Cutting surfaces and applications to periodic points and chaotic-like dynamics
aa r X i v : . [ m a t h . D S ] A p r CUTTING SURFACES AND APPLICATIONS TO PERIODICPOINTS AND CHAOTIC − LIKE DYNAMICS MARINA PIREDDU AND FABIO ZANOLIN
Abstract.
In this paper we propose an elementary topological approach whichunifies and extends various different results concerning fixed points and periodicpoints for maps defined on sets homeomorphic to rectangles embedded in euclideanspaces. We also investigate the associated discrete semidynamical systems in viewof detecting the presence of chaotic − like dynamics. − Miranda theorem, Leray − Schauder continuation theorem and itsgeneralizations, zeros of parameter dependent vector fields, chaotic dynamics, topologicalhorseshoes. Introduction
The celebrated Smale horseshoe can be rightly considered as a prototypical examplein the study of complex systems. It deals with a homeomorphism ψ, defined on a setdiffeomorphic to a rectangle in a two − dimensional manifold, possessing an invariantset Λ such that ψ | Λ is conjugate to the two − sided Bernoulli shift on two symbols.Such striking model exploits a simple and elegant geometric description in order todisplay the main features associated to all the various different definitions of chaos.Since the beginning [41, 42], its clear and intuitive geometrical structure turned outto be very useful in the study of dynamical systems, allowing a rigorous proof of thepresence of a complex behavior in several significant ODE models [31, Ch.3]. Oneof the crucial assumptions regarding the implementation of this method concernsthe verification of hyperbolicity. Usually, this concept is meant as the existence ofa splitting of the domain into a part along which ψ is contracting and another onewhere ψ is expanding, the two parts being filled by graphs of lipschitzean functions(see [31, 48]). However, in certain applications to differential systems, the verificationof hyperbolicity requires the smoothness of the involved maps and conditions ontheir jacobian matrix which lead sometimes to formidable and difficult computations(see [31, p.62]). This remark or similar ones carried various authors to look for aclass of relaxed assumptions in view of producing a structure as rich as before, butpossibly replacing the hyperbolicity hypothesis with topological conditions which, insome cases, require the knowledge of the behavior of ψ only on some subsets of itsdomain (for instance, at the boundary of certain sets). Results in this direction wereobtained by Easton [8], Burns and Weiss [3], Mischaikow and Mrozek [30], Szymczak[45], Zgliczy´nski [53], Srzednicki and W´ojcik [44] and further developed in subsequentworks (see, for instance, [32, 38, 43, 54, 55, 56] and the references therein, just toquote a few samples from a broad bibliography). In these papers usually the authorsprove the existence of a compact set Λ which is (forward) invariant (either for a THIS WORK HAS BEEN SUPPORTED BY THE PROJECT ”EQUAZIONI DIFFERENZIALI ORDI-NARIE E APPLICAZIONI” (PRIN 2005). given map ψ or for an iterate of it) and the existence of a continuous surjection g : Λ → Σ m := { , . . . , m − } Z (or g : Λ → Σ + m := { , . . . , m − } N ) which provides asemiconjugation of ψ | Λ with the two − sided (respectively one − sided) Bernoulli shift.In many cases the authors also show that the inverse image through g of a periodicsequence of m symbols contains a periodic point of ψ in Λ . The tools employed inthe related proofs are based on different topological methods, like the Conley indextheory (with associated homological or cohomological invariants) or some more or lesssophisticated fixed point methods (as the topological degree, the fixed point index orthe Lefschetz theory) which usually require the verification of suitable conditions (fora flow or for a map) at the boundary of a certain domain containing the invariant setin its interior.A different generalization of the Smale’s horseshoe has been obtained in recentyears by Kennedy and Yorke [20] and Kennedy, Ko¸cak and Yorke [18], who developedthe theory of the so-called topological horseshoes in the frame of metric spaces. Thistheory concerns the study of the behavior of a continuous map f : X ⊇ Q → X andof its iterates, where Q is a compact subset of a metric space X. Two disjoint compactsubsets end and end of Q are selected and the crossing number m is defined as thelargest number of disjoint preconnections contained in any connection. A connectionΓ ⊆ Q is a compact connected set (a continuum) with Γ ∩ end = ∅ and Γ ∩ end = ∅ , while a preconnection γ ⊆ Γ is a compact set such that f ( γ ) is a connection. Undera few more technical conditions (see [18, 20] for the precise statements) and if m ≥ , the authors prove the existence of a compact invariant set Q I ⊆ Q such that f | Q I issemiconjugate (via a surjective map g ) to a Bernoulli shift on m symbols. Applicationsof such results to different ODE models have been shown in [15, 50, 51, 52] and inrecent related papers. The great generality of the setting in [18, 20] does not lead tothe conclusion that the inverse image (through the surjection g ) of a periodic sequencein Σ m must necessarily contain a periodic point in Q I . In fact, a specific example ofan invariant set Q I without periodic points is presented in [20, p.2520].Another approach has been proposed by Papini and the second author in [34],motivated by the study of the Poincar´e map associated to a second order nonlineardifferential equation with sign changing weight [33]. In [34] and some subsequentmore general works [35, 36], the authors considered continuous planar maps whichpossess the property of stretching the paths joining two opposite sides of a topologicalrectangle. In this case the paths play a role analogous to that of the connections inKennedy and Yorke theory. Such a special configuration permits to obtain not onlychaotic dynamics, but also the existence of fixed points and periodic points (by meansof elementary topological arguments). In this manner one could complement someresults in the two − dimensional setting (like those in [52]) and provide the existenceof infinitely many periodic points (of any order) as well. Other applications of theresults in [34] can be found in [6, 35], dealing with second order ODEs. See also [37]for more information, although we have to warn that a precise relationship betweenthese works and the theory of topological horseshoes as presented in [18, 20] was notexplicitly stated therein. On the contrary, such relationship will be described in thepresent work (see Lemma 5.1), including some details elsewhere missing.The aim of this paper is twofold. In fact we provide an elementary topological toolthat, on the one hand, allows us to extend to the higher dimension the theory of[34, 35, 36] and to make a comparison with the above recalled results on chaotic − likedynamics, while, on the other hand, enables to generalize and unify some previous UTTING SURFACES 3 theorems about the existence of fixed points and periodic points for continuous mapsin euclidean spaces. In particular, we give a simplified proof of a recent result byKampen [17] as well as we present a generalization of some preceding theorems aboutperiodic points associated to Markov partitions [55]. As remarked above, one of thepurposes of our work is also that of obtaining a sufficiently rich structure (suitable, forinstance, to guarantee the existence of infinitely many periodic points), without theneed of too sophisticated methods. In fact, our main tools are, respectively, a mod-ified version of the classical Hurewicz − Wallman intersection lemma [16, D), p.40],[10, p.72] and the fundamental theorem of Leray − Schauder [26, Th´eor`eme Fonda-mental]. The former result (also referred to Eilenberg − Otto [9, Th.3], according to[7, Theorem on partitions, p.100]) is one of the basic lemmas in dimension theory andit is known to be one of the equivalent versions of the Brouwer fixed point theorem.It may be interesting to observe that extensions of the intersection lemma led togeneralizations of the Brouwer fixed point theorem to some classes of possibly non-continuous functions [47]. The other result we use concerns topological degree theoryand has found important applications in nonlinear analysis and differential equations(see [28]). Actually, in Theorem 4.1 of Section 4, we’ll employ a more general ver-sion of the Leray − Schauder continuation theorem due to Fitzpatrick, Massab´o andPejsachowicz [12]. However, we point out that for our applications to the study ofperiodic points and chaotic − like dynamics we rely on the classical Leray − Schauderfundamental theorem. Another particular feature of our approach consists in a com-bination of the Poincar´e − Miranda theorem [22, 29] with the properties of topologicalsurfaces cutting the arcs between two given sets. The corresponding details are widelydescribed in Section 2 and Section 3. In regard to the Poincar´e − Miranda theorem,which is an N − dimensional version of the intermediate value theorem (see Theorem3.1), we would like to recall also the recent work [1], where the authors show itseffectiveness in detecting chaotic dynamics in planar dynamical systems.This article is organized as follows. Section 2 is devoted to the presentation ofsome topological lemmas concerning the relationship between cutting surfaces andzero sets of continuous real valued functions. Analogous results can be found, oftenin a more implicit form, in different contexts. Since for our applications we need aspecific version of the statements, we give an independent proof with all the details.The remainder of the paper is divided in three parts. In Section 3 we present a variantof Hurewicz − Wallman theorem (Theorem 3.2), that (likewise [9, 16]) guarantees anonempty intersection among N closed topological surfaces which are in good posi-tion with respect to the faces of an N − dimensional hypercube. Such result is thenapplied in order to provide an extension of some recent theorems about fixed pointsand periodic points for continuous mappings in euclidean spaces. In particular, wegeneralize (with a simplified proof) a theorem by Kampen [17] (see Corollary 3.1)and obtain also an extension of a result by Zgliczy´nski [55] (see Theorem 3.3). InSection 4 we consider the case in which, roughly speaking, the number of intersectingsurfaces is smaller than the dimension of the space. After presenting a general situ-ation in Theorem 4.3, we restrict ourselves to the case of the intersection of N − N − dimensional hy-percube (see Corollary 4.1). This latter result turns out to be our main ingredientin Section 5 for the study of the dynamics of continuous maps which possess, in avery broad sense, a one − dimensional expansive direction. Indeed, it allows to provea fixed point theorem (see Theorem 5.1) for maps defined on topological rectangles. M. PIREDDU, F. ZANOLIN
The main hypothesis for our fixed point theorem requires that the map ψ stretches(across X ) the paths joining two disjoint subsets X ℓ and X r of the boundary of thetopological rectangle X. Formally, such stretching condition is expressed as follows:there exists a compact set
K ⊆ X on which ψ is continuous and such that, for anypath γ connecting in X the two sides X ℓ and X r , there exists a sub-path σ of γ contained in K , whose image through ψ is contained in X and joins the same twosides of the boundary (as shown in Figure 1). A special feature of our theorem isthat it ensures the existence of a fixed point for ψ in K , that, in turns, yields tothe existence of multiple fixed points when the stretching condition is satisfied withrespect to a certain number of pairwise disjoint subsets K i ’s of X. An application ofthe theorem to the iterates of ψ will then allow to obtain a set of infinitely manyperiodic points with a complex structure. More generally, we define, for a mapping ψ, a concept of stretching between two (possibly different) oriented N -dimensionalrectangles ( X, X − ) and ( Y, Y − ) , where the [ · ] − -sets are the union of two suitablychosen left and right sides of the boundary (see Definition 5.1 and Figure 1). Figure 1. The tubular sets X and Y in the picture represent two generalized3 − dimensional rectangles, in which we have put in evidence the compactset K and the boundary sets X ℓ and X r as well as Y ℓ and Y r (see Definition5.1 for more details). In this particular case the map ψ stretches the pathsof X not only across Y, but also across X itself and therefore the existenceof a fixed point for ψ inside K is ensured by Theorem 5.1. Note that,differently from the classical Rothe and Brouwer theorems, we don’t requirethat ψ ( ∂X ) ⊆ X. UTTING SURFACES 5
In this manner we fully extend to an arbitrary dimension the above recalled resultsby Papini and Zanolin [34, 35, 36]. As a consequence, also all the applications toordinary differential equations and dynamical systems contained in those articles andin related ones (like [4, 6, 33]) are, in principle, extendable to the higher dimension.The specific applications, which would require a detailed treatment, will be presentedin a future work.We end this introductory section with a list of basic definitions and notation. Somefurther concepts will be introduced later on in the paper when needed.As usual we denote by R , R + := [0 , + ∞ ) and R +0 := ]0 , + ∞ ) the sets of reals, aswell as the nonnegative and positive real numbers, respectively. The sets of integers Z and natural numbers N = { , , , . . . } are considered as well. For a subset M of a topological space Z, we denote by M o and M the interior and the closure of M, respectively. For a metric space ( X, d ) we indicate with B ( x , r ) the open ballof center x ∈ X and radius r > . Similarly, given M ⊆ X ( M = ∅ ), we define B ( M, r ) := { x ∈ X : ∃ w ∈ M, d ( x, w ) < r } . By a continuum we mean a compactand connected subset of a metric space (i.e. a metric continuum).Let Z be a topological space. By a path γ in Z we mean a continuous mapping(parameterized curve) γ : R ⊇ [ a, b ] → Z and we denote by γ its range, that is γ := γ ([ a, b ]) . A sub-path σ of γ is defined as σ := γ | [ c,d ] , for [ c, d ] ⊆ [ a, b ] , that is the restriction of γ to a closed subinterval of its domain. If Z, Y are topologicalspaces and ψ : Z ⊇ D ψ → Y is a map which is continuous on a set M ⊆ D ψ , thenfor any path γ in Z with γ ⊆ M , it follows that ψ ◦ γ is a path in Y with range equalto ψ ( γ ) . As usual in the theory of (continuous) parameterized curves, there is no lossof generality in assuming the paths to be defined on [0 , . In fact, if θ : [ a , b ] → Z and θ : [ a , b ] → Z, with a i < b i , i = 1 , , are two paths in Z, we set θ ∼ θ ifthere exists a homeomorphism h of [ a , b ] onto [ a , b ] (i.e., a change of variable inthe parameter) such that θ ( h ( t )) = θ ( t ) , ∀ t ∈ [ a , b ] . It is easy to check that “ ∼ ”is an equivalence relation and that if θ ∼ θ , then the ranges of θ and θ coincide.Hence, for any path γ we can find an equivalent path defined on [0 , . If γ , γ : [0 , → Z are two paths in Z with γ (1) = γ (0) , we define the gluing of γ with γ as the path γ ⋆ γ : [0 , → Z such that γ ⋆ γ ( t ) := (cid:26) γ (2 t ) for 0 ≤ t ≤ ,γ (2 t −
1) for ≤ t ≤ . Moreover, given a path γ : [0 , → Z, we denote by γ − : [0 , → Z the path having γ as support, but run with reverse orientation, i.e. γ − ( t ) := γ (1 − t ) , for all t ∈ [0 , . At last we recall a known definition. Let Z be a topological space. We say that Z is arcwise connected if, given any two different points p, q ∈ Z, there is a path γ : [0 , → Z such that γ (0) = p and γ (1) = q. In the case of a Hausdorff topologicalspace Z, the range γ of γ turns out to be a locally connected metric continuum (aPeano space according to [14]). Then, if Z is a metric space, the above definition ofarcwise connectedness is equivalent to the fact that, given any two points p, q ∈ Z with p = q, there exists an arc (that is the homeomorphic image of a compact interval)contained in Z and having p and q as extreme points (see, e.g., [14, pp.115–131]). M. PIREDDU, F. ZANOLIN Topological lemmas
As already mentioned in the Introduction, in this section we present some topo-logical lemmas in order to show the relationship between particular surfaces and zerosets of continuous real valued functions. First of all, we need a definition.
Definition 2.1.
Let X be an arcwise connected metric space. Let A, B, C ⊆ X beclosed and nonempty sets with A ∩ B = ∅ . We say that C cuts the arcs between A and B if for any path γ : [0 , → X, with γ ∩ A = ∅ and γ ∩ B = ∅ , it follows that γ ∩ C = ∅ . In the sequel, if X is a subspace of a larger metric space Z and we wishto stress the fact that we consider only paths contained in X, we make more preciseour definition by saying that C cuts the arcs between A and B in X .Such definition is a modification of the classical one regarding the cutting of a spacebetween two points in [25]. See also [2] for a more general concept in which the authorsconsider a set C which intersects every connected set meeting two nonempty sets A and B. In the case in which A and B are the opposite faces of an N − dimensionalcube, J. Kampen [17, p.512] says that C separates A and B. We prefer to use the“cutting” terminology in order to avoid misunderstanding with other definitions ofseparation which are more common in Topology. In particular (see [11]), we remarkthat our definition agrees with the usual one of cut when
A, B, C are pairwise disjoint.In the sequel, even when not explicitly mentioned, we assume that the basic space X is arcwise connected. In some of the next results the local arcwise connectednesswill be required too. With this respect, we recall that any connected and locallyarcwise connected metric space is arcwise connected (see [25, Th.2, p.253]). Lemma 2.1.
Let X be a connected and locally arcwise connected metric space andlet A, B, C ⊆ X be closed and nonempty sets with A ∩ B = ∅ . Then C cuts the arcsbetween A and B if and only if there exists a continuous function f : X → R suchthat f ( x ) ≤ , ∀ x ∈ A, f ( x ) ≥ , ∀ x ∈ B (2.1) and C = { x ∈ X : f ( x ) = 0 } . (2.2) Proof.
Assume there exists a continuous function f : X → R satisfying (2.1) and(2.2). Let γ : [0 , → X be a continuous path such that γ (0) ∈ A and γ (1) ∈ B. We want to prove that γ ∩ C = ∅ . Indeed, for the composite continuous function θ := f ◦ γ : [0 , → R , we have that θ (0) ≤ ≤ θ (1) and therefore Bolzano theoremensures the existence of t ∗ ∈ [0 ,
1] with θ ( t ∗ ) = 0 . This means that γ ( t ∗ ) ∈ C andtherefore γ ∩ C = ∅ . Thus we have proved that C cuts the arcs between A and B. Conversely, let us assume that C cuts the arcs between A and B. We define thefollowing auxiliary functions ρ : X → R + ,ρ ( x ) := dist( x, C ) , ∀ x ∈ X (2.3)and µ : X → {− , , } ,µ ( x ) := x ∈ C, − x C and there exists a path γ x : [0 , → X \ C such that γ x (0) ∈ A and γ x (1) = x, UTTING SURFACES 7
Observe that ρ is a continuous function with ρ ( x ) = 0 if and only if x ∈ C and also µ ( x ) = 0 if and only if x ∈ C. Moreover, µ is bounded.Let x C. We claim that µ is continuous in x . Actually, µ is locally constant on X \ C. Indeed, since x ∈ X \ C (an open set) and X is locally arcwise connected,there is a neighborhood U x of x with U x ⊆ X \ C, such that for each x ∈ U x thereexists a path σ x ,x joining x to x in U x . Clearly, if there is a path γ a,x in X \ C joining some point a ∈ A with x , then the path γ a,x ⋆ σ x ,x connects a to x in X \ C. This proves that if µ ( x ) = − , then µ ( x ) = − x ∈ U x . On the otherhand, if there is a path γ a,x in X \ C which connects some point a ∈ A to x ∈ U x , then, the path γ a,x ⋆ σ − x ,x connects a to x in X \ C. This shows that if µ ( x ) = 1(that is, it is not possible to connect x to any point of A in X \ C using a path), then µ ( x ) = 1 for every x ∈ U x (that is, it is not possible to connect any point x ∈ U x toany point of A in X \ C using a path). We can now define f : X → R , f ( x ) := ρ ( x ) µ ( x ) . (2.5)Clearly, f ( x ) = 0 if and only if x ∈ C and, moreover, f is continuous. Indeed, if x C, we have that f is continuous at x because both ρ and µ are continuous in x . If x ∈ C and x n → x (as n → ∞ ), then ρ ( x n ) → | µ ( x n ) | ≤ , so that f ( x n ) → f ( x ) . Finally, by the definition of µ in (2.4), it holds that µ ( a ) = − , for every a ∈ A \ C and therefore, for such an a , it holds that f ( a ) < . On thecontrary, if we suppose that b ∈ B \ C, we must have µ ( b ) = 1 . In fact, by thecutting condition, there is no path connecting in X \ C the point b to any point of A. Therefore, in this case we have f ( b ) > . The proof is complete. (cid:3)
Considering the functions µ and f as in (2.4), (2.5), we see that our definition,although adequate from the point of view of the proof of Lemma 2.1, perhaps doesnot represent an optimal choice. For instance, we would like the sign of f to coincidefor all the points located “at the same side” of A (respectively of B ) with respect to C. Having this request in mind, we propose a different definition for the function µ (see (2.8)). First of all, we introduce the following sets that we call the side of A in X and the side of B in X , respectively. S ( A ) := { x ∈ X : γ ∩ A = ∅ , ∀ path γ : [0 , → X, with γ (0) = x, γ (1) ∈ B } , S ( B ) := { x ∈ X : γ ∩ B = ∅ , ∀ path γ : [0 , → X, with γ (0) = x, γ (1) ∈ A } , that is, a point x belongs to the side of A (resp. to the side of B ) if whenever we tryto connect x to B (resp. to A ) by a path, we first meet A (resp., we first meet B ).By definition, it follows that A ⊆ S ( A ) , B ⊆ S ( B ) . Lemma 2.2.
Let X be a connected and locally arcwise connected metric space andlet A, B ⊆ X be closed and nonempty sets with A ∩ B = ∅ . Then S ( A ) and S ( B ) areclosed and, moreover, S ( A ) ∩ S ( B ) = ∅ . Proof.
First of all, we prove that S ( A ) is closed by checking that if w
6∈ S ( A ) thenthere is a neighborhood U w of w with U w ⊆ X \ S ( A ) . Indeed, if w
6∈ S ( A ) , thereexists a path γ : [0 , → X with γ (0) = w and γ (1) = b ∈ B and such that γ ( t ) A, for every t ∈ [0 , . Since A is a closed set and X is locally arcwise connected, thereexists an arcwise connected open set V w with w ∈ V w ⊆ X \ A. Hence, for every x ∈ V w , there is a path σ x connecting x to w in V w . As a consequence, we find thatthe path γ x := σ x ⋆ γ connects x ∈ V w to b ∈ B with γ x ⊆ X \ A. Clearly, the open
M. PIREDDU, F. ZANOLIN neighborhood U w := V w satisfies our requirement and this proves that X \ S ( A ) isopen. The same argument ensures that S ( B ) is closed.It remains to show that S ( A ) and S ( B ) are disjoint. Since A ∩ B = ∅ , it followsimmediately from the definition that A ∩ S ( B ) = ∅ , B ∩ S ( A ) = ∅ . Now assume, by contradiction, that there exists x ∈ S ( A ) ∩ S ( B ) , with x A ∪ B. Let γ : [0 , → X be a path such that γ (0) = x and γ (1) = b ∈ B (we know that there exists a path like that because X is arcwise connected). Thefact that x ∈ S ( A ) \ A implies that there is t ∈ ]0 ,
1[ such that γ ( t ) = a ∈ A. Onthe other hand, since x ∈ S ( B ) \ B, there exists s ∈ ]0 , t [ such that γ ( s ) = b ∈ B. Proceeding by induction and using repeatedly the definition of S ( A ) and S ( B ) weobtain a sequence t > s > t > · · · > t j > s j > t j +1 > · · · > γ ( t i ) = a i ∈ A and γ ( s i ) = b i ∈ B. For t ∗ = inf t n = inf s n ∈ [0 , , we have that γ ( t ∗ ) = lim a n = lim b n ∈ A ∩ B, a contradiction. The proof is complete. (cid:3) Lemma 2.3.
Let X be a connected and locally arcwise connected metric space andlet A, B, C ⊆ X be closed and nonempty sets with A ∩ B = ∅ . Then C cuts the arcsbetween A and B if and only if C cuts the arcs between S ( A ) and S ( B ) . Proof.
One direction of the inference is obvious. In fact, every path joining A with B is also a path joining S ( A ) with S ( B ) . Thus, if C cuts the arcs between S ( A ) and S ( B ) , then it also cuts the arcs between A and B. Conversely, let us assume that C cuts the arcs between A and B. We want to prove that C cuts the arcs between S ( A )and S ( B ) . By the definition of S ( A ) and S ( B ) , it is straightforward to check that C cuts the arcs between A and S ( B ) as well as it cuts the arcs between S ( A ) and B. Suppose now, by contradiction, that there exists a path γ : [ , → X \ C such that γ ( ) = a ∈ S ( A ) \ A and γ (1) = b ∈ S ( B ) \ B. We choose a point a ∈ A and connect it to a ∈ S ( A ) by a path σ : [0 , ] → X with σ (0) = a and σ ( ) = a. We define also the new path γ := σ ⋆ γ : [0 , → X, with γ (0) = a ∈ A and γ (1) = b ∈ S ( B ) . By the definition of S ( B ) we know that there exists s ∈ ]0 , γ ( s ) = b ∈ B. Note that b
6∈ S ( A ) (recall that B ∩ S ( A ) = ∅ ) and also0 < s < (in fact if < s < , then, recalling that γ ( ) = γ ( ) = a ∈ S ( A ) and γ ( s ) = γ ( s ) = b ∈ B, there must be a ˜ t ∈ [ , s ] such that γ (˜ t ) ∈ C, a contradictionto the assumption on γ ). The restriction of the path γ to the interval [ s , ] , definesa path joining b ∈ B to a ∈ S ( A ) . Therefore there exists t ∈ ] s , [ such that γ ( t ) = a ∈ A. The restriction of the path γ to the interval [ t , , defines a pathjoining a ∈ A to b ∈ S ( B ) . Hence there exists s ∈ ] t ,
1[ with γ ( s ) = b ∈ B. Asabove, we can also observe that b
6∈ S ( A ) and t < s < . Proceeding by induction,we obtain a sequence s < t < s < · · · < s j < t j < s j +1 < γ ( t i ) = a i ∈ A and γ ( s i ) = b i ∈ B. For t ∗ = sup t n = sup s n ∈ ]0 , ] , we have that γ ( t ∗ ) = σ ( t ∗ ) = lim a n = lim b n ∈ A ∩ B, a contradiction. The proof is complete. (cid:3) UTTING SURFACES 9
Lemma 2.4.
Let X be a connected and locally arcwise connected metric space andlet A, B, C ⊆ X be closed and nonempty sets with A ∩ B = ∅ . Then C cuts the arcsbetween A and B if and only if there exists a continuous function f : X → R suchthat f ( x ) ≤ , ∀ x ∈ S ( A ) , f ( x ) ≥ , ∀ x ∈ S ( B ) (2.6) and C = { x ∈ X : f ( x ) = 0 } . (2.7) Proof.
Clearly, if there exists a continuous function f : X → R satisfying (2.6) and(2.7), then (2.1) and (2.2) hold too. Hence Lemma 2.1 implies that C cuts the arcsbetween A and B. Conversely, if C cuts the arcs between A and B, then, by Lemma 2.3, C cuts thearcs between S ( A ) and S ( B ) as well. Therefore we can apply Lemma 2.1 with respectto the triple ( S ( A ) , S ( B ) , C ) . In particular, the function f will be defined as in (2.5),with ρ like in (2.3) and µ : X → {− , , } as follows: µ ( x ) := x ∈ C, − x C and there exists a path γ x : [0 , → X \ C such that γ x (0) ∈ S ( A ) and γ x (1) = x, (cid:3) Figure 2. The figure on the left-hand side gives an illustration of the sit-uation described in Definition 2.1. In the present example the space X isthe figure itself as a subset of the plane. The set C (darker region) cutsthe arcs between A (light) and B (grey). We allow a nonempty intersectionbetween B and C as well as between A and C (the singleton { a } ). Noticethat the only manner to connect with a path the points of A to the pointsof the “appendix” D is passing through the point a ∈ A ∩ C. The figure on the right-hand side provides an interpretation of Lemma 2.4.With respect to a function f having its factor µ defined like in (2.8), wehave painted with a light color the points where f < f > . Note that the region D has been painted in light color,because D ⊆ S ( A ) . Until now we have considered only the case of paths connecting two disjoint sets A and B. This choice is motivated by the foregoing examples for subsets of N − dimensionalspaces. For sake of completeness we end this section by discussing the situation inwhich A and B are joined by a continuum. We confine ourselves to the followinglemma which will find an application in Theorem 5.1. Lemma 2.5.
Let X be a connected and locally arcwise connected metric space andlet A, B ⊆ X be closed and nonempty sets with A ∩ B = ∅ . Let Γ ⊆ X be a compactconnected set such that Γ ∩ A = ∅ , Γ ∩ B = ∅ . Then, for every ε > there exists a path γ = γ ε : [0 , → X with γ (0) ∈ A, γ (1) ∈ B and γ ⊆ B (Γ , ε ) . Moreover, if X is locally compact and C ⊆ X is a closed set which cuts the arcsbetween A and B, then Γ ∩ C = ∅ . Proof.
Let ε > p ∈ Γ , a radius δ p ∈ ]0 , ε [ such thatany two points in B ( p, δ p ) can be joined by a path in B ( p, ε ) . By the compactness ofΓ we can find a finite number of points p , p , . . . , p k ∈ Γ , such that Γ ⊆ B := k [ i =1 B ( p i , δ i ) ⊆ B (Γ , ε ) , with δ i := δ p i . As a consequence of the hypothesis of connectedness of Γ , the following propertyholds: For every partition of { , . . . , k } into two nonempty disjoint subsets J and J , there exist i ∈ J and j ∈ J such that B ( p i , δ i ) ∩ B ( p j , δ j ) = ∅ . This, in turns,implies that we can rearrange the p i ’s (possibly changing their order in the labelling)so that B ( p i , δ i ) ∩ B ( p i +1 , δ i +1 ) = ∅ , ∀ i = 1 , . . . , k − . Hence we can conclude that for any pair of points ( w, z ) ∈ B , with w = z, thereis a path γ = γ w,z joining w with z and such that γ ⊆ B (Γ , ε ) . In particular, taking a ∈ A ∩ Γ and b ∈ B ∩ Γ , we have that there exists a path γ = γ ε : [0 , → B (Γ , ε ) , with γ (0) = a and γ (1) = b and this proves the first part of the statement.Assume now that X is locally compact (i.e., for any p ∈ X and η > , there exists0 < µ p ≤ η such that B ( p, µ p ) is compact). By the compactness of Γ we can find afinite number of points q , q , . . . , q l ∈ Γ and corresponding radii µ i := µ q i such thatΓ ⊆ A := l [ i =1 B ( q i , µ i )and A = S li =1 B ( q i , µ i ) is compact. Since A is an open neighborhood of the compactset Γ , there exists ε > B (Γ , ε ) ⊆ A . Hence, for each 0 < ε ≤ ε we havethat the set B (Γ , ε ) is compact.Taking now ε = n , we know that for every n ∈ N there exists a path γ n : [0 , → X, with γ n (0) ∈ A, γ n (1) ∈ B and γ n ⊆ B (Γ , n ) . But, since C cuts the arcs between A and B, it follows that for every n ∈ N , there is a point c n ∈ C ∩ B (Γ , n ) . For
UTTING SURFACES 11 n ≥ ˆ n large enough (ˆ n > /ε ), the sequence ( c n ) n ≥ ˆ n is contained in the compact set B (Γ , ε ) and therefore it admits a converging subsequence c n k → c ∗ ∈ B (Γ , ε ) . Since d ( c n k , Γ) < n k and the sets C, Γ are closed, the limit point c ∗ ∈ Γ ∩ C. The proof iscomplete. (cid:3) Fixed points in generalized rectangles
We present here some applications of the topological lemmas obtained in Section2 to the intersection of generalized surfaces which separate the opposite edges of an N − dimensional cube. Such generalized surfaces (see Definition 3.1) will be describedas zero-sets of continuous scalar functions and therefore a nonempty intersection willbe obtained as a zero of a suitably defined vector field. To this aim, we recall a classicalresult about the existence of zeros for continuous maps in R N , the Poincar´e − Mirandatheorem.
Theorem 3.1.
Let I N := [0 , N be the N − dimensional unit cube in R N for whichwe denote by [ x i = k ] := { x = ( x , . . . , x N ) ∈ I N : x i = k } . Let F = ( F , . . . , F N ) : I N → R N be a continuous mapping such that, for each i ∈ { , . . . , N } ,F i ( x ) ≤ , ∀ x ∈ [ x i = 0] and F i ( x ) ≥ , ∀ x ∈ [ x i = 1] or F i ( x ) ≥ , ∀ x ∈ [ x i = 0] and F i ( x ) ≤ , ∀ x ∈ [ x i = 1] . Then there exists ¯ x ∈ I N such that F (¯ x ) = 0 . We introduce now the spaces we are going to consider.
Definition 3.1.
Let Z be a metric space and h : R N ⊇ I N → X ⊆ Z be a homeomorphism of I N onto its image X. We call the pair b X := ( X, h )a generalized N − dimensional rectangle (or, simply, a generalized rectangle) of Z . Wealso set X ℓi := h ([ x i = 0]) , X ri := h ([ x i = 1])and call them the left and the right i − faces of X .Finally, we define ϑX := h ( ∂I N )and call it the contour of X. Our main result is the following theorem which can be considered as a variant of theHurewicz − Wallman lemma about dimension [16]. The statements of the two resultsare in fact very similar, but the lemma in [16] concerns, instead of our concept ofcutting, the stronger idea of separation and requires the sets
A, B, C in Definition 2.1to be pairwise disjoint (see [25]). Hence, because of some technical differences whichare crucial in view of our applications, we have chosen to provide all the details.
Theorem 3.2.
Let b X := ( X, h ) be a generalized rectangle in a metric space Z. Assume that, for each i ∈ { , . . . , N } , there exists a compact set S i ⊆ X such that S i cuts the arcs between X ℓi and X ri in X. Then N \ i =1 S i = ∅ . Proof.
Through the inverse of the homeomorphism h : R N ⊇ I N → X ⊆ Z we candefine the compact sets C i := h − ( S i ) , which cut the arcs between [ x i = 0] and [ x i = 1] in I N (for i = 1 , . . . , N ). Clearly, itwill be sufficient to prove that N \ i =1 C i = ