w ∈ W such that (g ◦ f)(w) ↔µ (h ◦ f)(w). I.e., for x = f(w) ∈ X we have g(x) ↔µ h(x). Since h was arbitrarily chosen, the assertion follows. � c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 166 Digital fixed and approximate fixed points Theorem 5.8. If g : (U,µ) → (X,κ) and h : (Y,λ) → (V,ν) are digital isomorphisms and f : X → Y is (κ,λ)-continuous, then the following are equivalent. (1) f is a universal function for (X,Y ). (2) f ◦g is universal. (3) h◦f is universal. Proof. (1 implies 2): Let k : U → Y be (µ,λ)-continuous. Since f is universal, there exists x ∈ X such that (k ◦ g−1)(x) ↔ f(x). By substituting x = g(g−1(x)), we have k(g−1(x)) ↔ (f◦g)(g−1(x)). Since k was arbitrarily chosen and g−1(x) ∈ U, it follows that f ◦g is universal. (2 implies 1): This follows from Theorem 5.7. (1 implies 3): Let m : X → V be (κ,ν)-continuous. Since f is universal, there exists x ∈ X such that (h−1 ◦ m)(x) ↔ f(x). Then m(x) = h((h−1 ◦ m)(x)) ↔ν (h◦f)(x). Since m was arbitrarily chosen, it follows that h◦f is universal. (3 implies 1): Suppose h ◦ f is universal. Then given a (κ,λ)-continuous r : X → Y , there exists x ∈ X such that h ◦ f(x) ↔ν h ◦ r(x). Therefore, f(x) = (h−1 ◦ h ◦ f)(x) ↔λ (h−1 ◦ h ◦ r)(x) = r(x). Since r was arbitrarily chosen, it follows that f is universal. � Corollary 5.9. Let f : (X,κ) → (Y,λ) be a digital isomorphism. Then f is universal for (X,Y ) if and only if (X,κ) has the AFPP. Proof. The function f is universal, by Theorem 5.8, if and only if f◦f−1 = 1X is universal, which, by Proposition 5.5, is true if and only if (X,κ) has the AFPP. � It may be useful to remind the reader for the following theorem that points that are cn-adjacent in Z n may differ in every coordinate. Concerning products, we have the following. Theorem 5.10. Let (Xi,cni ) ⊂ Zni , i = 1, 2, . . . ,m. Let s = ∑m i=1 ni. Con- sider the digital image X = Πmi=1 Xi ⊂ Z s. If (X,cs) has the AFPP then each (Xi,cni ) has the AFPP. Proof. Suppose (X,cs) has the AFPP. Let fi : Xi → Xi be (cni,cni )-continuous. Then the function f : X → X defined by f(x1,x2, . . . ,xm) = (f1(x1),f2(x2), . . . ,fm(xm)) is (cs,cs)-continuous. By Proposition 5.5, 1X is universal for (X,X). Therefore, there is a point x∗ = (x1,∗,x2,∗, . . . ,xm,∗) ∈ X with xi,∗ ∈ Xi such that x∗ ↔cs f(x∗). Therefore, xi,∗ ↔cni fi(xi,∗) for all i. Since fi was arbitrarily chosen, it follows that (Xi,cni ) has the AFPP. � c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 167 L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma 6. Corrections of published assertions In this section, we correct some assertions that appear in [14, 16]. We show below that the function F : [0, 1]Z → [0, 1]Z defined by F(x) = 1−x (i.e., F(0) = 1, F(1) = 0) provides a counterexample to several of the assertions of [14]. Clearly this function is (2, 2)-continuous and does not have a fixed point. We will need the following. Definition 6.1 ([15]). Let (X,κ) be a digital image whose digital homology groups are finitely generated and vanish above some dimension n. Let f : X → X be a (κ,κ)-continuous map. The Lefschetz number of f, denoted λ(f), is defined as λ(f) = n∑ i=0 (−1)i tr(fi,∗), where fi,∗ : H κ i (X) → H κ i (X) is the map induced by f on the i th homology group of (X,κ) and tr(fi,∗) is the trace of fi,∗. In studying digital maps from a sphere to itself, there is a question of how to represent a Euclidean sphere digitally. • As in [16], we will represent S1 by the set S1 = [−1, 1]2Z\{(0, 0)}⊂ Z 2 and c1-adjacency with points {xj}7j=0 labeled circularly. • More generally, as in [16], we will represent Sn by the set Sn = [−1, 1]n+1Z \{0n+1}⊂ Z n+1 and c1-adjacency, where 0n+1 is the origin in Zn+1. Definition 6.2 ([16, 24]). Suppose a continuous function f : (Sn,κ) → (Sn,κ) induces a homomorphism on the n-th homology group, f∗ : H κ n(Sn) → Hκn(Sn), such that f∗([x]) = m[x] for some fixed m ∈ Z, where [x] is a generator of Hκn(Sn). The value of m is the degree