Epsilon Nielsen coincidence theory

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1 Cent. Eur. J. Math. 12(9) DOI: /s Central European Journal of Mathematics Epsilon Nielsen coincidence theory Research Article Marcio Colombo Fenille 1 1 Faculdade de Matemática, Universidade Federal de Uberlândia, Av. João Naves de Ávila 2121, Santa Mônica, , Uberlândia MG, Brazil Received 18 March 2013; accepted 9 December 2013 Abstract: MSC: We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in Given two maps f, g: X Y from a well-behaved topological space into a metric space, we define µ ε (f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ε 1 -homotopic to f, g 1 is ε 2 -homotopic to g and ε 1 + ε 2 < ε. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ε (f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and id Y is its identity map, then µ ε (f, id Y ) is equal to the ε-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ε-nielsen coincidence number N ε (f, g) as a lower bound for µ ε (f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach. 55M20, 53B20 Keywords: Nielsen number Riemannian manifold Epsilon homotopy Epsilon coincidence Minimum Theorem Versita Sp. z o.o. 1. Introduction Starting from the idea that sufficiently close functions on well-behaved spaces are homotopic and the Nielsen fixed point number is a homotopy invariant, Robert Brown developed in [3] the so-called epsilon Nielsen fixed point theory. Thinking about the possibility of applications and disability of machines to achieve accuracy, Brown suggested that, instead of running for accuracy, a specific tolerance for error may be respected, as it is common in numerical analysis, for example. The purpose of this article is to develop a type of Nielsen coincidence theory similar to that presented by Brown and to generalize it in some aspects. We start with some definitions and illustrative examples. Let f : X Y be a map, where X is a Hausdorff, compact, connected and locally path connected space and Y is a metric space with metric d. Given a real number ε > 0, a homotopy {h t }: X Y is called an ε-homotopy provided d(h t, h t ) < ε for every t, t I, where mcfenille@gmail.com 1337

2 Epsilon Nielsen coincidence theory d(h t, h t ) = max {d(h t (x), h t (x)) : x X}. As already observed in [3], the concept of ε-homotopic maps does not give an equivalence relation. When there exists an ε-homotopy between two functions f 0 and f 1, we say that f 0 and f 1 are ε-homotopic and write f 0 ε f 1. Given two maps f, g: X Y, let Coin(f, g) denote the set of the coincidence points of the pair (f, g), that is, Coin(f, g) = {x X : f(x) = g(x)}. Given ε > 0, we say that a pair of maps (f, g) is ε-homotopic to a pair (f, g ), and write (f, g) ε (f, g ), provided f ε1 f and g ε2 g for some ε 1, ε 2 0 with ε 1 + ε 2 < ε. Note that we do not require that ε 1 and ε 2 are both positive. Now we are ready to define the ε-minimum coincidence number of the pair (f, g): µ ε (f, g) = min { # Coin(f, g ) : (f, g) ε (f, g ) }. Example 1.1. Let I be the closed interval [0, 1] with the metric induced from the line. Let f and g be self-maps of I defined as shown in Figure 1. Figure 1. µ(f, g) = 0 but µ ε (f, g) = 1 Then we have Coin(f, g) = {c 1, c 2, c 3, c 4 }, but only the coincidence point c 3 is ε-essential (it cannot be annihilated by an ε-homotopy starting at the pair (f, g)). Therefore, µ ε (f, g) = 1, despite µ(f, g) = 0. Here, µ(f, g) denotes the minimum number of coincidence points of pairs (f, g ) when f and g vary on the set of all maps homotopic to f and g, respectively. See [6] for details. Example 1.2. Consider the 1-sphere S 1 = I/{0, 1}. Let f and g be self-maps of S 1 defined as shown in Figure 2. Figure 2. µ(f, g) = 1 but µ ε (f, g) = 3 Then we have Coin(f, g) = {c 1, c 2, c 3, c 4, c 5 }, but only the coincidence points c 1, c 4 and c 5 are ε-essential. Therefore, µ ε (f, g) = 3. However, since f has degree 1 and g has degree zero, and so f is homotopic to the identity map and g is homotopic to a constant map, we have µ(f, g) =

