aa r X i v : . [ m a t h . G T ] F e b Fixed Point Sets in Digital Topology, 2
Laurence Boxer ∗ Abstract
We continue the work of [10], studying properties of digital imagesdetermined by fixed point invariants. We introduce pointed versions ofinvariants that were introduced in [10]. We introduce freezing sets andcold sets to show how the existence of a fixed point set for a continuousself-map restricts the map on the complement of the fixed point set.
As stated in [10]:Digital images are often used as mathematical models of real-worldobjects. A digital model of the notion of a continuous function,borrowed from the study of topology, is often useful for the studyof digital images. However, a digital image is typically a finite,discrete point set. Thus, it is often necessary to study digital imagesusing methods not directly derived from topology. In this paper, weexamine some properties of digital images concerned with the fixedpoints of digitally continuous functions; among these properties arediscrete measures that are not natural analogues of properties ofsubsets of R n .In [10], we studied rigidity, pull indices, fixed point spectra for digital imagesand for digitally continuous functions, and related notions. In the current work,we study pointed versions of notions introduced in [10]. We also study suchquestions as when a set of fixed points Fix( f ) determines that f is an identityfunction, or is “approximately” an identity function.Some of the results in this paper were presented in [6]. Much of this section is quoted or paraphrased from [10]. ∗ Department of Computer and Information Sciences, Niagara University, Niagara Univer-sity, NY 14109, USA; and Department of Computer Science and Engineering, State Universityof New York at Buffalo email: [email protected] N denote the set of natural numbers; N ∗ = { }∪ N , the set of nonnegativeintegers; and Z , the set of integers. X will be used for the number of membersof a set X . A digital image is a pair (
X, κ ) where X ⊂ Z n for some n and κ is an adjacencyon X . Thus, ( X, κ ) is a graph for which X is the vertex set and κ determines theedge set. Usually, X is finite, although there are papers that consider infinite X . Usually, adjacency reflects some type of “closeness” in Z n of the adjacentpoints. When these “usual” conditions are satisfied, one may consider the digitalimage as a model of a black-and-white “real world” image in which the blackpoints (foreground) are represented by the members of X and the white points(background) by members of Z n \ { X } .We write x ↔ κ y , or x ↔ y when κ is understood or when it is unnecessaryto 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 c u adjacencies, defined asfollows. Let X ⊂ Z n and let u ∈ Z , 1 ≤ u ≤ n . Then for points x = ( x , . . . , x n ) = ( y , . . . , y n ) = y we have x ↔ c u y if and only if • for at most u indices i we have | x i − y i | = 1, and • for all indices j , | x j − y j | 6 = 1 implies x j = y j .The c u -adjacencies are often denoted by the number of adjacent points apoint can have in the adjacency. E.g., • in Z , c -adjacency is 2-adjacency; • in Z , c -adjacency is 4-adjacency and c -adjacency is 8-adjacency; • in Z , c -adjacency is 8-adjacency, c -adjacency is 18-adjacency, and c -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 productadjacency or strong adjacency on X × Y , N P ( κ, λ ), is defined as follows. Given x , x ∈ X , y , y ∈ Y such that p = ( x , y ) = ( x , y ) = p , we have p ↔ NP ( κ,λ ) p if and only if one of the following is valid: • x ↔ κ x and y = y , or 2 x = x and y ↔ λ y , or • x ↔ κ x and y ↔ λ y .Building on the normal product adjacency, we have the following. Definition 2.2. [4] Given u, v ∈ N , 1 ≤ u ≤ v , and digital images ( X i , κ i ),1 ≤ i ≤ v , let X = Π vi =1 X i . The adjacency N P u ( κ , . . . , κ v ) for X is defined asfollows. Given x i , x ′ i ∈ X i , let p = ( x , . . . , x v ) = ( x ′ , . . . , x ′ v ) = q. Then p ↔ NP u ( κ ,...,κ v ) q if for at least 1 and at most u indices i we have x i ↔ κ i x ′ i and for all other indices j we have x j = x ′ j .Notice N P ( κ, λ ) = N P ( κ, λ ) [4].Let x ∈ ( X, κ ). We use the notations N ( x ) = N κ ( x ) = { y ∈ X | y ↔ κ x } and N ∗ ( x ) = N ∗ κ ( x ) = N κ ( x ) ∪ { x } . We denote by id or id X 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 areunderstood, if for every κ -connected subset X ′ of X , f ( X ′ ) is a λ -connectedsubset 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 continuousfunctions 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 relatedmeanings. • A path from x to y in a digital image ( X, κ ) is a set { x i } mi =0 ⊂ X suchthat x = x , x m = y , and x i - κ x i +1 for i = 0 , , . . . , m −
1. If the x i aredistinct, 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 thisusage, { P (0) , . . . , P ( m ) } is a path in the previous sense.3 efinition 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 • for all x ∈ X , h ( x,
0) = f ( x ) and h ( x, m ) = g ( x ); • 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 (2 , κ ′ ) − continuous. That is, h x is a path in Y . • 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 ( 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 ′ areunderstood. 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 x ∈ X such that f ( x ) = g ( x ) = y ∈ Y and h ( x , t ) = y for all t ∈ [0 , m ] Z , then h is a pointed homotopy and f and g are pointedhomotopic [3].If there exist continuous f : ( X, κ ) → ( Y, λ ) and g : ( Y, λ ) → ( X, κ ) suchthat g ◦ f ∼ κ,κ id X and f ◦ g ∼ λ,λ id Y , then ( X, κ ) and (
Y, λ ) are homotopyequivalent .If there is a κ -homotopy between id X and a constant map, we say X is κ -contractible , or just contractible when κ is understood. Theorem 2.7. [4]
Let ( X i , κ i ) and ( Y i , λ i ) be digital images, ≤ i ≤ v . Let f i : X i → Y i . Then the product map f : Q vi =1 X i → Q vi =1 Y i defined by f ( x , . . . , x v ) = ( f ( x ) , . . . , f v ( x v )) for x i ∈ X i , is ( N P v ( κ , . . . , κ v ) , N P v ( λ , . . . , λ v )) -continuous if and only ifeach f i is ( κ i , λ i ) -continuous. Definition 2.8.
