@ Appl. Gen. Topol. 21, no. 1 (2020), 111-133 doi:10.4995/agt.2020.12101 c© AGT, UPV, 2020 Fixed point sets in digital topology, 2 Laurence Boxer Department of Computer and Information Sciences, Niagara University, Niagara University, NY 14109, USA; and Department of Computer Science and Engineering, State University of New York at Buffalo (boxer@niagara.edu) Communicated by V. Gregori Abstract We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set. 2010 MSC: 54H25. Keywords: digital topology; digital image; fixed point; reducible image; re- tract; wedge; tree. 1. Introduction As stated in [10]: Digital images are often used as mathematical models of real- world objects. A digital model of the notion of a continuous function, borrowed from the study of topology, is often useful for the study of digital images. However, a digital image is typically a finite, discrete point set. Thus, it is often necessary to study digital images using methods not directly derived from topology. In this paper, we examine some properties of digital images concerned with the fixed points of digitally continuous functions; among these properties are discrete measures that are not natural analogues of properties of subsets of Rn. Received 19 July 2019 – Accepted 28 January 2020 http://dx.doi.org/10.4995/agt.2020.12101 L. Boxer In [10], we studied rigidity, pull indices, fixed point spectra for digital images and for digitally continuous functions, and related notions. In the current work, we study pointed versions of notions introduced in [10]. We also study such questions as when a set of fixed points Fix(f) determines that f is an identity function, or is “approximately” an identity function. Some of the results in this paper were presented in [6]. 2. Preliminaries Much of this section is quoted or paraphrased from [10]. Let N denote the set of natural numbers; N∗ = {0}∪N, the set of nonnegative integers; and Z, the set of integers. #X will be used for the number of members of a set X. 2.1. Adjacencies. A digital image is a pair (X,κ) where X ⊂ Zn for some n and κ is an adjacency on X. Thus, (X,κ) is a graph for which X is the vertex set and κ determines the edge set. Usually, X is finite, although there are papers that consider infinite X. Usually, adjacency reflects some type of “closeness” in Zn of the adjacent points. When these “usual” conditions are satisfied, one may consider the digital image as a model of a black-and-white “real world” image in which the black points (foreground) are represented by the members of X and the white points (background) by members of Zn\{X}. We write x ↔κ y, or x ↔ y when κ is understood or when it is unnecessary to mention κ, to indicate that x and y are κ-adjacent. Notations x -κ y, or x - y when κ is understood, indicate that x and y are κ-adjacent or are equal. The most commonly used adjacencies are the cu adjacencies, defined as follows. Let X ⊂ Zn and let u ∈ Z, 1 ≤ u ≤ n. Then for points x = (x1, . . . ,xn) 6= (y1, . . . ,yn) = y we have x ↔cu y if and only if • for at most u indices i we have |xi −yi| = 1, and • for all indices j, |xj −yj| 6= 1 implies xj = yj. The cu-adjacencies are often denoted by the number of adjacent points a point can have in the adjacency. E.g., • in Z, c1-adjacency is 2-adjacency; • in Z2, c1-adjacency is 4-adjacency and c2-adjacency is 8-adjacency; • in Z3, c1-adjacency is 8-adjacency, c2-adjacency is 18-adjacency, and c3-adjacency is 26-adjacency. The literature also contains several adjacencies to exploit properties of Carte- sian products of digital images. These include the following. Definition 2.1 ([1]). Let (X,κ) and (Y,λ) be digital images. The normal product adjacency or strong adjacency on X×Y , NP(κ,λ), is defined as follows. Given x0,x1 ∈ X, y0,y1 ∈ Y such that p0 = (x0,y0) 6= (x1,y1) = p1, we have p0 ↔NP(κ,λ) p1 if and only if one of the following is valid: c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 112 Fixed point sets in digital topology, 2 • x0 ↔κ x1 and y0 = y1, or • x0 = x1 and y0 ↔λ y1, or • x0 ↔κ x1 and y0 ↔λ y1. Building on the normal product adjacency, we have the following. Definition 2.2 ([4]). Given u,v ∈ N, 1 ≤ u ≤ v, and digital images (Xi,κi), 1 ≤ i ≤ v, let X = Πvi=1Xi. The adjacency NPu(κ1, . . . ,κv) for X is defined as follows. Given xi,x ′ i ∈ Xi, let p = (x1, . . . ,xv) 6= (x′1, . . . ,x ′ v) = q. Then p ↔NPu(κ1,...,κv) q if for at least 1 and at most u indices i we have xi ↔κi x′i and for all other indices j we have xj = x ′ j. Notice NP(κ,λ) = NP2(κ,λ) [4]. Let x ∈ (X,κ). We use the notations N(x) = Nκ(x) = {y ∈ X |y ↔κ x} and N∗(x) = N∗κ(x) = Nκ(x) ∪{x}. 2.2. Digitally continuous functions. We denote by id or idX the identity map id(x) = x for all x ∈ X. Definition 2.3 ([16, 3]). Let (X,κ) and (Y,λ) be digital images. A function f : X → Y is (κ,λ)-continuous, or digitally continuous when κ and λ are understood, if for every κ-connected subset X′ of X, f(X′) is a λ-connected subset of Y . If (X,κ) = (Y,λ), we say a function is κ-continuous to abbreviate “(κ,κ)-continuous.” Theorem 2.4 ([3]). A function f : X → Y between digital images (X,κ) and (Y,λ) is (κ,λ)-continuous if and only if for every x,y ∈ X, if x ↔κ y then f(x) -λ f(y). Theorem 2.5 ([3]). Let f : (X,κ) → (Y,λ) and g : (Y,λ) → (Z,µ) be con- tinuous functions between digital images. Then g ◦ f : (X,κ) → (Z,µ) is continuous. It is common to use the term path with the following distinct but related meanings. • A path from x to y in a digital image (X,κ) is a set {xi}mi=0 ⊂ X such that x0 = x, xm = y, and xi -κ xi+1 for i = 0, 1, . . . ,m− 1. If the xi are distinct, then m is the length of this path. • A path from x to y in a digital image (X,κ) is a (2,κ)-continuous function P : [0,m]Z → X such that P(0) = x and P(m) = y. Notice that in this usage, {P(0), . . . ,P(m)} is a path in the previous sense. Definition 2.6 ([3]; see also [14]). 