3 M.C. Fenille In the next section, we introduce the ε-nielsen coincidence number N ε (f, g) for a pair of maps (f, g) between orientable and closed Riemannian manifolds of the same dimension, and we prove that this number has the property N ε (f, g) µ ε (f, g), just as in the classical coincidence theory and in the epsilon Nielsen fixed point theory introduced by Brown. We anticipate that many of the results of this section are similar to those of [3, Section 2]. In Section 3 we get the ε-minimum number µ ε (f, g) deforming the maps f and g only on a particular open neighborhood of the set Coin(f, g). This result is quite technical but very important, because it is used in Section 4 to prove that when the range Y of the maps f and g is a closed Riemannian manifold, it is possible to attain the number µ ε (f, g) by deforming only one, rather than both of the maps f and g. This is an important result for epsilon coincidence theory analogous to the main result of Brooks in [1] for classical coincidence theory. The proof of this result is done using methods both of topology and Riemannian geometry. As an application, we show, Theorem 4.6, that, in some aspects, the epsilon coincidence theory is an extension of the epsilon fixed point theory presented by Brown in [3], thus these two theories are compatible. We point out that the result fails for manifolds with boundary, even for a closed interval. Theorem 4.6 and other general results automatically lead to an epsilon root theory. We describe this in brief at the beginning of Section 4 and then prove a Minimum Theorem in this context, that is, we prove that given a map f : X Y between orientable closed Riemannian manifolds of the same dimension and a fixed point a Y, there exists a map f : X Y having exactly N ε (f, a) roots at the point a and such that f is ε-homotopic to f. 2. Epsilon Nielsen coincidence number Henceforward, we consider X a Hausdorff, compact, connected and locally path connected space, and Y a compact, connected Riemannian manifold (possibly with boundary, for while), with metric d. By [7, Corollary 10.8] there exists a number ε > 0 such that two points of Y with distance less than ε are joined by a unique geodesic of length less than ε. Furthermore, this geodesic is minimal and depends differentiably on its endpoints. We view such geodesic between two points p and q as a path c pq (t) in Y such that c pq (0) = p and c pq (1) = q. The function (p, q) c pq is continuous. If y c pq then d(p, y) d(p, q) because c pq is the shortest path from p to q. Let f, g: X Y be two maps and consider ε > 0 as above. The following proposition provides an equivalent condition for (f, g) ε (f, g ). Again, we emphasize that whenever we assume ε > 0 small enough that any two point with less distance than ε may be joined by a unique geodesic. Proposition 2.1. (f, g) ε (f, g ) if and only if d(f, f ) + d(g, g ) < ε. Proof. The only if part is trivial. In order to prove the if part, let us write ε 1 = d(f, f ) and ε 2 = d(g, g ) and suppose that ε 1 +ε 2 < ε. Define { ht} f : X Y by h f t (x) = c f(x)f (x)(t) and {h g t }: X Y by h g t (x) = c g(x)g (x)(t). Then { } h f t is an ε 1 -homotopy between f and f and {h g t } is an ε 2 -homotopy between g and g. It follows that (f, g) ε (f, g ). This proposition provides an equivalent definition of the ε-minimum coincidence number between f, g: X Y, namely, µ ε (f, g) = min { # Coin(f, g ) : d(f, f ) + d(g, g ) < ε }. Given two maps f, g: X Y, let ε (f, g) = {x X : d(f(x), g(x)) < ε}. Theorem 2.2. The set ε (f, g) is open in X. Proof. Let R + be the subspace of R of non-negative real numbers. Define the (continuous) map D f,g : X R + by D f,g (x) = d(f(x), g(x)). Since [0, ε) is open in R +, it follows that ε (f, g) = Df,g 1 ([0, ε)) is an open subset of X. 1339

4 Epsilon Nielsen coincidence theory For two maps f, g: X Y, we define an equivalence relation on Coin(f, g) as follows: x 0, x 1 Coin(f, g) are ε-equivalent if there exists a path γ : I X from x 0 to x 1 such that f(γ) is ε-homotopic to g(γ), relative to the end points, that is f(γ) ε g(γ) rel {0, 1}. The equivalence classes under this relation will be called the ε-coincidence classes between f and g. Proposition 2.3. Two coincidence points x 0, x 1 Coin(f, g) are ε-equivalent if and only if there is a path γ : I X from x 0 to x 1 such that d(f(γ), g(γ)) < ε. Proof. The only if part is trivial. In order to prove the if part, suppose that γ : I X is a path with γ(0) = x 0 and γ(1) = x 1 such that d(f(γ), g(γ)) < ε. Then {h t }: I Y defined by h t (s) = c f(γ(s))g(γ(s)) (t) is a homotopy between f(γ) and g(γ). It remains to prove that this homotopy is relative to the end points. For this, let y 0 = f(x 0 ) = g(x 0 ) and y 1 = f(x 1 ) = f(x 2 ). Then, for every t I, we have h t (0) = c f(γ(0))g(γ(0)) (t) = c y0 y 0 (t) = y 0 and h t (1) = c f(γ(1))g(γ(1)) (t) = c y1 y 1 (t) = y 1. It proves that {h t } is a homotopy relative to the end points starting at f(γ) and ending at g(γ). Because of this proposition, we have the following alternative definition for the ε-equivalence: x 0, x 1 Coin(f, g) are ε-equivalent if there exists a path γ : I X from x 0 to x 1 such that d(f(γ), g(γ)) < ε. Theorem 2.4. Two coincidence points x 1, x 2 contains both of them. Coin(f, g) are ε-equivalent if and only if there exists a component of ε (f, g) that Proof. Suppose that x 1, x 2 Coin(f, g) are ε-equivalent and let γ : I X be a path from x 1 to x 2 such that d(f(γ), g(γ)) < ε. Thus, for each s I, we have d(f(γ(s)), g(γ(s))) < ε and we see that γ(i) ε (f, g). Since γ(i) is connected, it is contained in some component of ε (f, g). Conversely, suppose that x 1, x 2 Coin(f, g) are in the same component of ε (f, g). Since X is connected and locally path connected, the components of ε (f, g) are path connected (see [8, Theorem 25.5]), so there exists a path γ in it from x 1 to x 2. Since γ is in ε (f, g), d(f(γ), g(γ)) < ε and it follows that x 1 and x 2 are ε-equivalent. Corollary 2.5. The ε-coincidence class is open and closed in Coin(f, g). Proof. Let C be an ε-coincidence class and let x C Coin(f, g) be a coincidence point. Let U be the component of ε (f, g) which contains such point. By Theorem 2.2, U is an open neighborhood of x in X. By Theorem 2.4, the intersection U Coin(f, g) is exactly the class C. Therefore, C is open in Coin(f, g). Now, let C be an ε-coincidence class and let x Coin(f, g) be a coincidence point not belonging to C. Then there exists an ε-coincidence class C x such that x C x. By the first part of the proof, C x is open in Coin(f, g) and it is obvious that C C x =. It proves that C is closed in Coin(f, g). Corollary 2.6. For two maps f, g: X Y, there is only a finite number of nonempty ε-coincidence classes each of which is a compact subset of X. Proof. Certainly, the set Coin(f, g) is closed in X. Since X is Hausdorff compact, it follows that Coin(f, g) is compact as a subset of X. Now, by Corollary 2.5, all ε-coincidence classes between f and g form an open covering of Coin(f, g). By compactness, it has a finite sub-covering, that is, f and g have only a finite number of nonempty ε-coincidence classes. Again by Corollary 2.5, every ε-coincidence class is closed in Coin(f, g), and thus is compact in X. 1340

5 M.C. Fenille Henceforward, we assume that X and Y are both orientable and closed Riemannian manifolds of the same dimension. Let C ε 1,..., Cε r be ε-coincidence classes between f and g. We denote the component of ε (f, g) that contains C ε j by ε j (f, g). Since each component ε j (f, g) is open in X and C ε j = Coin(f, g) ε j (f, g) is compact, we have defined a coincidence index ind ( f, g, ε j (f, g) ). This index is an integer defined as in [6] and [9]. An ε-coincidence class C ε j is said essential if ind ( f, g, ε j (f, g) ) 0. The Nielsen ε-coincidence number between f and g, denoted by N ε (f, g), is the number of essential ε-coincidence classes between f and g. Theorem 2.7. If two coincident points x 0 and x 1 between the maps f, g: X Y are ε-equivalent, then they are in the same Nielsen coincidence class. So each Nielsen coincidence class is a union of ε-coincidence classes and N ε (f, g) N(f, g). Proof. If x 0 and x 1 are ε-equivalent by means of a path γ : I X between them such that f(γ) ε g(γ) rel {0, 1}, then f(γ) is homotopy to g(γ) by means of a homotopy relative to the end points. It proves that x 1 and x 2 are the same Nielsen coincidence class. Therefore, a Nielsen coincidence class C between f and g is a union of ε-coincidence classes. If C is essential, the additivity property of the coincidence index [9, Lemma 7.1, p. 190] implies that at least one of the ε-coincidence classes it contains must be an essential ε-coincidence class. Thus N ε (f, g) N(f, g). Before showing that N ε (f, g) is a lower bound for