Continuous Maps on Digital Simple Closed Curves

We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S ; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, such that 5 |> | Y (where | | Y is the number of points in Y ). Let Y X f  : be a digitally continuous antipodal map. Then f is not nullhomotopic in Y ; 3) Let S be a digital simple closed curve that is symmetric with respect to the origin. Let Z S f  : be a digitally continuous map. Then there is a pair of antipodes S x x   } , { such that 1 | ) ( ) ( |    x f x f .

be a digitally continuous antipodal map.Then f is not nullhomotopic in Y ; 3) Let S be a digital simple closed curve that is symmetric with respect to the origin.Let Z S f  : be a digitally continuous map.Then there is a pair of antipodes

Introduction
A digital image is a set X of lattice points that model a "continuous object" Y , where Y is a subset of a Euclidean space.Digital topology is concerned with developing a mathematical theory of such discrete objects so that, as much as possible, digital images have topological properties that mirror those of the Euclidean objects they model; however, in digital topology we view a digital image as a graph, rather than, e.g., a metric space, as the latter would, for a finite digital image, result in a discrete topological space.Therefore, the reader is reminded that in digital topology, our "nearness" notion is the graphical notion of adjacency, rather than a neighborhood system as in classical topology; usually, we use one of the natural l c -adjacencies (see Section 2).Early papers in the field, e.g., [1][2][3][4][5][6][7][8], noted that this notion of nearness allows us to express notions borrowed from classical topology, e.g., connectedness, continuous function, homotopy, and fundamental group, such that these often mirror their analogs with respect to Euclidean objects modeled by respective digital images.Applications of digital topology have been found shape description and in image processing operations such as thinning and skeletonization [9].
A. Rosenfeld wrote the following: "The discrete grid (of pixels or voxels) used in digital topology can be regarded as a 'digitization' of (two or three-dimensional) Euclidean space; from this viewpoint, it is of interest to study conditions under which this digitization process preserves topological (or other geometric) properties" [3].
In this spirit, we obtain in this paper several properties of continuous maps on digital simple closed curves, inspired by analogs for Euclidean simple closed curves.In particular, we show that digital simple closed curves of more than 4 points are not contractible, and we obtain several results for continuous maps and antipodal points on digital simple closed curves.

Preliminaries
Let Z be the set of integers.Then d Z is the set of lattice points in d -dimensional Euclidean space.Let d Z X  and let  be some adjacency relation for the members of X .Then the pair ) , (  X is said to be a (binary) digital image.A variety of adjacency relations are used in the study of digital images.Well known adjacencies include the following.
For a positive integer l with 1 l d   and two distinct points = ( , , , ), = ( , , , ) , d d d p p p p q q q q Z    p and q are l c -adjacent [10] if • there are at most l indices i such that 1 |= | i i q p  , and • for all other indices j such that 1 | |   j j q p , j j q p = .See Figure 1. is  -connected [11] if and only if for every pair of different points and i x and This characterization of continuity is what is called an immersion, gradually varied operator, or gradually varied mapping in [12,13].
Given digital images ) , ( [14] (this was called ) , ( 1 0   -homeomorphic in [7]) and f is a ) , ( By a digital  -path from x to y in a digital image X , we mean a . We say m is the length of this path.A simple closed  -curve of 4  m points (for some adjacencies, the minimal value of m may be greater than 4; see below) in a digital image X is a sequence { (0), (1), , , we say the points of S are circularly ordered.
Digital simple closed curves are often examples of digital images for which it is desirable to consider X Z d \ as a digital image with some adjacency (not necessarily the same adjacency as used by X ).For example, by analogy with Euclidean topology, it is desirable that a digital simple closed curve 2 Z X  satisfy the "Jordan curve property" of separating 2  Z into two connected components (one "inside" and the other "outside" X ).An example that fails to satisfy this property [10] , where is 2 c -connected.However, this anomaly is essentially due to the "smallness" of X as a simple closed curve; it is known [1,15,16] [18].
, we say H holds the endpoints fixed [18].