Let A ⊂ X . A κ -continuous function r : X → A is a retraction ,and A is a retract of X , if r ( a ) = a for all a ∈ A . If such a map r satisfies i ◦ r ∼ κ id X where i : A → X is the inclusion map, then A is a κ -deformationretract 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 − 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 . Wedenote by Fix( f ) the set { x ∈ X | x is a fixed point of f } . If x ∈ X \ Fix( f ), wesay f moves x . 4igure 1: (Figure 1 of [9].) The image X discussed in Example 3.1. Thecoordinates are ordered according to the axes in this figure. A function f : ( X, κ ) → ( Y, λ ) is rigid [10] when no continuous map is homotopicto 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 ( x ) = y , then f is pointed rigid [12] ifno continuous map is pointed homotopic to f other than f itself. When theidentity map id : ( X, x ) → ( X, x ) is pointed rigid, we say ( X, x ) is pointedrigid.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 homotopicfunctions are homotopic, but the converse is not always true.) We show in thefollowing that the converses of these assertions are not generally true. Example 3.1. [9] Let X = ([0 , Z × [0 , Z ) \ { (1 , , } . Let x = (0 , , ∈ X .See Figure 1. It was shown in [9] that X is 6-contractible (i.e., c -contractible)but ( X, x ) is not pointed 6-contractible. The proof of the latter uses an argu-ment that is easily modified to show that any homotopy of id X that moves somepoint must move x . It follows that id X is not rigid but is x -pointed rigid, i.e.,that X is not c -rigid but ( X, x ) is c -pointed rigid. Definition 3.2. [12] A finite image X is reducible if it is homotopy equivalentto 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 id X is homotopicto a nonsurjective map. Let (
X, κ ) be reducible. By Lemma 3.3, there exist x ∈ X and f ∈ C ( X, κ )such that id X ≃ κ f and x 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 latterpaper uses the notation “ N ( x )” for what we call “ N ∗ ( x )” or “ N ∗ κ ( x )”.5 emma 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 adeformation retract of X . Remark 3.5. [12]
A finite rigid image is irreducible.
Theorem 3.6.
Let ( X, c ) be a digital image in Z . Suppose there exists x ∈ X such that N c ( x ) is c -connected and N c ( x ) ∈ { , , } . Then ( X, c ) isreducible.Proof. We first show that in all cases, there exists y ∈ N c ( x ) such that N ∗ c ( x ) ⊂ N ∗ c ( y ).1. Suppose N c ( x ) = 1. Then there exists y ∈ X such that { y } = N c ( x ).Clearly, then, N ∗ c ( x ) ⊂ N ∗ c ( y ).2. Suppose N c ( x ) = 2. Then there exist distinct y, y ′ ∈ X such that { y, y ′ } = N c ( x ), which by hypothesis is connected. Therefore, { x , y ′ } ⊂ N c ( y ), so N ∗ c ( x ) ⊂ N ∗ c ( y ).3. Suppose N c ( x ) = 3. Then there exist distinct y, y , y ∈ X such that { y, y , y } = N c ( x ), which by hypothesis is connected. Therefore, oneof the members of N c ( x ), say, y , is adjacent to the other two. Thus, { x , y , y } ⊂ N c ( y ), so N ∗ c ( x ) ⊂ N ∗ c ( y ).Since in all cases we have N ∗ c ( x ) ⊂ N ∗ c ( y ), the assertion follows from Lemma 3.4. Remark 3.7.
If instead we use the c -adjacency, the analog of the previoustheorem is simpler, since if ( X, c ) is a digital image in Z and x ∈ X suchthat N c ( x ) is nonempty and c -connected, then N c ( x ) = 1 . This case issimilar to the case N c ( x ) = 1 of Theorem 3.6 above, so ( X, c ) is reducible. In this section, we define pointed versions of the homotopy fixed point spectrum 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 ) = { g ) | g ∼ κ f } . • Given f ∈ C ( X, κ ) and x ∈ Fix( f ), the pointed homotopy fixed pointspectrum of f is S ( f, x ) = { g ) | g ∼ κ f holding x fixed } . efinition 4.2. Let (
X, κ ) be a digital image. • [10] The fixed point spectrum of ( X, κ ) is F ( X ) = F ( X, κ ) = { f ) | f ∈ C ( X, κ ) } . • Given x ∈ X , the pointed fixed point spectrum of ( X, κ, x ) is F ( X, x ) = F ( X, κ, x ) = { f ) | f ∈ C ( X, κ ) , x ∈ Fix( f ) } . 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 thefollowing.
Theorem 4.4.
Let ( A, κ, x ) be a retract of ( X, κ, x ) . Then F ( A, κ, x ) ⊆ F ( X, κ, x ) . Theorem 4.5. [10]
Let X = [1 , a ] Z × [1 , b ] Z . Let κ ∈ { c , c } . Then S (id X , κ ) = F ( X, κ ) = { i } abi =0 . Example 4.6.
Consider the pointed digital image (
X, c , x ) of Example 3.1.Since f ∈ C ( X, c ) and x ∈ Fix( f ) imply f = id X , S (id X , c , x ) = { X } = { } . However, (
X, c ) is not rigid. It is easily seen that there is a c -deformationretraction of X to { ( x, y, ∈ X } , which is isomorphic to [1 , Z . It follows fromTheorem 4.3 and Theorem 4.5 that { i } i =0 ⊂ S (id X ). Since every f ∈ C ( X, c )such that f ≃ c id X and f = id X moves every point q of X such that p ( q ) = 1,it follows easily that S (id X , c ) = F ( X, c ) = { , , , , , , , , , , } . In this section, we consider subsets of Fix( f ) that determine that f ∈ C ( X, κ )must be the identity function id X . Interesting questions include what propertiessuch sets have, and how small they can be.In classical topology, given a connected set X ⊂ R n and a continuous self-map f on X , knowledge of a finite subset A of the fixed points of f rarely tellsus much about the behavior of f on X \ A . By contrast, we see in this sectionthat knowledge of a subset of the fixed points of a continuous self-map f on adigital image can completely characterize f as an identity map.7 .1 Definition and basic properties Definition 5.1.