µ ε (f, g), we prove the following two lemmas. Lemma 2.8. If (f, g) ε (f, g ), then Coin(f, g ) ε (f, g). Proof. Suppose that (f, g) ε (f, g ) and suppose that x X is a point not belonging to ε (f, g) such that d(f(x), g(x)) ε. We have ε d(f(x), g(x)) d(f(x), f (x)) + d(f (x), g (x)) + d(g (x), g(x)) < ε + d(f (x), g (x)). Consequently, d(f (x), g (x)) > 0 and so x / Coin(f, g ). Lemma 2.9. Let { ht} f : X Y be an ε1 -homotopy starting at f and let {h g t }: X Y be an ε 2 -homotopy starting at g. For each t I, let K t = Coin ( h f t, h g ) t. Then K = t I K t is a compact subset of X contained in ε (f, g). Proof. Write F : X I Y and G : X I Y to be the homotopies { } h f g t and {h t } respectively, that is, F(x, t) = h f t(x) and G(x, t) = h g t (x). Let K = Coin(F, G), π : X I X be the projection map π(x, t) = x. We have K = π( K), since x K t iff (x, t) K. Now, since K = DF,G 1 (0), where D F,G : X I R is the continuous map given by D F,G (x, t) = d(f(x, t), G(x, t)), it follows that K is closed in X, so K is a compact subset of X, because X is Hausdorff compact. Finally, since π is continuous and K = π( K), we conclude that K is compact. In order to prove that K is ( contained in ε (f, g), note that for each t I, we have (f, g) ε h f t, h g ) t. Thus, by Lemma 2.8, each Kt is contained in ε (f, g), what prove that K is contained in ε (f, g). Theorem Let f, g: X Y be maps between orientable and closed Riemannian manifolds of the same dimension. Then N ε (f, g) µ ε (f, g). Proof. Given maps f, g : X Y with d(f, f ) = ε 1 and d(g, g ) = ε 2 satisfying ε 1 + ε 2 < ε, let { h f t} : X Y be the ε 1 -homotopy with h f 0 = f and hf 1 = f defined by h f t(x) = c f(x)f (x)(t) and let {h g t }: X Y be the ε 2 -homotopy with h g 0 = g and hg 1 = g defined by h g t (x) = c g(x)g (x)(t). Theorem 2.2 implies that d(f(x), g(x)) ε for all x in the boundary of ε j (f, g). Thus, for x in the boundary of ε j (f, g) and t I we have d ( f(x), h f t(x) ) + d ( h f t(x), h g t (x) ) + d(h g t (x), g(x)) d(f(x), g(x)) ε. 1341

6 Epsilon Nielsen coincidence theory Since { ht} f is an ε1 -homotopy and {h g t } is an ε 2 -homotopy, d ( f(x), h f t(x) ) < ε 1 and d(h g t (x), g(x)) < ε 2. Moreover, since ε 1 + ε 2 < ε, we have d ( h f t(x), h g t (x) ) > 0, that is, h f t and h g t have no coincidence point in the boundary of ε j. Hence, the intersection ε j Coin ( h f t, h g ) t is exactly one ε-coincidence class between h f t and h g t. Furthermore, by the previous lemma, the set t I Coin( h f t, h g ) t is compact in X. Then, the homotopy property of the coincidence index [9, Lemma 7.4, p. 191] implies ind ( f, g, ε j (f, g) ) = ind ( f, g, ε j (f, g) ). Consequently, if C ε j = Coin(f, g) ε j is an essential ε-coincidence class, then the index ind ( f, g, ε j (f, g) ) 0 and so f and g have a coincident point in ε j (f, g). We conclude that f and g have at least N ε (f, g) coincident points. Although Theorem 2.10 tells us that N ε (f, g) is a lower bound for the number of coincidence points of all pairs of maps (f, g ) that are ε-homotopic to the pair (f, g), the number N ε (f, g) is not itself invariant under ε-homotopy. 3. Moving the maps f and g only on ε (f, g) It seems clear that given a pair (f, g) of maps between manifolds, it is possible to get the ε-minimum number µ ε (f, g) by deforming the maps f and g only on ε (f, g). In fact, it holds true and it is proved in this section. This result is simple and technical, but it is important to applications in some next results. Proposition 3.1. Let f, g: X Y be maps. There exists a pair (f, g ) providing µ ε (f, g) such that (f, g ) coincides with (f, g) on the closed set X \ ε (f, g). Proof. Let (f 1, g 1 ) be a pair of maps from X into Y providing µ ε (f, g). Let { ht} f : X Y be an ε1 -homotopy from f to f 1 and let { h g } t : X Y be ε2 -homotopy from g to g 1, so that ε 1 + ε 2 < ε. For each t I, we define K t = Coin ( h f t, h g ) t and K = t I K t, as in Lemma 2.9. In particular, Coin(f, g) = K 0 and Coin(f 1, g 1 ) = K 1. By Lemma 2.9, K is compact and K ε (f, g). Since X is Hausdorff, K is a closed subset of X contained in ε (f, g). Thus, K and X \ ε (f, g) are two disjoint closed subsets of X. Let ξ : X [0, 1] be a Urysohn function satisfying ξ(k) = 1 and ξ(x \ ε (f, g)) = 0. Let us define ht f, h g t : X Y by setting ht f (x) = hξ(x)t f (x) and h g t (x) = h g ξ(x)t (x). Then h t f and h g t are ε 1 - and ε 2 -homotopies, respectively, and so the pair (f, g ) = ( h1 f, h g ) 1 is ε-homotopic to (f, g). Moreover, it is easy to see that (f, g ) provide µ ε (f, g) and coincides with (f, g) on X \ ε (f, g). In other words, Proposition 3.1 tells us that it is possible to realize the ε-minimum number µ ε (f, g) by deforming the maps f and g only on ε (f, g). 