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 h : X × [0,m]Z → Y such that c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 113 L. Boxer • for all x ∈ X, h(x, 0) = f(x) and h(x,m) = g(x); • for all x ∈ X, the induced function hx : [0,m]Z → Y defined by hx(t) = h(x,t) for all t ∈ [0,m]Z is (2,κ′)−continuous. That is, hx is a path in Y . • for all t ∈ [0,m]Z, the induced function ht : X → Y defined by ht(x) = h(x,t) for all x ∈ X is (κ,κ′)−continuous. Then h is a digital (κ,κ′)−homotopy between f and g, and f and g are digi- tally (κ,κ′)−homotopic in Y , denoted f ∼κ,κ′ g or f ∼ g when κ and κ′ are understood. If (X,κ) = (Y,κ′), we say f and g are κ-homotopic to abbrevi- ate “(κ,κ)-homotopic” and write f ∼κ g to abbreviate “f ∼κ,κ g”. If further h(x,t) = x for all t ∈ [0,m]Z, we say h holds x fixed. If there exists x0 ∈ X such that f(x0) = g(x0) = y0 ∈ Y and h(x0, t) = y0 for all t ∈ [0,m]Z, then h is a pointed homotopy and f and g are pointed homotopic [3]. If there exist continuous f : (X,κ) → (Y,λ) and g : (Y,λ) → (X,κ) such that g ◦ f ∼κ,κ idX and f ◦ g ∼λ,λ idY , then (X,κ) and (Y,λ) are homotopy equivalent. If there is a κ-homotopy between idX and a constant map, we say X is κ-contractible, or just contractible when κ is understood. Theorem 2.7 ([4]). Let (Xi,κi) and (Yi,λi) be digital images, 1 ≤ i ≤ v. Let fi : Xi → Yi. Then the product map f : ∏v i=1 Xi → ∏v i=1 Yi defined by f(x1, . . . ,xv) = (f1(x1), . . . ,fv(xv)) for xi ∈ Xi, is (NPv(κ1, . . . ,κv),NPv(λ1, . . . ,λv))-continuous if and only if each fi is (κi,λi)-continuous. Definition 2.8. Let A ⊂ X. A κ-continuous function r : X → A is a retrac- tion, and A is a retract of X, if r(a) = a for all a ∈ A. If such a map r satisfies i◦ r ∼κ idX where i : A → X is the inclusion map, then A is a κ-deformation retract of X. A function f : (X,κ) → (Y,λ) is an isomorphism (called a homeomorphism in [2]) if f is a continuous bijection such that f−1 is continuous. We use the following notation. For a digital image (X,κ), C(X,κ) = {f : X → X |f is continuous}. Given f ∈ C(X,κ), a point x ∈ X is a fixed point of f if f(x) = x. We denote by Fix(f) the set {x ∈ X |x is a fixed point of f}. If x ∈ X \ Fix(f), we say f moves x. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 114 Fixed point sets in digital topology, 2 Figure 1. (Figure 1 of [9].) The image X discussed in Ex- ample 3.1. The coordinates are ordered according to the axes in this figure. 3. Rigidity and reducibility A function f : (X,κ) → (Y,λ) is rigid [10] when no continuous map is homotopic to f except f itself. When the identity map id : X → X is rigid, we say X is rigid [12]. If f : X → Y with f(x0) = y0, then f is pointed rigid [12] if no continuous map is pointed homotopic to f other than f itself. When the identity map id : (X,x0) → (X,x0) is pointed rigid, we say (X,x0) is pointed rigid. Rigid maps and digital images are discussed in [12, 10]. Clearly, a rigid map is pointed rigid, and a rigid digital image is pointed rigid. (Note these assertions may seem counterintuitive as, e.g., pointed homotopic functions are homotopic, but the converse is not always true.) We show in the following that the converses of these assertions are not generally true. Example 3.1 ([9]). Let X = ([0, 2]2Z × [0, 1]Z)\{(1, 1, 1)}. Let x0 = (0, 0, 1) ∈ X. See Figure 1. It was shown in [9] that X is 6-contractible (i.e., c1- contractible) but (X,x0) is not pointed 6-contractible. The proof of the latter uses an argument that is easily modified to show that any homotopy of idX that moves some point must move x0. It follows that idX is not rigid but is x0-pointed rigid, i.e., that X is not c1-rigid but (X,x0) is c1-pointed rigid. Definition 3.2 ([12]). A finite image X is reducible if it is homotopy equivalent to an image of fewer points. Otherwise, we say X is irreducible. Lemma 3.3 ([12]). A finite image X is reducible if and only if idX is homotopic to a nonsurjective map. Let (X,κ) be reducible. By Lemma 3.3, there exist x ∈ X and f ∈ C(X,κ) such that idX 'κ f and x 6∈ f(X). We will call such a point a reduction point. In Lemma 3.4 below, we have changed the notation of [12], since the latter paper uses the notation “N(x)” for what we call “N∗(x)” or “N∗κ(x)”. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 115 L. Boxer Lemma 3.4 ([12]). If there exist distinct x,y ∈ X so that N∗(x) ⊂ N∗(y), then X is reducible. In particular, x is a reduction point of X, and X \{x} is a deformation retract of X. Remark 3.5 ([12]). A finite rigid image is irreducible. Theorem 3.6. Let (X,c2) be a digital image in Z2. Suppose there exists x0 ∈ X such that Nc2 (x0) is c2-connected and #Nc2 (x0) ∈ {1, 2, 3}. Then (X,c2) is reducible. Proof. We first show that in all cases, there exists y ∈ Nc2 (x0) such that N∗c2 (x0) ⊂ N ∗ c2 (y). (1) Suppose #Nc2 (x0) = 1. Then there exists y ∈ X such that {y} = Nc2 (x0). Clearly, then, N ∗ c2 (x0) ⊂ N∗c2 (y). (2) Suppose #Nc2 (x0) = 2. Then there exist distinct y,y ′ ∈ X such that {y,y′} = Nc2 (x0), which by hypothesis is connected. Therefore, {x0,y′}⊂ Nc2 (y), so N∗c2 (x0) ⊂ N ∗ c2 (y). (3) Suppose #Nc2 (x0) = 3. Then there exist distinct y,y0,y1 ∈ X such that {y,y0,y1} = Nc2 (x0), which by hypothesis is connected. There- fore, one of the members of Nc2 (x0), say, y, is adjacent to the other two. Thus, {x0,y0,y1}⊂ Nc2 (y), so N∗c2 (x0) ⊂ N ∗ c2 (y). In all cases we have N∗c2 (x0) ⊂ N ∗ c2 (y). The assertion follows from Lemma 3.4. � Remark 3.7. If instead we use the c1-adjacency, the analog of the previous theorem is simpler, since if (X,c1) is a digital image in Z2 and x0 ∈ X such that Nc1 (x0) is nonempty and c1-connected, then #Nc1 (x0) = 1. This case is similar to the case #Nc2 (x0) = 1 of Theorem 3.6 above, so (X,c1) is reducible. 