Let (
X, κ ) be a digital image. We say A ⊂ X is a freezing setfor X if given g ∈ C ( X, κ ), A ⊂ Fix( g ) implies g = id X . Theorem 5.2.
Let ( X, κ ) be a digital image. Let A ⊂ X . The following areequivalent.1. A is a freezing set for X .2. id X is the unique extension of id A to a member of C ( X, κ ) .3. For every isomorphism F : X → ( Y, λ ) , if g : X → Y is ( κ, λ ) -continuousand F | A = g | A , then g = F .4. Any continuous g : A → Y has at most one extension to an isomorphism ¯ g : X → Y .Proof. ⇔ ⇒ 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 − ◦ g | A = F − ◦ F | A = id X | A = id A . Since the composition of digitally continuous functions is continuous, it followsby hypothesis that F − ◦ g = id X , and therefore that g = F ◦ ( F − ◦ g ) = F ◦ id X = F. ⇒ 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 = id X | A , so since id X is an isomorphism, g = id X .3) ⇒ ⇒ g to be the inclusion of A into X , whichextends to id X .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 ) = id Y | F ( A ) . Then g ◦ F | A = g | F ( A ) ◦ F | A = id Y | F ( A ) ◦ F | A = F | A . By Theorem 5.2, g ◦ F = F . Thus g = ( g ◦ F ) ◦ F − = F ◦ F − = id Y . By Definition 5.1, F ( A ) is a freezing set for ( Y, λ ).We will use the following. 8 roposition 5.4. [10]
Let ( X, κ ) be a digital image and f ∈ C ( X, κ ) . Suppose 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 p i : Z n → Z be the projection to the i th coordinate: p i ( z , . . . , z n ) = z i .The following assertion can be interpreted to say that in a c u -adjacency, acontinuous function that moves a point p also moves a point that is “behind” p . E.g., in Z , if q and q ′ are c - or c -adjacent with q left, right, above, orbelow q ′ , and a continuous function f moves q to the left, right, higher, orlower, respectively, then f also moves q ′ to the left, right, higher, or lower,respectively. Lemma 5.5.
Let ( X, c u ) ⊂ Z n be a digital image, ≤ u ≤ n . Let q, q ′ ∈ X besuch that q ↔ c u q ′ . Let f ∈ C ( X, c u ) .1. If p i ( f ( q )) > p i ( q ) > p i ( q ′ ) then p i ( f ( q ′ )) > p i ( q ′ ) .2. If p i ( f ( q )) < p i ( q ) < p i ( q ′ ) then p i ( f ( q ′ )) < p i ( q ′ ) .Proof.
1. Suppose p i ( f ( q )) > p i ( q ) > p i ( q ′ ). Since q ↔ c u q ′ , if p i ( q ) = m then p i ( q ′ ) = m −
1. Then p i ( f ( q )) > m . By continuity of f , we musthave f ( q ′ ) - c u f ( q ), so p i ( f ( q ′ )) ≥ m > p i ( 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 ′ = id X ′ , but f = id X . Theassertion follows. Corollary 5.7.
Let ( X, κ ) be a reducible digital image. Let x be a reductionpoint 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, c ) be a connected digital image in Z . Suppose x ∈ X is such that N c ( x ) is connected and N c ( x ) ∈ { , , } . If A is afreezing set for ( X, c ) , then x ∈ A .Proof. By the proof of Theorem 3.6, we can use Lemma 3.4 to conclude that x is a reduction point. The assertion follows from Corollary 5.7.Proposition 5.8 cannot in general be extended to permit N c ( x ) = 4, asshown in the following. 9igure 2: Illustration for Example 5.9 Example 5.9.
Let X = { x i } i =0 ⊂ Z , where x = (0 , , x = (0 , − , x = (1 , , x = (0 , , x = ( − , . See Figure 2. Then N c ( x ) is c -connected and N c ( x ) = 4. It is easily seenthat X \ { x } is a freezing set for ( X, c ). For any digital image (
X, κ ), clearly X is a freezing set. An interesting questionis how small A ⊂ X can be for A to be a freezing set for X . We say a freezingset A is minimal if no proper subset of A is a freezing set for X . Definition 5.10.
Let X ⊂ Z n . • The boundary of X [15] is Bd ( X ) = { x ∈ X | there exists y ∈ Z n \ X such that y ↔ c x } . • The interior of X is int ( X ) = X \ Bd ( X ). Proposition 5.11.
Let [ a, b ] Z ⊂ [ c, d ] Z and let f : [ a, b ] Z → [ c, d ] Z be c -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 = Fix( f ), then we have at least one of the following: • For some smallest t satisfying a < t < b , f ( t ) > t . But then f ( t − ≤ t −
1, so f ( t − - c f ( t ), contrary to the continuity of f . • For some largest t satisfying a < t < b , f ( t ) < t . But then f ( t + 1) ≥ t + 1, so f ( t + 1) - c f ( t ), contrary to the continuity of f .10t 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 , c ) such that { a, b } ⊂ Fix( f ), we conclude that { a, b } is a freezing set for [ a, b ] Z .To establish minimality, observe that all proper subsets B of { a, b } allowconstant functions c that are c -continuous non-identities with c | B = id B . Proposition 5.12.