4. Compatibility with epsilon fixed point theory In this section we prove that the epsilon coincidence theory constructed in this article is a kind of generalization of the epsilon fixed point theory introduced by Brown in [3], in the case in which Y, the range of the considered maps, is assumed to be a closed (compact and without boundary) Riemannian manifold, that is, if this is the case, then the study of epsilon fixed points of a map f : Y Y is a particular case of the study of the epsilon coincidence theory of maps f, g: Y Y in the case in which g is the identity map of Y. On the other hand, we present a simple example to show that the same is not necessarily true if Y has boundary. Given two maps f and g, it is natural to ask what happens with the coincidence number between them if we deform only one of the two maps. In the classical coincidence theory, the minimal coincidence number of a pair (f, g) of maps from X into a manifold Y (without boundary), which we denote µ(f, g), may be achieved by deforming only one of the maps, that is, µ(f, g) = min {# Coin(f, g) : f f } = min {# Coin(f, g ) : g g }. 1342

7 M.C. Fenille This result was first proved by Brooks in [1]. In order to simplify the notation, given two maps f, g: X Y, we denote µ ε (f, g] = min {Coin(f, g) : f ε f }, µ ε [f, g) = min {Coin(f, g ) : g ε g }. We will prove that if Y is a closed Riemannian manifold, then the numbers µ ε (f, g), µ ε (f, g] and µ ε [f, g) are all equal. Let Y be a compact Riemannian n-manifold with metric d. Given ε > 0 (again as in Section 2), we consider the following subset of the Cartesian product Y Y : ε Y = {(y 1, y 2 ) Y Y : d(y 1, y 2 ) < ε}. It is an open subset of Y Y, since ε Y = D 1 Y ([0, ε)), where D Y : Y Y R + is the continuous map given by D Y (y 1, y 2 ) = d(y 1, y 2 ). In fact, ε Y is the ε-neighborhood of the diagonal Y = {(y, y) : y Y } in Y Y. We consider Y Y with its natural product 2n-manifold structure. Definition 4.1. We say that Y is an ε-parallel manifold if there exists a continuous map θ : Y ε Y Y satisfying the following properties: P1. θ(y 1, y 2, y 3 ) = y 2 y 1 = y 3, for all (y 1, y 2, y 3 ) ε Y Y ; P2. d(y 3, θ(y 1, y 2, y 3 )) d(y 1, y 2 ), for all (y 1, y 2, y 3 ) ε Y Y. If the map θ may be defined on the whole Cartesian product Y Y Y satisfying P1. and P2. then Y is said a parallel manifold. Every parallel manifold is an ε-parallel manifold; in fact, if θ : Y Y Y Y is a continuous map satisfying P1. and P2., then the restriction of θ on Y ε Y is a continuous map and also satisfies P1. and P2. Property P2. shows that if y 1 = y 2 then θ(y 1, y 2, y 3 ) = y 3. The reciprocal is true if θ satisfies a stronger condition than P2., namely, d(y 3, θ(y 1, y 2, y 3 )) = d(y 1, y 2 ). The following proposition provides examples of parallel manifolds. Proposition 4.2. Every Lie group with left-invariant metric is a parallel manifold. Proof. Let Y be a Lie group with a left-invariant metric d. We define the map θ : Y Y Y Y by setting θ(y 1, y 2, y 3 ) = y 3 y 1 1 y 2. We have θ(y 1, y 2, y 3 ) = y 2 y 3 y 1 1 y 2 = y 2 y 3 y 1 1 = 1 y 1 = y 3, d(y 3, θ(y 1, y 2, y 3 )) = d(y 3, y 3 y 1 1 y 2) = d(1, y 1 1 y 2) = d(y 1, y 2 ). The proposition is proved. Now, we prove that every compact Riemannian manifold without boundary is an ε-parallel manifold, what is not true for manifolds with boundary, as we will see. Proposition 4.3. Every closed Riemannian manifold is an ε-parallel manifold. Proof. Let consider the open subset Yε 3 of Y ε Y defined as Y ε 3 = {(y 1, y 2, y 3 ) Y ε Y : d(y 1, y 3 ) < ε}. We start defining the map θ for points in Yε 3. Consider the map β : Yε 3 R + given by d(y 1, y 2 ) d(y 1, y 3 ) 2 β(y 1, y 2, y 3 ) = (ε d(y 1, y 3 ))d(y 1, y 2 ) 2(ε d(y 1, y 2 )) if d(y 1, y 3 ) d(y 1, y 2 ), if d(y 1, y 3 ) > d(y 1, y 2 ). 1343