of f. Theorem 6.3 ([8]). Let S be a digital simple closed curve. For an isomorphism of S and a continuous non-surjective self-map of S to be homotopic, we must have |S| = 4. We state the following corrections. • Incorrect assertion stated as Theorem 3.3 of [14]: If (X,κ) is a finite digital image and f : X → X is a (κ,κ)-continuous function with λ(f) 6= 0, then f has a fixed point. In fact, the function F defined above is a counterexample to this assertion, since it is easily seen that λ(F) 6= 0. • Incorrect assertion stated as Theorem 3.4 of [14]: Every (c1,c1)- continuous function f : [0, 1]Z → [0, 1]Z has a fixed point. In fact, [23] shows that this assertion is false, and the function F defined above is a counterexample. • Incorrect assertion stated as Theorem 3.5 of [14]: Let X = {(0, 0), (1, 0), (0, 1), (1, 1)}⊂ Z2. Then every (c1,c1)-continuous function f : X → X has a fixed point. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 168 Digital fixed and approximate fixed points In fact, we can use the function F above to obtain a counterexam- ple. Let G : X → X be defined by G(x,y) = (x,F(y)). Then G is (c1,c1)-continuous and has no fixed point. Alternately, it follows from Theorem 3.1 that the assertion is incorrect. • Incorrect assertion stated as Theorem 3.8 of [14]: Let (X,κ) be a κ- contractible digital image. Then every (κ,κ)-continuous map f : X → X has a fixed point. In fact, since [0, 1]Z is c1-contractible, the function F above pro- vides a counterexample to this assertion. Alternately, it follows from Theorem 3.1 that the assertion is incorrect. • Incorrect assertion stated as Example 3.9 of [14]: Let X = {(0, 0), (0, 1), (1, 1)}⊂ Z2. Then (X,c2) has the FPP. In fact, the map f : X → X defined by f(0, 0) = (0, 1), f(0, 1) = (1, 1), f(1, 1) = (0, 0) is (c2,c2)-continuous and has no fixed points. Alternately, it follows from Theorem 3.1 that the assertion is incorrect. • Incorrect assertion stated as Corollary 3.10 of [14]: Any digital image with the same digital homology groups as a single point image has the FPP. To show this assertion is incorrect, observe that if X = {(0, 0), (0, 1), (1, 1)}⊂ Z2 and Y is a digital image of one point in Z2, then [1, 9] H8q (X) = H 8 q (Y ) = { Z if q = 0; 0 if q 6= 0. It follows from Theorem 3.1 that the assertion is incorrect. • Incorrect assertions stated as Example 3.17 and Corollary 3.18 of [14]: The digital images (MSS′6, 6) and (P 2, 6), each with more than one point, have the FPP. It follows from Theorem 3.1 that these assertions are incorrect. • Incorrect assertion stated as Theorem 3.5 of [16]: If (X,κ) is a finite digital image and f : X → X is a (κ,κ)-continuous function with λ(f) 6= 0, then any map homotopic to f has a fixed point. In fact, we observed above that the function F, which is homotopic to itself and has λ(F) 6= 0, does not have a fixed point. • Incorrect assertion stated as Theorem 3.7 of [16]: If (X,κ) is a digital image such that χ(X,κ) 6= 0, then any map homotopic to the identity has a fixed point. In fact, we can take X = [0, 1]Z, for which χ(X,c1) = (−1)1(2) + (−1)2(1) 6= 0, and the function F discussed above is homotopic to 1X and does not have a fixed point. • Incorrect assertion stated as Theorem 3.11 of [16]: Let (Sn,c1) ⊂ Zn+1 be a digital n-sphere as described above, where n ∈{1, 2}. If f : Sn → Sn is a continuous map of degree m 6= 1, then f has a fixed point. In fact, we have the following. Elementary calculations show that Hc11 (S1) ≈ Z; also, H c1 1 (S2) ≈ Z 23 [13]. For n ∈ {1, 2}, as in the proof of Theorem 3.1, we can choose distinct and adjacent x0 and x1 c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 169 L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma in Sn and let f : Sn → Sn be given by f(x) = x0 for x 6= x0 and f(x0) = x1. Clearly, f is continuous and does not have a fixed point. Since f∗ : H1(Sn) → H1(Sn) is 0, the degree of f is 0. • Proposition 3.12 of [16] depends on an unstated