Let X ⊂ Z n be finite. Let ≤ u ≤ n . Let A ⊂ X . Let f ∈ C ( X, c u ) . If Bd ( A ) ⊂ Fix( f ) , then A ⊂ Fix( f ) .Proof. By hypothesis, it suffices to show int ( A ) ⊂ Fix( f ). Let x = ( x , . . . , x n ) ∈ int ( A ). Suppose, in order to obtain a contradiction, x Fix( f ). Then for someindex j , p j ( f ( x )) = x j . (1)Since X is finite, there exists a path P = { y i = ( x , . . . , x j − , a i , x j +1 . . . , x n ) } mi =1 in X such that a < x j < a m and a i +1 = a i + 1; y , y m ∈ Bd ( A ); and { y i } m − i =2 ⊂ int ( A ). Note x ∈ P . Now, (1) implies either p j ( f ( x )) < x j or p j ( f ( x )) > x j . If the former, then by Lemma 5.5, y m Fix( f ); and if the lat-ter, then by Lemma 5.5, y Fix( f ); so in either case, we have a contradiction.We conclude that x ∈ Fix( f ). The assertion follows. Theorem 5.13.
Let X ⊂ Z n be finite. Then for ≤ u ≤ n , Bd ( X ) is a freezingset for ( X, c u ) .Proof. The assertion follows from Proposition 5.12.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 ) ∈ Z | y ≥ } . Consider the function f : X → X defined by f ( x, y ) = (cid:26) ( x,
0) if y = 0;( x + 1 , y ) if y > . Then f ∈ C ( X, c ), Bd ( X ) = Z × { } , and f | Bd ( X ) = id Bd ( X ) , but X Fix( f ),so Bd ( X ) is not a c -freezing set for X . c In this section, we consider freezing sets for digital cubes using the c adjacency. Theorem 5.15.
Let X = Π ni =1 [0 , m i ] Z . Let A = Π ni =1 { , m i } . • Let Y = Π ni =1 [ a i , b i ] Z be such that [0 , m i ] ⊂ [ a i , b i ] Z for all i . Let f : X → Y be c -continuous. If A ⊂ Fix( f ) , then X ⊂ Fix( f ) . • A is a freezing set for ( X, c ) ; minimal for n ∈ { , } . roof. 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 , andwe now assume the assertion is established for n ≤ k where k ≥ n = k + 1 and f : X → Y is c -continuous with A ⊂ Fix( f ).Let X = Π ki =1 [0 , m i ] Z × { } , X = Π ki =1 [0 , m i ] Z × { m k +1 } . We have that f | X and f | X are c -continuous, A ∩ X ⊂ Fix( f | X ), and A ∩ X ⊂ Fix( f | X ). Since X and X are isomorphic to k -dimensional digital cubes, byTheorem 5.3 and the inductive hypothesis, we have (cid:0) Π ki =1 [0 , m i ] Z × { } (cid:1) ∪ (cid:0) Π ki =1 [0 , m i ] Z × { m n } (cid:1) ⊂ Fix( f ) . Then given x = ( x , . . . , x n ) ∈ X , x is a member of the unique shortest c -path { ( x , x , . . . , x k , t ) } m t =0 from ( x , x , . . . , x k , ∈ A to ( x , x , . . . , x k , m n ) ∈ A .By Proposition 5.4, x ∈ Fix( f ). Since x was taken arbitrarily, this completesthe induction proof that X ⊂ Fix( f ).By taking Y = X and applying the above to all f ∈ C ( X, c ) such that A ⊂ Fix( f ), we conclude that A is a freezing set for ( X, c ).Minimality of A for n = 1 was established at Proposition 5.11. To showminimality of A for n = 2, consider a proper subset A ′ of A . Without loss ofgenerality, (0 , ∈ A \ A ′ , m >
0, and m >
0. For x ∈ X , let g : X → X bethe function g ( x ) = (cid:26) x if x = (0 , ,
1) if x = (0 , . Suppose y ∈ X is such that y ↔ c (0 , y = (1 ,
0) or y = (0 , g ( y ) = y ↔ c (1 ,
1) = g (0 , . Thus g ∈ C ( X, c ), A ′ ⊂ Fix( g ), and g = id X . Therefore, A ′ is not a freezingset for ( X, c ), so A is minimal.The minimality assertion of Theorem 5.15 does not extend to n = 3, asshown in the following. Example 5.16.
Let X = [0 , Z . Let A = { (0 , , , (0 , , , (1 , , , (1 , , } . See Figure 3. Then A is a minimal freezing set for ( X, c ). Proof.
Note if x ∈ X \ A then for each index i ∈ { , , } , x is c -adjacent to y i ∈ A such that x and y i differ in the i th coordinate. Therefore, if f ∈ C ( X, c )such that f ( x ) = x , then c -continuity requires that for some i we have f ( y i ) = y i . It follows that A is a freezing set for ( X, c ).Minimality is shown as follows. Let A ′ be a proper subset of A . Without lossof generality, (0 , , ∈ A \ A ′ . Let g : X → X be the function (see Figure 3) g ( x ) = (1 , ,
0) if x = (0 , , , ,
1) if x = (0 , , x otherwise.12igure 3: The function g in the proof of Example 5.16. Members of A \{ (0 , , } are circled. Straight line segments indicate c adjacencies. Curved arrows showthe mapping for points in X \ Fix( g ).Then g ∈ C ( X, c ), g | A ′ = id A ′ , and g = id X . Therefore, A ′ is not a freezingset for ( X, c ). c n In this section, we consider freezing sets for digital cubes in Z n , using the c n adjacency. Theorem 5.17.