4. Pointed homotopy fixed point spectrum In this section, we define pointed versions of the homotopy fixed point spec- trum of f ∈ C(X,κ) and the fixed point spectrum of a digital image (X,κ). Definition 4.1. Let (X,κ) be a digital image. • [10] Given f ∈ C(X,κ), the homotopy fixed point spectrum of f is S(f) = {# Fix(g) |g ∼κ f}. • Given f ∈ C(X,κ) and x0 ∈ Fix(f), the pointed homotopy fixed point spectrum of f is S(f,x0) = {# Fix(g) |g ∼κ f holding x0 fixed}. Definition 4.2. Let (X,κ) be a digital image. • [10] The fixed point spectrum of (X,κ) is F(X) = F(X,κ) = {# Fix(f) |f ∈ C(X,κ)}. • Given x0 ∈ X, the pointed fixed point spectrum of (X,κ,x0) is F(X,x0) = F(X,κ,x0) = {# Fix(f) |f ∈ C(X,κ),x0 ∈ Fix(f)}. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 116 Fixed point sets in digital topology, 2 Theorem 4.3 ([10]). Let A be a retract of (X,κ). Then F(A) ⊆ F(X). The argument used to prove Theorem 4.3 is easily modified to yield the following. Theorem 4.4. Let (A,κ,x0) be a retract of (X,κ,x0). Then F(A,κ,x0) ⊆ F(X,κ,x0). Theorem 4.5 ([10]). Let X = [1,a]Z × [1,b]Z. Let κ ∈{c1,c2}. Then S(idX,κ) = F(X,κ) = {i}abi=0. Example 4.6. Consider the pointed digital image (X,c1,x0) of Example 3.1. Since f ∈ C(X,c1) and x0 ∈ Fix(f) imply f = idX, S(idX,c1,x0) = {#X} = {17}. However, (X,c1) is not rigid. It is easily seen that there is a c1-deformation retraction of X to {(x,y, 0) ∈ X}, which is isomorphic to [1, 3]2Z. It follows from Theorem 4.3 and Theorem 4.5 that {i}9i=0 ⊂ S(idX). Since every f ∈ C(X,c1) such that f 'c1 idX and f 6= idX moves every point q of X such that p3(q) = 1, it follows easily that S(idX,c1) = F(X,c1) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17}. 5. Freezing sets In this section, we consider subsets of Fix(f) that determine that f ∈ C(X,κ) must be the identity function idX. Interesting questions include what properties such sets have, and how small they can be. In classical topology, given a connected set X ⊂ Rn and a continuous self- map f on X, knowledge of a finite subset A of the fixed points of f rarely tells us much about the behavior of f on X \A. By contrast, we see in this section that knowledge of a subset of the fixed points of a continuous self-map f on a digital image can completely characterize f as an identity map. 5.1. Definition and basic properties. Definition 5.1. Let (X,κ) be a digital image. We say A ⊂ X is a freezing set for X if given g ∈ C(X,κ), A ⊂ Fix(g) implies g = idX. Theorem 5.2. Let (X,κ) be a digital image. Let A ⊂ X. The following are equivalent. (1) A is a freezing set for X. (2) idX is the unique extension of idA to a member of C(X,κ). (3) For every isomorphism F : X → (Y,λ), if g : X → Y is (κ,λ)- continuous and F|A = g|A, then g = F . (4) Any continuous g : A → Y has at most one extension to an isomor- phism ḡ : X → Y . c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 117 L. Boxer Proof. 1) ⇔ 2): This follows from Definition 5.1. 1) ⇒ 3): Suppose A is a freezing set for X. Let F : X → Y be a (κ,λ)- isomorphism. Let g : X → Y be (κ,λ)-continuous, such that g|A = F |A. Then F−1 ◦g|A = F−1 ◦F|A = idX |A = idA . Since the composition of digitally continuous functions is continuous, it follows by hypothesis that F−1 ◦g = idX, and therefore that g = F ◦ (F−1 ◦g) = F ◦ idX = F. 3) ⇒ 1): Suppose for every isomorphism F : X → (Y,λ), if g : X → Y is (κ,λ)-continuous and F|A = g|A, then g = F. For g ∈ C(X,κ), A ⊂ Fix(g) implies g|A = idX |A, so since idX is an isomorphism, g = idX. 3) ⇒ 4): This is elementary. 4) ⇒ 2): This follows by taking g to be the inclusion of A into X, which extends to idX. � Freezing sets are topological invariants in the sense of the following. Theorem 5.3. Let A be a freezing set for the digital image (X,κ) and let F : (X,κ) → (Y,λ) be an isomorphism. Then F(A) is a freezing set for (Y,λ). Proof. Let g ∈ C(Y,λ) such that g|F(A) = idY |F(A). Then g ◦F|A = g|F(A) ◦F|A = idY |F(A) ◦F|A = F|A. By Theorem 5.2, g ◦F = F. Thus g = (g ◦F) ◦F−1 = F ◦F−1 = idY . By Definition 5.1, F(A) is a freezing set for (Y,λ). � We will use the following. Proposition 5.4 ([10]). Let (X,κ) be a digital image and f ∈ C(X,κ). Sup- pose x,x′ ∈ Fix(f) are such that there is a unique shortest κ-path P in X from x to x′. Then P ⊂ Fix(f). Let pi : Zn → Z be the projection to the ith coordinate: pi(z1, . . . ,zn) = zi. The following assertion can be interpreted to say that in a cu-adjacency, a continuous function that moves a point p also moves a point that is “behind” p. E.g., in Z2, if q and q′ are c1- or c2-adjacent with q left, right, above, or below q′, and a continuous function f moves q to the left, right, higher, or lower, respectively, then f also moves q′ to the left, right, higher, or lower, respectively. Lemma 5.5. Let (X,cu) ⊂ Zn be a digital image, 1 ≤ u ≤ n. Let q,q′ ∈ X be such that q ↔cu q′. Let f ∈ C(X,cu). (1) If pi(f(q)) > pi(q) > pi(q ′) then pi(f(q ′)) > pi(q ′). (2) If pi(f(q)) < pi(q) < pi(q ′) then pi(f(q ′)) < pi(q ′). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 118 Fixed point sets in digital topology, 2 Figure 2. Illustration for Example 5.9 Proof. (1) Suppose pi(f(q)) > pi(q) > pi(q ′). Since q ↔cu q′, if pi(q) = m then pi(q ′) = m− 1. Then pi(f(q)) > m. By continuity of f, we must have f(q′) -cu f(q), so pi(f(q ′)) ≥ m > pi(q′). (2) This case is proven similarly. � Theorem 5.6. Let (X,κ) be a digital image. Let X′ be a proper subset of X that is a retract of X. Then X′ does not contain a freezing set for (X,κ). Proof. Let r : X → X′ be a retraction. Then f = i ◦ r ∈ C(X,κ), where i : X′ → X is the inclusion map. Then f|X′ = idX′ , but f 6= idX. The assertion follows. � Corollary 5.7. Let (X,κ) be a reducible digital image. Let x be a reduction point for X. Let A be a freezing set for X. Then x ∈ A. Proof. Since x is a reduction point for X, by Lemma 3.4, there is a retraction r : X → X \ {x}. It follows that X \ {x} does not contain a freezing set for (X,κ). � Proposition 5.8. Let (X,c2) be a connected digital image in Z2. Suppose x0 ∈ X is such that Nc2 (x0) is connected and #Nc2 (x0) ∈ {1, 2, 3}. If A is a freezing set for (X,c2), then x0 ∈ A. Proof. By the proof of Theorem 3.6, we can use Lemma 3.4 to conclude that x0 is a reduction point. The assertion follows from Corollary 5.7. � Proposition 5.8 cannot in general be extended to permit #Nc2 (x0) = 4, as shown in the following. Example 5.9. Let X = {xi}4i=0 ⊂ Z 2, where x0 = (0, 0), x1 = (0,−1), x2 = (1, 0), x3 = (0, 1), x4 = (−1, 1). See Figure 2. Then Nc2 (x0) is c2-connected and #Nc2 (x0) = 4. It is easily seen that X \{x0} is a freezing set for (X,c2). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 119 L. Boxer 5.2. Boundaries and freezing sets. For any digital image (X,κ), clearly X is a freezing set. An interesting question is how small A ⊂ X can be for A to be a freezing set for X. We say a freezing set A is minimal if no proper subset of A is a freezing set for X. Definition 5.10. Let X ⊂ Zn. • The boundary of X [15] is Bd(X) = {x ∈ X | there exists y ∈ Zn \X such that y ↔c1 x}. • The interior of X is int(X) = X \Bd(X). Proposition 5.11. Let a < b, [a,b]Z ⊂ [c,d]Z, and let f : [a,b]Z → [c,d]Z be c1-continuous. • If {a,b}⊂ Fix(f), then [a,b]Z = Fix(f). • Bd([a,b]Z) = {a,b} is a minimal freezing set for [a,b]Z. Proof. If [a,b]Z 6= Fix(f), then we have at least one of the following: • For some smallest t0 satisfying a < t0 < b, f(t0) > t0. But then f(t0 − 1) ≤ t0 − 1, so f(t0 − 1) 6-c1 f(t0), contrary to the continuity of f. • For some largest t1 satisfying a < t1 < b, f(t1) < t1. But then f(t1 + 1) ≥ t1 + 1, so f(t1 + 1) 6-c1 f(t1), contrary to the continuity of f. It follows that f|[a,b]Z is an inclusion function, as asserted. By taking [c,d]Z = [a,b]Z and considering all f ∈ C([a,b]Z,c1) such that {a,b}⊂ Fix(f), we conclude that {a,b} is a freezing set for [a,b]Z. To establish minimality, observe that the proper nonempty subsets B of {a,b} allow constant functions c that are c1-continuous non-identities with c|B = idB. � Proposition 5.12. Let X ⊂ Zn be finite. Let 1 ≤ u ≤ n. Let A ⊂ X. Let f ∈ C(X,cu). If Bd(A) ⊂ Fix(f), then A ⊂ Fix(f). Proof. By hypothesis, it suffices to show int(A) ⊂ Fix(f). Let x = (x1, . . . ,xn) ∈ int(A). Suppose, in order to obtain a contradiction, x 6∈ Fix(f). Then for some index j, (5.1) pj(f(x)) 6= xj. Since X is finite, there exists a path P = {yi = (x1, . . . ,xj−1,ai,xj+1 . . . ,xn)}mi=1 in X such that a1 < xj < am and ai+1 = ai + 1; y1,ym ∈ Bd(A); and {yi}m−1i=2 ⊂ int(A). Note x ∈ P . Now, (5.1) implies either pj(f(x)) < xj or pj(f(x)) > xj. If the former, then by Lemma 5.5, ym 6∈ Fix(f); and if the lat- ter, then by Lemma 5.5, y1 6∈ Fix(f); so in either case, we have a contradiction. We conclude that x ∈ Fix(f). The assertion follows. � Theorem 5.13. Let X ⊂ Zn be finite. Then for 1 ≤ u ≤ n, Bd(X) is a freezing set for (X,cu). Proof. The assertion follows from Proposition 5.12. � c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 120 Fixed point sets in digital topology, 2 Without the finiteness condition used in Proposition 5.12 and in Theo- rem 5.13, the assertions would be false, as shown in the following. Example 5.14. Let X = {(x,y) ∈ Z2 |y ≥ 0}. Consider the function f : X → X defined by f(x,y) = { (x, 0) if y = 0; (x + 1,y) if y > 0. Then f ∈ C(X,c2), Bd(X) = Z×{0}, and f|Bd(X) = idBd(X), but X 6⊂ Fix(f), so Bd(X) is not a c2-freezing set for X. 5.3. Digital cubes and c1. In this section, we consider freezing sets for digital cubes using the c1 adjacency. Theorem 5.15. Let X = Πni=1[0,mi]Z. Let A = Π n i=1{0,mi}. • Let Y = Πni=1[ai,bi]Z be such that [0,mi] ⊂ [ai,bi]Z for all i. Let f : X → Y be c1-continuous. If A ⊂ Fix(f), then X ⊂ Fix(f). • A is a freezing set for (X,c1); minimal for n ∈{1, 2}. Proof. The first assertion has been established for n = 1 at Proposition 5.11. We can regard this as a base case for an argument based on induction on n, and we now assume the assertion is established for n ≤ k where k ≥ 1. Now suppose n = k + 1 and f : X → Y is c1-continuous with A ⊂ Fix(f). Let X0 = Π k i=1[0,mi]Z ×{0}, X1 = Π k i=1[0,mi]Z ×{mk+1}. We have that f|X0 and f|X1 are c1-continuous, A ∩ X0 ⊂ Fix(f|X0 ), and A∩X1 ⊂ Fix(f|X1 ). Since X0 and X1 are isomorphic to k-dimensional digital cubes, by Theorem 5.3 and the inductive hypothesis, we have( Πki=1[0,mi]Z ×{0} ) ∪ ( Πki=1[0,mi]Z ×{mn} ) ⊂ Fix(f). Then given x = (x1, . . . ,xn) ∈ X, x is a member of the unique shortest c1-path {(x1,x2, . . . ,xk, t)}m1t=0 from (x1,x2, . . . ,xk, 0) ∈ A to (x1,x2, . . . ,xk,mn) ∈ A. By Proposition 5.4, x ∈ Fix(f). Since x was taken arbitrarily, this completes the induction proof that X ⊂ Fix(f). By taking Y = X and applying the above to all f ∈ C(X,c1) such that A ⊂ Fix(f), we conclude that A is a freezing set for (X,c1). Minimality of A for n = 1 was established at Proposition 5.11. To show minimality of A for n = 2, consider a proper subset A′ of A. Without loss of generality, (0, 0) ∈ A\A′, m1 > 0, and m2 > 0. For x ∈ X, let g : X → X be the function g(x) = { x if x 6= (0, 0); (1, 1) if x = (0, 0). Suppose y ∈ X is such that y ↔c1 (0, 0). Then y = (1, 0) or y = (0, 1), hence g(y) = y ↔c1 (1, 1) = g(0, 0). Thus g ∈ C(X,c1), A′ ⊂ Fix(g), and g 6= idX. Therefore, A′ is not a freezing set for (X,c1), so A is