assumption that (recall Definition 2.7) ακq (X) is finite for all q, a condition that is satisfied if and only if X is finite; after all, one can study infinite digital images (X,κ), as in [7], for which, e.g., ακ0 (X) = ∞. E.g., we could take X = Z; according to Definition 2.7, χ(Z,c1) is undefined, since α c1 1 (Z) = αc10 (Z) = ∞. Therefore, the proposition should be stated as follows. Let (X,κ) be a finite digital image and suppose f : (X,κ) → (X,κ) is continuous. If f∗ : H κ ∗ (X) → Hκ∗ (X) is defined by f∗(z) = kz where k ∈ Z, i.e., if there exists k ∈ Z such that in every dimension i we have f inducing the homomorphism fi∗ : H κ i (X) → H κ i (X) defined by fi∗(z) = kz, then λ(f) = k χ(X). • A theoretically minor, but possibly confusing, error in Theorem 3.14 of [16]: In discussing an antipodal map f : X → X, one needs the property that for every x ∈ X we have −x ∈ X; this property does not characterize the version of S2 used in Theorem 3.14 of [16]. In the following, we use S2 = [−1, 1]3Z \{(0, 0, 0)}, as described above. Theorem 3.14 of [16] asserts that If αi : (Si,c1) → (Si,c1) is the antipodal map between two digital i-spheres Si ⊂ Zi+1, for i ∈{1, 2}, then αi has degree (−1)i+1. In fact, we show that this assertion is correct for i = 1, although an argument different from that of [16] must be given, as the argument of [16] makes use of Theorem 3.4 of [16] (= Theorem 3.3 of [14]), which, as noted above, is incorrect. For i = 2, we show the assertion is not well defined. – For i = 1, we have the following. Let the points {ej}7j=0 of S1 be circularly ordered. For notational convenience, let e8 = e0, and, more generally, index arithmetic is assumed to be mod- ulo 8. The generators of the 1-chains of S1 are the members of {< ejej+1 >}7j=0. We have 0 = δ( 7∑ j=0 uj < ejej+1 >) = 7∑ j=0 uj(ej −ej+1) = 8∑ j=1 (uj −uj−1)ej implies u0 = u1 = · · · = u7. Therefore, Z1(S1) is generated by ∑7 j=0 < ejej+1 >. Since, clearly, B1(S1) = {0}, we have H1(S1) = Z1(S1)/B1(S1) is isomorphic to Z. Therefore, the ho- momorphism (α1)∗ : H1(S1) → H1(S1) induced by α1 must satisfy (α1)∗(x) = kx for some integer k. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 170 Digital fixed and approximate fixed points Indeed, since the antipode of ej is ej+4, we have (α1)∗( 7∑ j=0 < ejej+1 >) = 7∑ j=0 < ej+4ej+5 >= 7∑ j=0 < ejej+1 >, so k = 1 = (−1)1+1, as asserted. – For i=2, we observe that, using the c1 adjacency, there is no triple of distinct, mutually adjacent points in S2. Therefore, H2(S2) = {0}. Therefore, the degree of α2 is not well defined since for any integer k we have (α2)∗(x) = kx for all x = 0 ∈ H2(S2). • Incorrect assertion stated as Theorem 3.15 of [16]: Let S1 be a digital simple closed curve in Z2. If h : (S1,c1) → (S1,c1) is a continuous function that is homotopic to a constant function in S1, then h has a fixed point. In fact, we can take S1 as above with its points ordered circularly, S1 = {xj}7j=0 where distinct points xu,xv are adjacent if and only if u + 1 = v mod 8 or u − 1 = v mod 8. Then, as in the proof of Theorem 3.1, the function h : S1 → S1 given by h(x) = { x0 if x 6= x0; x1 if x = x0, is continuous and homotopic to the constant function x0 in S1 but has no fixed point. • Correct (for |S1| > 4) assertion incorrectly “proven” as Corollary 3.16 of [16]: Let S1 be as above. Let h : (S1,c1) → (S1,c1) be given by h(xi) = x(i+1) mod m, where m = |S1|. Then h is not homotopic in S1 to a constant map. The argument given for this assertion in [16] depends on Theorem 3.15 of [16], which, we have shown above, is incorrect. However, since h is easily seen to be an isomorphism, by Theorem 6.3, the current assertion is true if and only if |S1| > 4. 7. Summary We have shown that only single-point digital images have the fixed point property. However, digital n-cubes have the approximate fixed point property with respect to the cn-adjacency [23]. 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