Let X = Q ni =1 [0 , m i ] Z ⊂ Z n , where m i > for all i . Then Bd ( X ) is a minimal freezing set for ( X, c n ) .Proof. That Bd ( X ) is a freezing set for ( X, c n ) follows from Theorem 5.13.To show Bd ( X ) is a minimal freezing set, it suffices to show that if A is aproper subset of Bd ( X ) then A is not a freezing set for ( X, c n ). We must showthat there exists f ∈ C ( X, c n ) such that f | A = id A but f = id X . (2)By hypothesis, there exists y = ( y , . . . , y n ) ∈ Bd ( X ) \ A .Since y ∈ Bd ( X ), for some index j , y j ∈ { , m j } . • If y j = 0 the function f : X → X defined by f ( y ) = ( y , . . . , y j − , , y j +1 , . . . , y n ) , f ( x ) = x for x = y, satisfies (2). • If y j = m j the function f : X → X defined by f ( y ) = ( y , . . . , y j − , m j − , y j +1 , . . . , y n ) , f ( x ) = x for x = y, satisfies (2).The assertion follows. 13 .5 Freezing sets and the normal product adjacency In the following, p j : Q vi =1 X i → X j is the map p j ( x , . . . , x v ) = x j where x i ∈ X i . Theorem 5.18.
Let ( X i , κ i ) be a digital image, i ∈ [1 , v ] Z . Let X = Q vi =1 X i .Let A ⊂ X . Suppose A is a freezing set for ( X, N P v ( κ , . . . , κ v )) . Then foreach i ∈ [1 , v ] Z , p i ( A ) is a freezing set for ( X i , κ i ) .Proof. Let f i ∈ C ( X i , κ i ). Let F : X → X be defined by F ( x , . . . , x v ) = ( f ( x ) , . . . , f v ( x v )) . Then by Theorem 2.7, F ∈ C ( X, N P v ( κ , . . . , κ v )).Suppose for all a = ( a , . . . , a v ) ∈ A , F ( a ) = a , hence f i ( a i ) = a i for all a i ∈ p i ( A ). Since A is a freezing set of X , we have that F = id X , and therefore, f i = id X i . The assertion follows. A cycle or digital simple closed curve of n distinct points is a digital image( C n , κ ) with C n = { x i } n − i =0 such that x i ↔ κ x j if and only if j = i + 1 mod n or j = i − n .Given indices i < j , there are two distinct paths determined by x i and x j in C n , consisting of the sets P i,j = { x k } jk = i and P ′ i,j = C n \ { x k } j − k = i +1 . If one ofthese has length less than n/
2, it is the shorter path from p i to p j and the otheris the longer path ; otherwise, both have length n/
2, and each is a shorter path and a longer path from p i to p j .In this section, we consider minimal fixed point sets for f ∈ C ( C n ) that force f to be an identity map. Theorem 5.19.
Let n > . Let x i , x j , x k be distinct members of C n be suchthat C n is a union of unique shorter paths determined by these points. Let f ∈ C ( C n , κ ) . Then f = id C n if and only if { x i , x j , x k } ⊂ Fix( f ) ; i.e., { x i , x j , x k } is a freezing set for C n . Further, this freezing set is minimal.Proof. Clearly f = id C n implies { x i , x j , x k } ⊂ Fix( f ).Suppose { x i , x j , x k } ⊂ Fix( f ). By hypothesis, there are unique shorterpaths P from x i to x j , P from x j to x k , and P from x k to x i , in C n . ByProposition 5.4, each of P , P , and P is contained in Fix( f ). By hypothesis C n = P ∪ P ∪ P , so f = id C n . Hence { x i , x j , x k } is a freezing set.For any distinct pair x i , x j ∈ C n , there is a non-identity continuous self-map on C n that takes a longer path determined by x i and x j to a shorter pathdetermined by x i and x j . Thus, { x i , x j } is not a freezing set for C n , so the set { x i , x j , x k } discussed above is minimal. Remark 5.20.
In Theorem 5.19, we need the assumption that n >
4, as thereis a continuous self-map f on C with 3 fixed points such that f = id C [10].14 .7 Wedges Let (
X, κ ) ⊂ Z n be such that X = X ∪ X , where X ∩ X = { x } ; and if x ∈ X , y ∈ X , and x ↔ κ y , then x ∈ { x, y } . We say X is the wedge of X and X , denoted X = X ∨ X . We say x is the wedge point . Theorem 5.21.
Let A be a freezing set for ( X, κ ) , where X = X ∨ X ⊂ Z n , X > , and X > . Let X ∩ X = { x } . Then A must include points of X \ { x } and X \ { x } .Proof. Otherwise, either A ⊂ X or A ⊂ X .Suppose A ⊂ X . Then the function f : X → X given by f ( x ) = (cid:26) x if x ∈ X ; x if x ∈ X , belongs to C ( X, κ ) and satisfies f | A = id A , but f = id X . Thus A is not afreezing set for ( X, κ ).The case A ⊂ X is argued similarly. Example 5.22.
The wedge of two digital intervals is (isomorphic to) a digitalinterval. It follows from Theorem 5.3 and Proposition 5.11 that a freezing setfor a wedge need not include the wedge point.
Theorem 5.23.