8 Epsilon Nielsen coincidence theory It is not hard to check that β is a continuous map. Also, we have β(y 1, y 2, y 3 ) = 0 if and only if y 1 = y 2 and, moreover, β(y 1, y 2, y 3 ) d(y 1, y 2 ) for all (y 1, y 2, y 3 ) Y 3 ε, with the equality β(y 1, y 2, y 3 ) = d(y 1, y 2 ) if and only if y 1 = y 3. Let (y 1, y 2, y 3 ) Yε 3 be arbitrary. There exists a unique geodesic c y1 y 2 from y 1 to y 2 and a unique geodesic c y1 y 3 from y 1 to y 3. Let v T y1 Y be the velocity vector of the path c y1 y 2 at the point y 1, that is, v = dc y1 y 2 /dt t=0. Let v T y3 Y be the parallel transport of v through the geodesic path c y1 y 3. From classical results of Riemannian Geometry (see [4] for example) there is a unique geodesic cy v 3, of length β(y 1, y 2, y 3 ) (remember that β(y 1, y 2, y 3 ) < ε), starting at y 3 and having at this point velocity vector v. Moreover, the map (y 1, y 2, y 3 ) cy v 3 is continuous. We define θ (y 1, y 2, y 3 ) to be the end point of the geodesic path cy v 3. From the construction, θ : Yε 3 Y is well defined and continuous. Now, we observe that β(y 1, y 2, y 3 ) converges to zero when d(y 1, y 3 ) converges to ε, and so θ (y 1, y 2, y 3 ) converges to y 3 when (y 1, y 2, y 3 ) converges to a point in the boundary Yε 3 of the set Yε 3. It implies that the map θ : Y ε Y Y defined by θ(y 1, y 2, y 3 ) = { θ (y 1, y 2, y 3 ) if (y 1, y 2, y 3 ) Y 3 ε, y 3 if (y 1, y 2, y 3 ) / Y 3 ε, is continuous. Moreover, we have, (i) θ(y 1, y 2, y 3 ) = y 2 if and only if y 1 = y 3 and (ii) d(y 3, θ(y 1, y 2, y 3 )) = β(y 1, y 2, y 3 ) d(y 1, y 2 ), what proves that Y is an ε-parallel manifold. With the previous proposition, we present below the theorem that allows us to conclude on the compatibility between the epsilon coincidence theory presented in this article with that presented by Brown in [3]. Theorem 4.4. Let f, g: X Y be maps into an ε-parallel manifold and let (f 1, g 1 ) be a pair of maps ε-homotopic to (f, g). Define two maps f, g : X Y as f (x) = θ(g 1 (x), g(x), f 1 (x)) and g (x) = θ(f 1 (x), f(x), g 1 (x)), where θ : Y ε Y Y is a map as in Definition 4.1. Then f and g are continuous maps with f ε f and g ε g and such that Coin(f, g) = Coin(f 1, g 1 ) = Coin(f, g ). Proof. Since (f 1, g 1 ) ε (f, g) we have d(f 1, f) < ε and d(g 1, g) < ε. Thus, for every x X we have (f 1 (x), f(x)) ε Y and (g 1 (x), g(x)) ε Y, what shows that the maps f and g are well defined. Moreover, it is obvious that they are continuous. We will prove that Coin(f, g) = Coin(f 1, g 1 ) and f ε f. The proof that Coin(f 1, g 1 ) = Coin(f, g ) and g ε g is analogous and so it is omitted. From property P1. it follows that f (x) = g(x) if and only if f 1 (x) = g 1 (x), what proves that Coin(f, g) = Coin(f 1, g 1 ). Now, from the property P2. we have d(f 1, f ) = sup x X d ( f 1 (x), θ(g 1 (x), g(x), f 1 (x)) ) sup d(g(x), g 1 (x)) = d(g, g 1 ). x X It follows that d(f, f ) d(f, f 1 ) + d(f 1, f ) d(f, f 1 ) + d(g, g 1 ) < ε, since (f, g) is ε-homotopic to (f 1, g 1 ). Therefore, f is ε-homotopic to f. Corollary 4.5. If f, g: X Y are maps into an ε-parallel manifold, then there exist maps f, g : X Y such that both pairs (f, g) and (f, g ) provide µ ε (f, g). Proof. Let (f 1, g 1 ) be a pair of maps providing µ ε (f, g). By Theorem 4.4, there are maps f, g : X Y with f ε f and g ε g such that # Coin(f, g) = µ ε (f, g) = # Coin(f, g ). The following theorem concentrates all the results of this section specifically related with epsilon coincidence theory. 1344

9 M.C. Fenille Theorem 4.6. Let f, g: X Y be maps into a closed Riemannian manifold. Then µ ε [f, g) = µ ε (f, g) = µ ε (f, g], what implies that the ε-minimum fixed point number MF ε (f) of a map f : Y Y (defined by Brown in [3]) is a particular case of the ε-minimum coincidence number µ ε (f, g) of maps f, g: X Y in the case in which X = Y is a closed Riemannian manifold and g is the identity map of Y. Proof. The equality µ ε [f, g) = µ ε (f, g) = µ ε (f, g] is from the previous corollary. Now, let MR ε (φ) be the ε-minimum fixed point number of a map φ : Y Y, as defined in [3], and let id Y : Y Y be the identity map of Y. We have MR ε (φ) = min {# Fix(φ ) : d(φ, φ ) < ε} = min {# Coin(φ, id Y ) : d(φ, φ ) < ε} = µ ε (φ, id Y ] = µ ε (φ, id Y ), what proves the theorem. The following proposition shows that Theorem 4.6 is not necessarily true if the manifold Y has nonempty boundary. It also shows that the closed interval I is not an ε-parallel manifold. Proposition 4.7. Let I be a closed interval. distinguished. There are maps f, g: I I such that the numbers µ ε (f, g), µ ε [f, g) and µ ε (f, g] are all Proof. Figure 3 shows maps f, g: I I such that µ ε (f, g) = 0, µ ε [f, g) = 1 and µ ε (f, g] = 2. Figure 3. µ ε (f, g) = 0, µ ε [f, g) = 1 and µ ε (f, g] = 2 The following facts are decisive for the validity of the result stated in the previous proposition: Coin(f, g) I. Im(f, g) I, where Im(f, g) = f(coin(f, g)) = g(coin(f, g)). There exists x Coin(f, g)\ Y such that f(x) = g(x) Y. In fact, we can prove that, if one of these facts does not occur, then at least two of the numbers µ ε (f, g), µ ε [f, g) and µ ε (f, g] are equal. All this suggests the following result, which is a version of Theorem 4.6 for the case in which the manifold Y is not necessarily closed. Proposition 4.8. Let Y be a compact Riemannian manifold with nonempty boundary Y. If f, g: Y Y are maps such that both the sets Coin(f, g) and Im(f, g) are far enough from Y, then µ ε (f, g) = µ ε [f, g) = µ ε (f, g]. 1345

10 Epsilon Nielsen coincidence theory 5. Epsilon Nielsen root theory Beyond the compatibility with the fixed point theory, the results of Section 4 provide automatically an epsilon root theory. Indeed, let consider a map f : X Y between orientable and closed Riemannian manifolds of the same dimension and let a Y be a point. Denote also by a: X Y the constant map at a. Then f 1 (a) = Coin(f, a) and we may define the ε-minimum root number of f at a to be the ε-minimum coincidence number of the pair (f, a). Since by Section 4 we have µ ε (f, a) = µ ε (f, a] = min { # f 1 1 (a) : f 1 ε f }, the number µ ε (f, a) is in fact a number worthy to be called the ε-minimum root number of the map f at a. Furthermore, considering again a as a (constant) map, we may easily define an ε-nielsen root number of the map f at a simply by N ε (f, a). We will prove next a Minimum Theorem for epsilon root theory, as Brown proved an analogous theorem for epsilon fixed point theory. Our proof is, in some points, similar to the Brown s proof. It is obvious that the Minimum Theorems for roots and for fixed points are independent of each other, that is, none of them implies the other. On the other hand, a potential Minimum Theorem for epsilon coincidence theory would imply both, the Minimum Theorem for roots and the Minimum Theorem for fixed point. However, although we believe that the Minimum Theorem holds for epsilon coincidence theory, we still do not have a proof. Theorem 5.1 (Minimum Theorem for roots). Given a map f : X Y between orientable closed Riemannian manifold of the same dimension n and a point a Y, there exists a map f : X Y with d(f, f ) < ε and having exactly N ε (f, a) roots at the point a, that is, # f 1 (a) = N ε (f, a). Proof. Let C ε j = f 1 (a) ε (f, a) be an ε-root class. There exists an open and connected subset V j of ε j (f, a) containing C ε j whose closure cl(v j ) is contained in ε (f, a). For the map D f,a : X R + defined by D f,a (x) = d(f(x), a) we see that D f,a (cl(v j )) = [0, δ j ] where δ j < ε. Choose α j > 0 small enough so that δ j + 3α j < ε. We will define f outside ε (f, a) to be a simplicial approximation to f such that d(f, f ) < α, where α = min {α j }. The proof then consists of describing f on each component ε j (f, a) of ε (f, a) so, to simplify notation, we will assume for now that ε (f, a) is connected and thus we are able to suppress the subscript j. First, we use a Hopf-type construction and consider a map u: X Y such that d(u, f) < α and u 1 (a) is a finite set. See [2] for details on the Hopf construction for fixed points and see [5] for details on the Hopf construction for coincidences. Using simplicial technique as in [3], we may construct an open set W V containing u 1 (a) and a homeomorphism φ: W R n such that φ(u 1 (a)) int(b 1 ), where B 1 is the unit closed ball in R n. We denote B 1 = φ 1 (B 1 ), 0 = φ 1 (0). Define the retraction ρ: B 1 0 B 1 of B 1 0 onto the boundary B 1 by ( ) 1 ρ(x) = φ 1 φ(x) φ(x). For each x B1 0 we have ρ(x) B1 W V, then d(uρ(x), a) < δ. So, there exists a unique geodesic c auρ(x) from a to uρ(x), which we consider defined for t I = [0, 1]. For each t I, let cauρ(x) t be the length of that geodesic 0 from t = 0 to t = t. The (continuous!) map (B 1 0 ) I (x, t) ( c auρ(x), t ) c auρ(x) t 0 maps (B 1 0 ) {0} to zero. Thus, there exists t > 0 such that cauρ(x) t 0 < ε δ 2α, x B