minimal. � c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 121 L. Boxer Figure 3. The function g in the proof of Example 5.16. Mem- bers of A\{(0, 0, 0)} are circled. Straight line segments indicate c1 adjacencies. Curved arrows show the mapping for points in X \ Fix(g). The minimality assertion of Theorem 5.15 does not extend to n = 3, as shown in the following. Example 5.16. Let X = [0, 1]3Z. Let A = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}. See Figure 3. Then A is a minimal freezing set for (X,c1). Proof. Note if x ∈ X \ A then for each index i ∈ {1, 2, 3}, x is c1-adjacent to yi ∈ A such that x and yi differ in the ith coordinate. Therefore, if f ∈ C(X,c1) such that f(x) 6= x, then c1-continuity requires that for some i we have f(yi) 6= yi. It follows that A is a freezing set for (X,c1). Minimality is shown as follows. Let A′ be a proper subset of A. Without loss of generality, (0, 0, 0) ∈ A\A′. Let g : X → X be the function (see Figure 3) g(x) =   (1, 1, 0) if x = (0, 0, 0); (1, 1, 1) if x = (0, 0, 1); x otherwise. Then g ∈ C(X,c1), g|A′ = idA′ , and g 6= idX. Therefore, A′ is not a freezing set for (X,c1). � 5.4. Digital cubes and cn. In this section, we consider freezing sets for digital cubes in Zn, using the cn adjacency. Theorem 5.17. Let X = ∏n i=1[0,mi]Z ⊂ Z n, where mi > 1 for all i. Then Bd(X) is a minimal freezing set for (X,cn). Proof. That Bd(X) is a freezing set for (X,cn) follows from Theorem 5.13. To show Bd(X) is a minimal freezing set, it suffices to show that if A is a proper subset of Bd(X) then A is not a freezing set for (X,cn). We must show that there exists (5.2) f ∈ C(X,cn) such that f|A = idA but f 6= idX . c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 122 Fixed point sets in digital topology, 2 By hypothesis, there exists y = (y1, . . . ,yn) ∈ Bd(X) \A. Since y ∈ Bd(X), for some index j, yj ∈{0,mj}. • If yj = 0 the function f : X → X defined by f(y) = (y1, . . . ,yj−1, 1,yj+1, . . . ,yn), f(x) = x for x 6= y, satisfies (5.2). • If yj = mj the function f : X → X defined by f(y) = (y1, . . . ,yj−1,mj − 1,yj+1, . . . ,yn), f(x) = x for x 6= y, satisfies (5.2). The assertion follows. � 5.5. Freezing sets and the normal product adjacency. In the following, pj : ∏v i=1 Xi → Xj is the map pj(x1, . . . ,xv) = xj where xi ∈ Xi. Theorem 5.18. Let (Xi,κi) be a digital image, i ∈ [1,v]Z. Let X = ∏v i=1 Xi. Let A ⊂ X. Suppose A is a freezing set for (X,NPv(κ1, . . . ,κv)). Then for each i ∈ [1,v]Z, pi(A) is a freezing set for (Xi,κi). Proof. Let fi ∈ C(Xi,κi). Let F : X → X be defined by F(x1, . . . ,xv) = (f1(x1), . . . ,fv(xv)). Then by Theorem 2.7, F ∈ C(X,NPv(κ1, . . . ,κv)). Suppose for all a = (a1, . . . ,av) ∈ A, F(a) = a, hence fi(ai) = ai for all ai ∈ pi(A). Since A is a freezing set of X, we have that F = idX, and therefore, fi = idXi . The assertion follows. � 5.6. Cycles. A cycle or digital simple closed curve of n distinct points is a digital image (Cn,κ) with Cn = {xi}n−1i=0 such that xi ↔κ xj if and only if j = i + 1 mod n or j = i− 1 mod n. Given indices i < j, there are two distinct paths determined by xi and xj in Cn, consisting of the sets Pi,j = {xk} j k=i and P ′ i,j = Cn \{xk} j−1 k=i+1. If one of these has length less than n/2, it is the shorter path from pi to pj and the other is the longer path; otherwise, both have length n/2, and each is a shorter path and a longer path from pi to pj. In this section, we consider minimal fixed point sets for f ∈ C(Cn) that force f to be an identity map. Theorem 5.19. Let n > 4. Let xi,xj,xk be distinct members of Cn be such that Cn is a union of unique shorter paths determined by these points. Let f ∈ C(Cn,κ). Then f = idCn if and only if {xi,xj,xk}⊂ Fix(f); i.e., {xi,xj,xk} is a freezing set for Cn. Further, this freezing set is minimal. Proof. Clearly f = idCn implies {xi,xj,xk}⊂ Fix(f). Suppose {xi,xj,xk} ⊂ Fix(f). By hypothesis, there are unique shorter paths P0 from xi to xj, P1 from xj to xk, and P2 from xk to xi, in Cn. By c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 123 L. Boxer Proposition 5.4, each of P0, P1, and P2 is contained in Fix(f). By hypothesis Cn = P0 ∪P1 ∪P2, so f = idCn . Hence {xi,xj,xk} is a freezing set. For any distinct pair xi,xj ∈ Cn, there is a non-identity continuous self-map on Cn that takes a longer path determined by xi and xj to a shorter path determined by xi and xj. Thus, {xi,xj} is not a freezing set for Cn, so the set {xi,xj,xk} discussed above is minimal. � Remark 5.20. In Theorem 5.19, we need the assumption that n > 4, as there is a continuous self-map f on C4 with 3 fixed points such that f 6= idC4 [10]. 5.7. Wedges. Let (X,κ) ⊂ Zn be such that X = X0 ∪X1, where X0 ∩X1 = {x0}; and if x ∈ X0, y ∈ X1, and x ↔κ y, then x0 ∈ {x,y}. We say X is the wedge of X0 and X1, denoted X = X0 ∨X1. We say x0 is the wedge point. Theorem 5.21. Let A be a freezing set for (X,κ), where X = X0 ∨X1 ⊂ Zn, #X0 > 1, and #X1 > 1. Let X0 ∩X1 = {x0}. Then A must include points of X0 \{x0} and X1 \{x0}. Proof. Otherwise, either A ⊂ X0 or A ⊂ X1. Suppose A ⊂ X0. Then the function f : X → X given by f(x) = { x if x ∈ X0; x0 if x ∈ X1, belongs to C(X,κ) and satisfies f|A = idA, but f 6= idX. Thus A is not a freezing set for (X,κ). The case A ⊂ X1 is argued similarly. � Example 5.22. The wedge of two digital intervals is (isomorphic to) a digital interval. It follows from Theorem 5.3 and Proposition 5.11 that a freezing set for a wedge need not include the wedge point. Theorem 5.23. Let Cm and Cn be cycles, with m > 4, n > 4. Let x0 be the wedge point of X = Cm∨Cn. Let xi,xj ∈ Cm and x′k,x ′ p ∈ Cn be such that Cm is the union of unique shorter paths determined by xi,xj,x0 and Cn is the union of unique shorter paths determined by