Let C m and C n be cycles, with m > , n > . Let x be thewedge point of X = C m ∨ C n . Let x i , x j ∈ C m and x ′ i , x ′ j ∈ C n be such that C m is the union of unique shorter paths determined by x i , x j , x and C n is the unionof unique shorter paths determined by x ′ k , x ′ p , x . Then A = { x i , x j , x ′ k , x ′ p } is afreezing set for X .Proof. Let f ∈ C ( X, κ ) be such that A ⊂ Fix( f ). Let P be the unique shorterpath in C m from x i to x j ; let P be the unique shorter path in C m from x j to x ; let P be the unique shorter path in C m from x to x i ; let P ′ be the uniqueshorter path in C n from x ′ k to x ′ p ; let P ′ be the unique shorter path in C n from x ′ p to x ; let P ′ be the unique shorter path in C n from x to x ′ k .By Proposition 5.4, each of the following paths is contained in Fix( f ): P , P ∪ P ′ (from x j to x to x ′ p ), P ∪ P ′ (from x i to x to x ′ k ), and P ′ . Since X = P ∪ ( P ∪ P ′ ) ∪ ( P ∪ P ′ ) ∪ P ′ ⊂ Fix( f ) , the assertion follows. A tree is an acyclic graph ( X, κ ) that is connected, i.e., lacking any subgraphisomorphic to C n for n >
2. The degree of a vertex x in X is the number ofdistinct vertices y ∈ X such that x ↔ y . A vertex of a tree may be designatedas the root . We have the following. 15 emma 5.24. [10] Let ( X, κ ) be a digital image that is a tree in which theroot 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 > . Let E be the set of vertices of G that have degree1. 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 = { x , x } = E , and E is a freezing set. E must be minimal,since X admits constant functions that are identities on their restrictions toproper subsets of E .Otherwise, there exists x ∈ X such that x has degree of at least 2 in G .This implies X >
2, and since G is finite and acyclic, E >
0. Since G isacyclic, removal of any member of X \ E would disconnect X . If we take x tobe 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 . Thenthe function f : X → X defined by f ( x ) = (cid:26) y ′ if x = y ; x if x = y, satisfies f ∈ C ( X, κ ), f | E \{ y } = id E \{ y } , and f = id X . Thus E \ { y } is not afreezing set. Since y was arbitrarily chosen, E is minimal. s -Cold sets In this section, we generalize our focus from fixed points to approximate fixedpoints and, more generally, to points constrained in the amount they can bemoved by continuous self-maps in the presence of fixed point sets. We obtainsome analogues of our previous results for freezing sets.
In the following, we use the path-length metric d for connected digital images( X, κ ), defined [13] as d ( x, y ) = min { ℓ | ℓ 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 connecteddigital image ( X, κ ) if given g ∈ C ( X, κ ) such that g | A = id A , then for all x ∈ X , d ( x, g ( x )) ≤ s . A cold set is a 1-cold set.16ote a 0-cold set is a freezing set. Theorem 6.2.
Let ( X, κ ) be a connected digital image. Let A ⊂ X . Then A ⊂ F ix ( 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 forall 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 = id A and all x ∈ X , we have d ( x, f ( x )) ≤ s . Let F : ( X, κ ) → ( Y, λ ) be anisomorphism. Let g : X → Y be ( κ, λ )-continuous with F | A = g | A . Thenid A = F − ◦ F | A = F − ◦ g | A . Let x ∈ X . Then d ( x, F − ◦ g ( x )) ≤ s , i.e., there is a κ -path P in X of lengthat most s from x to F − ◦ g ( x ). Therefore, F ( P ) is a λ -path in Y of length atmost s from F ( x ) to F ◦ F − ◦ 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 = id A . Since id X 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 almostfixed 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 setfor ( 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 = id A , every x ∈ X is an approximate fixed point of f . • In a finite connected digital image ( X, κ ) , every nonempty subset of X isa diam ( X ) -cold set. • If s < s and A is an s -cold set for ( X, κ ) , then A is an s -cold set for ( X, κ ) . Note a freezing set is a cold set, but the converse is not generally true, asshown in the following.
Example 6.4.
It follows from Definition 6.1 that { } is a cold set, but not afreezing set, for X = [0 , Z , since the constant function g with value 0 satisfies g | { } = id { } , and g (1) = 0 ↔ c 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 setfor ( X, κ ) , and let F : ( X, κ ) → ( Y, λ ) be an isomorphism. Then F ( A ) is an s -cold set for ( Y, λ ) . roof. Let f ∈ C ( Y, λ ) such that f | F ( A ) = id F ( A ) . Then f ◦ F | A = f | F ( A ) ◦ F | A = id F ( A ) ◦ F | A = F | A . 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 setfor ( 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]. Thisnotion is somewhat analogous to that of a dense set in a topological space, andthe following is somewhat analogous to the fact that in topological spaces, acontinuous function is uniquely determined by its values on a dense subset ofthe domain. Theorem 6.6.
Let ( X, κ ) be a digital image and let A be κ -dominating in X .Then A is -cold in ( X, κ ) .Proof. Let f ∈ C ( X, κ ) such that f | A = id A . 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 havethe path { x, a, f ( x ) } ⊂ X from x to f ( x ) of length at most 2. The assertionfollows. Theorem 6.7.
Let ( X, κ ) be rigid. If A is a cold set for X , then A is a freezingset for X .Proof. Let f ∈ C ( X, κ ) be such that f | A = id A . Since A is cold, f ( x ) - x forall x ∈ X . Therefore, the map H : X × [0 , Z → X defined by H ( x,
0) = x , H ( x,
1) = f ( x ), is a homotopy. Since X is rigid, f = id X . The assertionfollows. In this section, we consider cold sets for digital cubes in Z n . Note the hypothesesof Proposition 6.8 imply A is c - and c -dominating in Bd ( X ). Proposition 6.8.
Let m, n ∈ N . Let X = [0 , m ] Z × [0 , n ] Z . Let A ⊂ Bd ( X ) besuch that no pair of c -adjacent members of Bd ( X ) belong to Bd ( X ) \ A . Then A is a cold set for ( X, c ) . Further, for all f ∈ C ( X, c ) , if f | A = id A then f | Int ( X ) = id | Int ( X ) .Proof. Let x = ( x , y ) ∈ X . Let f ∈ C ( X, c ) such that f | A = id A . Considerthe following. • If x ∈ A then f ( x ) = x . • If x ∈ Bd ( X ) \ A then both of the c -neighbors of x in Bd ( X ) belong to A . We will show f ( x ) - c x .Let K = { (0 , , (0 , n ) , ( m, , ( m, n ) } ⊂ Bd ( X ).18igure 4: Illustration of the proof of Proposition 6.8 for the case ( x , y ) ∈ Int ( X ). X = [0 , Z × [0 , 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 cannothave c -adjacent members of Bd ( X ) in Bd ( X ) \ A , we can have c -adjacentmembers of Bd ( X ) in Bd ( X ) \ A , e.g., (5 ,
4) and (6 , c -path P of length n = 4: P (0) = q L = (2 , P (1) = (1 , P (2) = ( x , y ) = (1 , P (3) = (1 , P (4) = q U = (1 , – For x ∈ K , consider the case x = (0 , { (0 , , (1 , } ⊂ A , sowe must have f ( x ) ∈ N ∗ c ((0 , ∩ N ∗ c ((1 , ⊂ N ∗ c ( x ) . For other x ∈ K , we similarly find f ( x ) - c x . – For x ∈ Bd ( X ) \ K , consider the case x = ( t, { ( t − , , ( t + 1 , } ⊂ A , so( t − ,
0) = f ( t − , - c f ( x ) - c f ( t + 1 ,
0) = ( t + 1 , . Therefore, f ( x ) ∈ { x, ( t, } , so f ( x ) - c x .For other x ∈ Bd ( X ) \ K , we similarly find f ( x ) - c x . • If x ∈ Int ( X ), let L = { ( z, } x +1 z = x − and U = { ( z, n ) } x +1 z = x − . We have L ∩ A = ∅ 6 = U ∩ A. Since no pair of c -adjacent members of Bd ( X ) belong to Bd ( X ) \ A , thereexist q L ∈ L ∩ A , q U ∈ U ∩ A such that | p ( q L ) − x | ≤ | p ( q U ) − x | ≤ . c -path P : [0 , n ] Z → X such that P ([0 , y )] Z )runs from q L to x and P ([ y , n ] Z ) runs from x to q U (note since we use c -adjacency, there can be steps of the path that change both coordinates- see Figure 4). Therefore, f ◦ P is a path from f ( q L ) = q L to f ( x ) to f ( q U ) = q U , and p ◦ f ◦ P is a path from p ( q L ) = 0 to p ( f ( x )) to p ( q U ) = n .If y ′ = p ( f ( x )) > y , then p ◦ f ◦ P | [0 ,y ] Z is a c -path of length y from0 to y ′ , which is impossible. Similarly, if y ′ < y , then p ◦ f ◦ P | [ y ,n ] Z is a c -path of length n − y from y ′ to n , which is impossible. Therefore, wemust have p ◦ f ( x ) = y . (3)Similarly, by replacing the neighborhoods of the projections of x on thelower and upper edges of the cube, L and U , by the neighborhoods ofthe projections of x on the the left and right edges of the cube, L ′ = { (0 , z ) } y +1 z = y − and R = { ( m, z ) } y +1 z = y − , and using an argument similar tothat used to obtain (3), we conclude that p ◦ f ( x ) = x . (4)It follows from (4) and (3) that f ( x ) = x .Thus, in all cases, f ( x ) - c x , and f | Int ( X ) = id Int ( X ) . Proposition 6.9.
Let m, n ∈ N . Let X = [0 , m ] Z × [0 , n ] Z . Let A ⊂ Bd ( X ) be c -dominating in Bd ( X ) . Then A is a 2-cold set for ( X, c ) . Further, for all f ∈ C ( X, c ) , if f | A = id A then f | Int ( X ) = id | Int ( X ) .Proof. Our argument is similar to that of Proposition 6.8. Let x = ( x , y ) ∈ X .Let f ∈ C ( X, c ) such that f | A = id A . Consider the following. • If x ∈ A then f ( x ) = x . • If x ∈ Bd ( X ) \ A then for some a ∈ A , x - c a . Therefore, f ( x ) - c f ( a ) = a . Thus, { x, a, f ( x ) } is a path in X from x to f ( x ) of length atmost 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 ) = id Int ( X ) .An example of a 2-cold set A that is not a 1-cold set, such that A is as inProposition 6.9, is given in the following. Example 6.10.
Let X = [0 , Z . Let A = { (0 , , (1 , , (2 , } ⊂ X. A is c -dominating in Bd ( X ), so by Proposition 6.9, is a 2-cold set for( X, c ). Let f : X → X be the function f (0 ,
0) = (2 , f (0 ,
1) = (1 , f ( x ) = x for all x ∈ X \{ (0 , , (0 , } . Then f ∈ C ( X, c ) but d ((0 , , f (0 , A is not a 1-cold set. Proposition 6.11.
Let X = Q ni =1 [0 , m i ] Z ⊂ Z n , where m i > for all i . Let A ⊂ Bd ( X ) be such that A is not c n -dominating in Bd ( X ) . Then A is not acold set for ( X, c n ) .Proof. By hypothesis, there exists y = ( y , . . . , y n ) ∈ Bd ( X ) \ A such that N ( y, c n ) ∩ A = ∅ .Since y ∈ Bd ( X ), for some index j we have y j ∈ { , m j } . Let x =( x , . . . , x n ) ∈ X , for x i ∈ [0 , m i ] Z . • If y j = 0, let f : X → X be defined as follows. f ( x ) = ( x , . . . , x j − , , x j +1 , . . . , x n ) if x = y ;( x , . . . , x j − , , x j +1 , . . . , x n ) if x ∈ N c n ( y ); x otherwise.If u, v ∈ X , u - c n v , then u and v differ by at most 1 in every coordinate.Consider the following cases. – If u = y , then v ∈ N c n ( y ), and clearly f ( u ) and f ( v ) differ by atmost 1 in every coordinate, hence are c n -adjacent. Similarly if v = y . – If u, v ∈ N c n ( y ), then clearly f ( u ) and f ( v ) differ by at most 1 inevery coordinate, hence are c n -adjacent. – If u ∈ N c n ( y ) and v N c n ( y ), then p j ( u ) ∈ { , } , so p j ( f ( v )) = p j ( v ) ∈ { , , } , and p j ( f ( u )) = 1. It follows easily that f ( u ) and f ( v ) differ by at most 1 in every coordinate, hence are c n -adjacent.Similarly if v ∈ N c n ( y ) and u N c n ( y ) – Otherwise, { u, v } ∩ N ∗ c n ( y ) = ∅ , so f ( u ) = u - c n v = f ( v ).Therefore, f ∈ C ( X, c n ). • If y j = m j , let f : X → X be defined by f ( x ) = ( x , . . . , x j − , m j − , x j +1 , . . . , x n ) if x = y ;( x , . . . , x j − , m j − , x j +1 , . . . , x n ) if x ∈ N c n ( y ); x otherwise.By an argument similar to that of the case y j = 0, we conclude that f ∈ C ( X, c n ).Further, in both cases, f | A = id A , and f ( y ) - c n y . The assertion follows.21 .3 s -cold sets for rectangles in Z The following generalizes the case n = 2 of Theorem 5.15. Proposition 6.12.
Let X = [ − m, m ] Z × [ − n, n ] Z ⊂ Z , 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 s -cold set for ( X, c ) .Proof. Let f ∈ C ( X, c ) such that f | A = id A . 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 )) ≤ s . This isseen as follows. For x ∈ X \ A ′ , there exists a c -path P of length at most 2 s from x to some y ∈ Bd ( A ′ ). Then f ( P ) is a c -path from f ( x ) to f ( y ) = y oflength at most 2 s . Therefore, P ∪ f ( P ) contains a path from x to y to f ( x ) oflength at most 4 s . The assertion follows.The following generalizes Proposition 6.8. Proposition 6.13.
Let X = [ − m, m ] Z × [ − n, n ] Z ⊂ Z , s ∈ N ∗ , where m − s ≥ , n − s ≥ . Let A = [ − m + s, m − s ] Z × [ − n + s, n − s ] Z ⊂ X. Let A ′ ⊂ Bd ( A ) such that no pair of c -adjacent members of Bd ( A ) belongs to Bd ( A ) \ A ′ .Then A ′ is a s -cold set for ( X, c ) .Further, if f ∈ C ( X, c ) and f | A ′ = id A ′ , then f | A = id A .Proof. Let f ∈ C ( X, c ) be such that f | A ′ = id A ′ . As in the proof of Proposi-tion 6.8, f | A = id A .Now consider x ∈ X \ A . There is a c -path P in X from x to some y ∈ A ′ oflength at most s . Then f ( P ) is a c -path in X from f ( x ) to f ( y ) = y of lengthat most s . Therefore, P ∪ f ( P ) contains a c -path in X from x to y to f ( x ) oflength at most 2 s . The assertion follows. s -cold sets for Cartesian products We modify the proof of Theorem 5.18 to obtain the following.
Theorem 6.14.
Let ( X i , κ i ) be a digital image, i ∈ [1 , v ] Z . Let X = Q vi =1 X i .Let s ∈ N ∗ . Let A ⊂ X . Suppose A is an s -cold set for ( X, N P v ( κ , . . . , κ v )) .Then for each i ∈ [1 , v ] Z , p i ( A ) is an s -cold set for ( X i , κ i ) . roof. Let f i ∈ C ( X i , κ i ). Let F : X → X be defined by F ( x , . . . , x v ) = ( f ( x ) , . . . , f v ( x v )) . Then by Theorem 2.7, F ∈ C ( X, N P v ( κ , . . . , κ v )).Suppose for all i , a i ∈ p i ( A ), we have f i ( a i ) = a i . Note this implies, for a = ( a , . . . , a v ), that F ( a ) = a . Since a is an arbitrary member of the s -set A of X , we have that d ( F ( x ) , x ) ≤ s , for all x = ( x , . . . , x v ) ∈ X , x i ∈ X i , andtherefore, d ( f i ( x i ) , x i ) ≤ s . The assertion follows. s -cold sets for infinite digital images In this section, we obtain properties of s -cold sets for some infinite digital images. Theorem 6.15.
Let ( Z n , c u ) be a digital image, ≤ u ≤ n . Let A ⊂ Z n . Let s ∈ N ∗ . If A is an s -cold set for ( Z n , c u ) , then for every index i , p i ( A ) is aninfinite 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 { p i ( a ) | a ∈ A } or M = max { p i ( a ) | a ∈ A } . If the former, then for z = ( z , . . . , z n ) ∈ Z n , define f : Z n → Z n by f ( z ) = (cid:26) ( z , . . . , z i − , m, z i +1 , . . . , z n ) if z i ≤ m ; z otherwise.Then f ∈ C ( Z n , c u ) and f | A = id A , but f = id Z n . 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 , c ) if and only if A containssequences { a i } ∞ i =1 and { a ′ i } ∞ i =1 such that lim i →∞ a i = ∞ and lim i →∞ a ′ i = −∞ .Proof. This follows from Lemma 5.5 and Theorem 6.15.The converse of Theorem 6.15 is not generally correct, as shown by thefollowing.
Example 6.17.
Let A = { ( z, z ) | z ∈ Z } ⊂ Z . Then although p ( A ) = p ( A ) = Z contains sequences tending to ∞ and to −∞ , A is not an s -cold set for ( Z , c ),for any s . Proof.
Consider f : Z → Z defined by f ( x, y ) = ( x, x ). We have f ∈ ( Z , c )and f | A = id A , but one sees easily that for all s there exist ( x, y ) ∈ Z suchthat d (( x, y ) , f ( x, y )) > s . 23 Further remarks
We have continued the work of [10] in studying fixed point invariants and relatedideas 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 showus 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 informationabout the nature of f | X \ Fix( f ) , if f ∈ C ( X, κ ) and A ⊂ Fix( f ) is a freezingset or, more generally, an s -cold set for ( X, κ ), then f | X \ Fix( f ) may be severelylimited. P. Christopher Staecker and an anonymous reviewer were most helpful. Theyeach suggested several of our assertions, and several corrections.
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