Homotopy Properties of Digital Simple Closed Curves
A classical theorem of Euclidean topology, due to L.E.J. Brouwer, states that a d -dimensional sphere d S is not contractible [19]

S S
, then the following are equivalent.
, the equivalence of a) and b) follows from the fact that 0 S is a finite set.That c) implies both a) and b) follows from the definition of isomorphism.Therefore, we can complete the proof by showing that b) implies c). Let , where the points of a S are circularly ordered, {0,1} , and the 0  -neighbors of Since u was taken as an arbitrary index, , where the points of S are circularly ordered.
There exists , and the continuity property of homotopy implies and , but this is impossible, for the following reasons. • x .The contradiction arose from the assumption that 4 > n .Therefore, we must have . Since a digital simple closed curve is assumed to have at least 4 points, we must have 4 = n .In [8], an example is given of a simple closed 2 ccurve , such that S is 2 c -contractible (see Figure 2).By contrast, we have the following.It is natural to ask whether we can obtain an analog of Theorem 3.3 for higher dimensions.In order to do so, we must decide what is an appropriate digital model for the k -dimensional Euclidean sphere k S .The literature contains the following.
• Let k X 2 be the set of all points Notice that this example generalizes the contractibility of a 4-point digital simple closed curve [8] (see Figure 2 for the planar version); the contractibility seems due to the smallness of the image, rather than its form.
• Let 1 .Then, for some integer j , we have be the induced map.The assertion follows from the following.
Claim 1: For each , there is an integer j such that To prove Claim 1, we argue by mathematical induction on t .For 0 = t , we can clearly take 0 = j .Now, suppose the claim is valid for . Then, in particular, there is an integer j such that . The continuity properties of homotopy imply . Without loss of generality, . This is the initial case of the following: Claim 2: By the continuity properties of homotopy, ) ( 1 and with must also be an isomorphism by Theorem 3.2, from Equation (3), x   .This completes the induction proof for the Claim 2, which, in turn, completes the induction proof for the Claim 1.Thus, the assertion is established.

Antipodal Maps
A classical theorem of Euclidean topology, due to K. Borsuk, states that a continuous antipodal map : from the d -dimensional unit sphere to itself is not homotopic to a constant map [19].In this section, we obtain a digital analog, Theorem 4. 16   antipodal map if be a digital simple closed l c -curve in d Z such that the points of S are circularly ordered.If S is symmetric with respect to the origin, then the origin is not a member of S .
Proof: Suppose the origin is a member of S .Without loss of generality, 0 x is the origin, and therefore is its own antipode.
By Lemma 4.2, 1 x and are antipodes; by Lemma 4.1, these points are not l c -adjacent.This establishes the base case of an induction argument: is a connected subset of S , such that 1 x and x are antipodes, and by Lemma 4.1, these points are not l c -adjacent.Thus, isomorphic to a digital interval, with endpoints 1 x and This completes an induction argument from which we conclude that -isomorphic to a digital arc, with the endpoints of S being x  are non-adjacent antipodes.
• If n is odd, then 1  n is even, so S S =  .This is a contradiction, since S is not a simple closed l ccurve.
• If n is even, then . Since S is symmetric with respect to the origin, we must have that x is the origin, this is a contradiction.
Whether n is even or odd, the assumption that the origin belongs to S yields a contradiction.Hence, the origin is not a member of S .Lemma 4.4 Let Z such that the points of S are circularly ordered.If S is symmetric with respect to the origin, then n is even.
Proof: By Lemma 4.3, the origin is not a member of S , so every member of S is distinct from its antipode.Therefore, n must be even.
Lemma 4.5 Let x is antipodal to either . Without loss of generality, 1 x and  Proof: , where the points of 1 S are circularly ordered.Without loss of generality, there exists . Since f is anti-podal, it follows from Lemma 4.5 that follows that one of the 1  -paths in 1 , there exists x q f  , and from Lemma 4.5, , each symmetric with respect to the origin.Let Proof: By Theorem 4.6, f is onto.The assertion follows from Proposition 3.1.
Proposition 4.8 Let , so that the image under G of 0 x at time t is constant with respect to t .We show below that G is a homotopy.In particular, 0 ; and, for all then we have the following.
• By the continuity of t H , we have . To simplify the following, let 1 0 ( 1) = ( ) = ( ), . Note that the continuity of , then , then x i G t Therefore, the induced function be digital images and let Y y  .We denote by : y X Y  (or, for short, y ) the constant map ( ) = y x y for all X x  .Lemma 4.9 Let ) , ( . The assertion follows from Proposition 4.8.
Given a digital image ) , (  X and a positive integer n , for X x  we define [20] ( , The covering space and the lifting of maps are notions borrowed from algebraic topology that have been important in digital algebraic topology.We have the following.
; and • the restriction map ,1) ( ,1) The following is a minor generalization of an example of a digital covering map given in [20].
Example 4.12 Let -covering map (see Figure 5).Definition 4.13 [20] , and there is a E  -homotopy in E from 0 g to 1 g that holds the endpoints fixed.Theorem 4. 16 Let be a simple closed X  -curve with points circularly ordered, that is symmetric with respect to the origin.Let be defined by   liftings with respect to p to paths 0 F and 1 F , respectively, in Z , each starting at (0))) ( ( 0 , so p is a radius 2 local isomorphism.From Theorem 4.15, 0 F and 1 F must end at the same point.Indeed, this point must be 0 , for the uniqueness of 1 F implies 1 F must be the constant map 0 .
Since f is an antipodal map, we must have Since F is continuous and 5 , a simple induction argument based on Equations ( 4) and (5) shows that , which contradicts Equation (5).The assertion follows from the contradiction.

Antipodes Mapped Together
A classical result of topology is that if f is a continuous map from the d -dimensional unit sphere d S to Euclidean d -space d R , then there is a pair of antipodes , where the points of S are circularly ordered.Suppose S  .Therefore, we assume for all i that 0. ) (  i g (6) Clearly, for all i , ).
The continuity of f implies that It follows from Equation (7) that g takes both positive and negative values, so from inequality (6), there is an index j such that ) ( j g and 1) (  j g have opposite sign; without loss of generality, 0 > ) ( j g and 0 < 1) (  j g . From inequality (8), it follows that given by

Further Remarks
In this paper, we have obtained several analogs of classical theorems of Euclidean topology concerning maps on digital simple closed curves.We have shown that digital simple closed curves of more than 4 points are not contractible; that a continuous antipodal map from a digital simple closed curve to itself is not nullhomotopic; and that a continuous map from a digital simple closed curve to the digital line must map a pair of antipodes within 1 of each other.Except as indicated concerning whether or not a digital model of a sphere is contractible, it is not known at the current writing whether these results extend to higher dimensional digital models of Euclidean spheres.
We thank the anonymous referees for their helpful suggestions.
the number of points in Y ).Let Y X f  :

Figure 1
Figure 1.In is not  -contractible.Proof: It follows from Theorem 3.2 that if 4 map in S .

Figure 4 -homotopic to 1 S
Figure 4.A digital simple closed 1 c -curve . But since S is a simple closed l c -curve in which 0 map.Then f is onto.
c  -homotopic paths in Y .From Theorem 4.14, these functions have unique

Figure 5 .
Figure 5.A simple closed 1 c -curve C and a covering by the digital line Z .Members of C are labeled by their respective indices.A point z Z  is labeled by the index of point of C to which the covering map sends z .

Figure 7 .
Figure 7. S and : f S Z  .Each number in the grid labels a point of S , showing the image of the grid point under f .Note for 0 = (1, 2) s , ].In this section, we study properties of continuous antipodal maps between digital simple closed curves.