x′k,x ′ p,x0. Then A = {xi,xj,x′k,x ′ p} is a freezing set for X. Proof. Let f ∈ C(X,κ) be such that A ⊂ Fix(f). Let P0 be the unique shorter path in Cm from xi to xj; let P1 be the unique shorter path in Cm from xj to x0; let P2 be the unique shorter path in Cm from x0 to xi; let P ′ 0 be the unique shorter path in Cn from x ′ k to x ′ p; let P ′ 1 be the unique shorter path in Cn from x′p to x0; let P ′ 2 be the unique shorter path in Cn from x0 to x ′ k. By Proposition 5.4, each of the following paths is contained in Fix(f): P0, P1 ∪P ′1 (from xj to x0 to x′p), P2 ∪P ′2 (from xi to x0 to x′k), and P ′ 0. Since X = P0 ∪ (P1 ∪P ′1) ∪ (P2 ∪P ′ 2) ∪P ′ 0 ⊂ Fix(f), the assertion follows. � c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 124 Fixed point sets in digital topology, 2 5.8. Trees. A tree is an acyclic graph (X,κ) that is connected, i.e., lacking any subgraph isomorphic to Cn for n > 2. The degree of a vertex x in X is the number of distinct vertices y ∈ X such that x ↔ y. A vertex of a tree may be designated as the root. We have the following. Lemma 5.24 ([10]). Let (X,κ) be a digital image that is a tree in which the root vertex has at least 2 child vertices. Then f ∈ C(X,κ) implies Fix(f) is κ-connected. Theorem 5.25. Let (X,κ) be a digital image such that the graph G = (X,κ) is a finite tree with #X > 1. Let E be the set of vertices of G that have degree 1. Then E is a minimal freezing set for G. Proof. First consider the case that each vertex has degree 1. Since X is a tree, it follows that X = {x0,x1} = E, and E is a freezing set. E must be minimal, since X admits constant functions that are identities on their restrictions to proper subsets of E. Otherwise, there exists x0 ∈ X such that x0 has degree of at least 2 in G. This implies #X > 2, and since G is finite and acyclic, #E > 0. Since G is acyclic, removal of any member of X \E would disconnect X. If we take x0 to be the root vertex, it follows from Lemma 5.24 that E is a freezing set. Since #E > 0, for any y ∈ E there exists y′ ∈ X\E such that y′ ↔ y. Then the function f : X → X defined by f(x) = { y′ if x = y; x if x 6= y, satisfies f ∈ C(X,κ), f|E\{y} = idE\{y}, and f 6= idX. Thus E \{y} is not a freezing set. Since y was arbitrarily chosen, E is minimal. � 6. s-Cold sets In this section, we generalize our focus from fixed points to approximate fixed points and, more generally, to points constrained in the amount they can be moved by continuous self-maps in the presence of fixed point sets. We obtain some analogues of our previous results for freezing sets. 6.1. Definition and basic properties. In the following, we use the path- length metric d for connected digital images (X,κ), defined [13] as d(x,y) = min{` |` is the length of a κ-path in X from x to y}. If X is finite and κ-connected, the diameter of (X,κ) is diam(X,κ) = max{d(x,y) |x,y ∈ X}. We introduce the following generalization of a freezing set. Definition 6.1. Given s ∈ N∗, we say A ⊂ X is an s-cold set for the connected digital image (X,κ) if given g ∈ C(X,κ) such that g|A = idA, then for all x ∈ X, d(x,g(x)) ≤ s. A cold set is a 1-cold set. Note a 0-cold set is a freezing set. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 125 L. Boxer Theorem 6.2. Let (X,κ) be a connected digital image. Let A ⊂ X. Then A ⊂ Fix(g) is an s-cold set for (X,κ) if and only if for every isomorphism F : (X,κ) → (Y,λ), if g : X → Y is (κ,λ)-continuous and F |A = g|A, then for all x ∈ X, d(F(x),g(x)) ≤ s. Proof. Suppose A is an s-cold set for (X,κ). Then for all f ∈ C(X,κ) such that f|A = idA and all x ∈ X, we have d(x,f(x)) ≤ s. Let F : (X,κ) → (Y,λ) be an isomorphism. Let g : X → Y be (κ,λ)-continuous with F |A = g|A. Then idA = F −1 ◦F|A = F−1 ◦g|A. Let x ∈ X. Then d(x,F−1 ◦g(x)) ≤ s, i.e., there is a κ-path P in X of length at most s from x to F−1 ◦g(x). Therefore, F(P) is a λ-path in Y of length at most s from F(x) to F ◦F−1 ◦g(x) = g(x), i.e., d(F(x),g(x)) ≤ s. Suppose A ⊂ X and for every isomorphism F : (X,κ) → (Y,λ), if g : X → Y is (κ,λ)-continuous and F|A = g|A, then for all x ∈ X, d(F(x),g(x)) ≤ s. Let f ∈ C(X,κ) with f|A = idA. Since idX is an isomorphism, for all x ∈ X, d(x,f(x)) ≤ s. Thus, A is an s-cold set for (X,κ). � Given a digital image (X,κ) and f ∈ C(X,κ), a point x ∈ X is an almost fixed point of f [16] or an approximate fixed point of f [7] if f(x) -κ x. Remark 6.3. The following are easily observed. • If A ⊂ A′ ⊂ X and A is an s-cold set for (X,κ), then A′ is an s-cold set for (X,κ). • A is a cold set (i.e., a 1-cold set) for (X,κ) if and only if given f ∈ C(X,κ) such that f|A = idA, every x ∈ X is an approximate fixed point of f. • In a finite connected digital image (X,κ), every nonempty subset of X is a diam(X)-cold set. • If s0 < s1 and A is an s0-cold set for (X,κ), then A is an s1-cold set for (X,κ). Note a freezing set is a cold set, but the converse is not generally true, as shown in the following. Example 6.4. It follows from Definition 6.1 that {0} is a cold set, but not a freezing set, for X = [0, 1]Z, since the constant function g with value 0 satisfies g|{0} = id{0}, and g(1) = 0 ↔c1 1. s-cold sets are invariant in the sense of the following. Theorem 6.5. Let (X,κ) be a connected digital image, let A be an s-cold set for (X,κ), and let F : (X,κ) → (Y,λ) be an isomorphism. Then F(A) is an s-cold set for (Y,λ). Proof. Let f ∈ C(Y,λ) such that f|F(A) = idF(A). Then f ◦F|A = f|F(A) ◦F|A = idF(A) ◦F|A = F|A. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 126 Fixed point sets in digital topology, 2 By Theorem 6.2, for all x ∈ X, d(f ◦F(x),F(x)) ≤ s. Substituting y = F(x), we have that y ∈ Y implies d(f(y),y) ≤ s. By Definition 6.1, F(A) is a cold set for (Y,λ). � A is a κ-dominating set (or a dominating set when κ is understood) for (X,κ) if for every x ∈ X there exists a ∈ A such that x -κ a [11]. This notion is somewhat analogous to that of a dense set in a topological space, and the following is somewhat analogous to the fact that in topological spaces, a continuous function is uniquely determined by its values on a dense subset of the domain. Theorem 6.6. Let (X,κ) be a digital image and let A be κ-dominating in X. Then A is 2-cold in (X,κ). Proof. Let f ∈ C(X,κ) such that f|A = idA. Since A is κ-dominating, for every x ∈ X there is an a ∈ A such that x - a. Then f(x) - f(a) = a. Thus, we have the path {x,a,f(x)} ⊂ X from x to f(x) of length at most 2. The assertion follows. � Theorem 6.7. Let (X,κ) be rigid. If A is a cold set for X, then A is a freezing set for X. Proof. Let f ∈ C(X,κ) be such that f|A = idA. Since A is cold, f(x) - x for all x ∈ X. Therefore, the map H : X × [0, 1]Z → X defined by H(x, 0) = x, H(x, 1) = f(x), is a homotopy. Since X is rigid, f = idX. The assertion follows. � 6.2. Cold sets for cubes. In this section, we consider cold sets for digital cubes in Zn. Note the hypotheses of Proposition 6.8 imply A is c1- and c2- dominating in Bd(X). Proposition 6.8. Let m,n ∈ N. Let X = [0,m]Z × [0,n]Z. Let A ⊂ Bd(X) be such that no pair of c1-adjacent members of Bd(X) belong to Bd(X) \ A. Then A is a cold set for (X,c2). Further, for all f ∈ C(X,c2), if f|A = idA then f|Int(X) = id |Int(X). Proof. Let x = (x0,y0) ∈ X. Let f ∈ C(X,c2) such that f|A = idA. Consider the following. • If x ∈ A then f(x) = x. • If x ∈ Bd(X) \A then both of the c1-neighbors of x in Bd(X) belong to A. We will show f(x) -c2 x. Let K = {(0, 0), (0,n), (m, 0), (m,n)}⊂ Bd(X). – For x ∈ K, consider the case x = (0, 0). Then {(0, 1), (1, 0)}⊂ A, so we must have f(x) ∈ N∗c2 ((0, 1)) ∩N ∗ c2 ((1, 0)) ⊂ N∗c2 (x). For other x ∈ K, we similarly find f(x) -c2 x. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 127 L. Boxer Figure 4. Illustration of the proof of Proposition 6.8 for the case (x0,y0) ∈ Int(X). X = [0, 6]Z × [0, 4]Z. Members of the set A ⊂ Bd(X) are marked “a”. Corner points such as (0, 4) need not belong to A; also, although we cannot have c1-adjacent members of Bd(X) in Bd(X)\A, we can have c2- adjacent members of Bd(X) in Bd(X)\A, e.g., (5, 4) and (6, 3). The heavy polygonal line illustrates a c2-path P of length n = 4: P(0) = qL = (2, 0), P(1) = (1, 1), P(2) = (x0,y0) = (1, 2), P(3) = (1, 3), P(4) = qU = (1, 4). – For x ∈ Bd(X) \ K, consider the case x = (t, 0). For this case, {(t− 1, 0), (t + 1, 0)}⊂ A, so (t− 1, 0) = f(t− 1, 0) -c2 f(x) -c2 f(t + 1, 0) = (t + 1, 0). Therefore, f(x) ∈{x, (t, 1)}, so f(x) -c2 x. For other x ∈ Bd(X) \K, we similarly find f(x) -c2 x. • If x ∈ Int(X), let L = {(z, 0)}x0+1z=x0−1 and U = {(z,n)} x0+1 z=x0−1. We have L∩A 6= ∅ 6= U ∩A. Since no pair of c1-adjacent members of Bd(X) belong to Bd(X)\A, there exist qL ∈ L∩A, qU ∈ U ∩A such that |p1(qL) −x0| ≤ 1 and |p1(qU ) −x0| ≤ 1. Thus, there is an injective c2-path P : [0,n]Z → X such that P([0,y0)]Z) runs from qL to x and P([y0,n]Z) runs from x to qU (note since we use c2-adjacency, there can be steps of the path that change both coordi- nates - see Figure 4). Therefore, f ◦ P is a path from f(qL) = qL to f(x) to f(qU ) = qU , and p2◦f◦P is a path from p2(qL) = 0 to p2(f(x)) to p2(qU ) = n. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 128 Fixed point sets in digital topology, 2 If y′ = p2(f(x)) > y0, then p2 ◦f ◦P |[0,y0]Z is a c2-path of length y0 from 0 to y′, which is impossible. Similarly, if y′ < y0, then p2 ◦ f ◦ P |[y0,n]Z is a c2-path of length n−y0 from y ′ to n, which is impossible. Therefore, we must have (6.1) p2 ◦f(x) = y0. Similarly, by replacing the neighborhoods of the projections of x on the lower and upper edges of the cube, L and U, by the neighborhoods of the projections of x on the the left and right edges of the cube, L′ = {(0,z)}y0+1z=y0−1 and R = {(m,z)} y0+1 z=y0−1, and using an argument similar to that used to obtain (6.1), we conclude that (6.2) p1 ◦f(x) = x0. It follows from (6.2) and (6.1) that f(x) = x. Thus, in all cases, f(x) -c2 x, and f|Int(X) = idInt(X). � Proposition 6.9. Let m,n ∈ N. Let X = [0,m]Z × [0,n]Z. Let A ⊂ Bd(X) be c1-dominating in Bd(X). Then A is a 2-cold set for (X,c2). Further, for all f ∈ C(X,c2), if f|A = idA then f|Int(X) = id |Int(X). Proof. Our argument is similar to that of Proposition 6.8. Let x = (x0,y0) ∈ X. Let f ∈ C(X,c2) such that f|A = idA. Consider the following. • If x ∈ A then f(x) = x. • If x ∈ Bd(X) \A then for some a ∈ A, x -c1 a. Therefore, f(x) -c1 f(a) = a. Thus, {x,a,f(x)} is a path in X from x to f(x) of length at most 2. • If x ∈ Int(X), then as in the proof of Proposition 6.8 we have that f(x) = x. Thus, in all cases, d(f(x),x) ≤ 2, and f|Int(X) = idInt(X). � An example of a 2-cold set A that is not a 1-cold set, such that A is as in Proposition 6.9, is given in the following. Example 6.10. Let X = [0, 2]2Z. Let A = {(0, 2), (1, 0), (2, 2)}⊂ X. Then A is c1-dominating in Bd(X), so by Proposition 6.9, is a 2-cold set for (X,c2). Let f : X → X be the function f(0, 0) = (2, 0), f(0, 1) = (1, 1), and f(x) = x for all x ∈ X \ {(0, 0), (0, 1)}. Then f ∈ C(X,c2) but d((0, 0),f(0, 0)) = 2, so A is not a 1-cold set. Proposition 6.11. Let X = ∏n i=1[0,mi]Z ⊂ Z n, where mi > 1 for all i. Let A ⊂ Bd(X) be such that A is not cn-dominating in Bd(X). Then A is not a cold set for (X,cn). Proof. By hypothesis, there exists y = (y1, . . . ,yn) ∈ Bd(X) \ A such that N(y,cn) ∩A = ∅. Since y ∈ Bd(X), for some index j we have yj ∈ {0,mj}. Let x = (x1, . . . ,xn) ∈ X, for xi ∈ [0,mi]Z. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 129 L. Boxer • If yj = 0, let f : X → X be defined as follows. f(x) =   (x1, . . . ,xj−1, 2,xj+1, . . . ,xn) if x = y; (x1, . . . ,xj−1, 1,xj+1, . . . ,xn) if x ∈ Ncn (y); x otherwise. If u,v ∈ X, u -cn v, then u and v differ by at most 1 in every coordi- nate. Consider the following cases. – If u = y, then v ∈ Ncn (y), and clearly f(u) and f(v) differ by at most 1 in every coordinate, hence are cn-adjacent. Similarly if v = y. – If u,v ∈ Ncn (y), then clearly f(u) and f(v) differ by at most 1 in every coordinate, hence are cn-adjacent. – If u ∈ Ncn (y) and v 6∈ Ncn (y), then pj(u) ∈ {0, 1}, so pj(f(v)) = pj(v) ∈{0, 1, 2}, and pj(f(u)) = 1. It follows easily that f(u) and f(v) differ by at most 1 in every coordinate, hence are cn-adjacent. Similarly if v ∈ Ncn (y) and u 6∈ Ncn (y) – Otherwise, {u,v}∩N∗cn (y) = ∅, so f(u) = u -cn v = f(v). Therefore, f ∈ C(X,cn). • If yj = mj, let f : X → X be defined by f(x) =   (x1, . . . ,xj−1,mj − 2,xj+1, . . . ,xn) if x = y; (x1, . . . ,xj−1,mj − 1,xj+1, . . . ,xn) if x ∈ Ncn (y); x otherwise. By an argument similar to that of the case yj = 0, we conclude that f ∈ C(X,cn). Further, in both cases, f|A = idA, and f(y) 6-cn y. The assertion follows. � 6.3. s-cold sets for rectangles in Z2. The following generalizes the case n = 2 of Theorem 5.15. Proposition 6.12. Let X = [−m,m]Z × [−n,n]Z ⊂ Z2, s ∈ N∗, where s ≤ min{m,n}. Let A = {(−m + s,−n + s), (−m + s,n−s), (m−s,−n + s), (m−s,n−s)}. Then A is a 4s-cold set for (X,c1). Proof. Let f ∈ C(X,c1) such that f|A = idA. Let A′ = [−m + s,m−s]Z × [−n + s,n−s]Z. By Proposition 5.4, Bd(A′) ⊂ Fix(f). It follows from Proposition 5.12 that A′ ⊂ Fix(f). Thus it remains to show that x ∈ X \ A′ implies d(x,f(x)) ≤ 4s. This is seen as follows. For x ∈ X \A′, there exists a c1-path P of length at most 2s from x to some y ∈ Bd(A′). Then f(P) is a c1-path from f(x) to f(y) = y of length at most 2s. Therefore, P ∪f(P) contains a path from x to y to f(x) of length at most 4s. The assertion follows. � The following generalizes Proposition 6.8. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 130 Fixed point sets in digital topology, 2 Proposition 6.13. Let X = [−m,m]Z×[−n,n]Z ⊂ Z2, s ∈ N∗, where m−s ≥ 0, n−s ≥ 0. Let A = [−m + s,m−s]Z × [−n + s,n−s]Z ⊂ X. Let A′ ⊂ Bd(A) such that no pair of c1-adjacent members of Bd(A) belongs to Bd(A) \A′. Then A′ is a 2s-cold set for (X,c2). Further, if f ∈ C(X,c2) and f|A′ = idA′ , then f|A = idA. Proof. Let f ∈ C(X,c2) be such that f|A′ = idA′ . As in the proof of Proposi- tion 6.8, f|A = idA. Now consider x ∈ X \A. There is a c2-path P in X from x to some y ∈ A′ of length at most s. Then f(P) is a c2-path in X from f(x) to f(y) = y of length at most s. Therefore, P ∪f(P) contains a c2-path in X from x to y to f(x) of length at most 2s. The assertion follows. � 6.4. s-cold sets for Cartesian products. We modify the proof of Theo- rem 5.18 to obtain the following. Theorem 6.14. Let (Xi,κi) be a digital image, i ∈ [1,v]Z. Let X = ∏v i=1 Xi. Let s ∈ N∗. Let A ⊂ X. Suppose A is an s-cold set for (X,NPv(κ1, . . . ,κv)). Then for each i ∈ [1,v]Z, pi(A) is an s-cold set for (Xi,κi). Proof. Let fi ∈ C(Xi,κi). Let F : X → X be defined by F(x1, . . . ,xv) = (f1(x1), . . . ,fv(xv)). Then by Theorem 2.7, F ∈ C(X,NPv(κ1, . . . ,κv)). Suppose for all i, ai ∈ pi(A), we have fi(ai) = ai. Note this implies, for a = (a1, . . . ,av), that F(a) = a. Since a is an arbitrary member of the s-cold set A of X, we have that d(F(x),x) ≤ s, for all x = (x1, . . . ,xv) ∈ X, xi ∈ Xi, and therefore, d(fi(xi),xi) ≤ s. The assertion follows. � 6.5. s-cold sets for infinite digital images. In this section, we obtain pro- perties of s-cold sets for some infinite digital images. Theorem 6.15. Let (Zn,cu) be a digital image, 1 ≤ u ≤ n. Let A ⊂ Zn. Let s ∈ N∗. If A is an s-cold set for (Zn,cu), then for every index i, pi(A) is an infinite set, with sequences of members tending both to ∞ and to −∞. Proof. Suppose otherwise. Then for some i, there exist m or M in Z such that m = min{pi(a) |a ∈ A} or M = max{pi(a) |a ∈ A}. If the former, then for z = (z1, . . . ,zn) ∈ Zn, define f : Zn → Zn by f(z) = { (z1, . . . ,zi−1,m,zi+1, . . . ,zn) if zi ≤ m; z otherwise. Then f ∈ C(Zn,cu) and f|A = idA, but f 6= idZn . Thus, A is not an s-cold set. Similarly, if M < ∞ as above exists, we conclude A is not an s-cold set. � Corollary 6.16. A ⊂ Z is a freezing set for (Z,c1) if and only if A contains sequences {ai}∞i=1 and {a ′ i} ∞ i=1 such that limi→∞ai = ∞ and limi→∞a ′ i = −∞. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 131 L. Boxer Proof. This follows from Lemma 5.5 and Theorem 6.15. � The converse of Theorem 6.15 is not generally correct, as shown by the following. Example 6.17. Let A = {(z,z) |z ∈ Z} ⊂ Z2. Then although p1(A) = p2(A) = Z contains sequences tending to ∞ and to −∞, A is not an s-cold set for (Z2,c2), for any s. Proof. Consider f : Z2 → Z2 defined by f(x,y) = (x,x). We have f ∈ (Z2,c2) and f|A = idA, but one sees easily that for all s there exist (x,y) ∈ Z2 such that d((x,y),f(x,y)) > s. � 7. Further remarks We have continued the work of [10] in studying fixed point invariants and related ideas in digital topology. We have introduced pointed versions of rigidity and fixed point spectra. We have introduced the notions of freezing sets and s-cold sets. These show us that although knowledge of the fixed point set Fix(f) of a continuous self- map f on a connected topological space X generally gives us little information about the nature of f|X\Fix(f), if f ∈ C(X,κ) then f|X\Fix(f) may be severely limited if A ⊂ Fix(f) is a freezing set or, more generally, an s-cold set for (X,κ). Acknowledgements. P. Christopher Staecker and an anonymous reviewer were most helpful. 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