11 M.C. Fenille Define a (continuous!) map u 1 : B 1 Y by setting u 1(0 ) = a and, for x 0, u 1 (x) = c auρ(x) ( φ(x) t ). Since u 1 (a) B 1 =, the map u 1 checks u 1 1 (a) = {0 }. Moreover, for x B 1 0 we have d(u(x), u 1 (x)) d(u(x), a) + d(a, u 1 (x)) δ + cvρ(x)uρ(x) φ(x) t 0 < δ + ε δ 2α = ε 2α. It is clear that for x = 0 we have the same inequality, namely, d(u(0 ), u 1 (0 )) = d(u(0 ), a) δ < ε 2α. Therefore, the map u 1 satisfies u 1 1 (a) = {0 }, d(u(x), u 1 (x)) < ε 2α, x B 1. Next, we will extend u 1 on a specific set containing B1. We start by constructing this set. By straightforward arguments of general topology, we may choose r > 1 small enough so that d ( uφ 1 (z), uφ 1 (z/ z ) ) < α for all z in the annulus A = {z R n : 1 z r }. It is obvious that B r = B 1 A and B 1 = B r A. We define A = φ 1 (A), Br = φ 1 (B r ). Thus, we have Br = B1 A and B1 = B r A. We extend ρ on Br 0 defining again ρ: Br 0 B1 by By construction, for each x Br int(b1 ) we have ( ) 1 ρ(x) = φ 1 φ(x) φ(x). d(u(x), uρ(x)) < α, d(u(x), a) < δ, d(a, uρ(x)) < δ. Thus, there exists a unique geodesic c auρ(x) from a to uρ(x) and there exists a unique geodesic c uρ(x)u(x) from uρ(x) to u(x), both defined on I = [0, 1]. We define H(x, ): [0, 1] Y and K(x, ): [1, 2] Y by H(x, t) = c auρ(x) (t), K(x, t) = c uρ(x)u(x) (1 t). We also define (H K)(x, ): [0, 2] Y by (H K)(x, t) = { H(x, t) if 0 t 1, K(x, t) if 1 t 2. This map is clearly well defined and continuous. Finally, we define u 2 : B r int(b 1 ) Y by ( u 2 (x) = (H K) x, t 2 φ(x) + 2 r ) t. 1 r 1 r We remark that for each x Br int(b1 ), the point u 2(x) lies in one of the geodesics c auρ(x) or c uρ(x)u(x). Hence, for each such x we have d(u(x), u 2 (x)) cauρ(x) 1 + cuρ(x)u(x) 1 δ + α < ε 2α. 0 0 Moreover, x B 1 = u 2 (x) = (H K)(x, t ) = (H K)(x, φ(x) t ) = u 1 (x), x B r = u 2 (x) = (H K)(x, 2) = u(x). Thus, we may use the maps u, u 1 and u 2 to define the continuous map u : X Y by u 1 (x) if x B1, u (x) = u 2 (x) if x Br B1 u(x) if x / Br. 1347

12 Epsilon Nielsen coincidence theory By construction we have u 1 (a) = {0 }, d(u, u ) < ε 2α. Now, if ind(f, a, ε (f, a)) 0, we let f = u : X Y. On the other hand, if ind(f, a, ε (f, a)) = 0, it is well known from the classical root theory that there exists a map f : X Y, identical to u outside B 1, such that f has no root in B 1 and d(f, u ) < α. Moreover, we have d(f, f ) d(f, u) + d(u, u ) + d(u, f ) < α + ε 2α + α = ε. Thus concludes the proof in the case in which ε (f, a) is connected. We return now to the general case in which ε (f, a) may not be connected. Applying the construction above to each component ε j (f, a) of ε (f, a) gives us maps f j : X Y which coincide with u outside ε j (f, a). These maps may be put together to define a map f : X Y that coincides with u outside ε (f, a) and with f j on each ε j (f, a). Moreover, d(f, f ) < ε and f 1 (a) = N ε (f, a). Acknowledgements This work is sponsored by Fundação de Amparo a Pesquisa do Estado de Minas Gerais FAPEMIG Grant CEX- APQ References [1] Brooks R.B.S., On removing coincidences of two maps when only one, rather than both of them, may be deformed by a homotopy, Pacific J. Math., 1972, 40, [2] Brown R.F., The Lefschetz Fixed Point Theorem, Scott, Foresman, Glenview London, 1971 [3] Brown R.F., Epsilon Nielsen fixed point theory, Fixed Point Theory Appl., 2006, Special Issue, # [4] do Carmo M.P., Riemannian Geometry, Math. Theory Appl., Birkhäuser, Boston, 1992 [5] Cotrim F.S., Homotopias Finitamente Fixadas e Pares de Homotopias Finitamente Coincidentes, M.Sc. thesis, Universidade Federal de São Carlos, São Carlos, 2011 [6] Gonçalves D.L., Coincidence theory, In: Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 3 42 [7] Milnor J., Morse Theory, Ann. of Math. Stud., 51, Princeton University Press, Princeton, 1963 [8] Munkres J.R., Topology, 2nd ed., Prentice-Hall, Englewood Cliffs, 2000 [9] Vick J.W., Homology Theory, 2nd ed., Grad. Texts in Math., 145, Springer, New York,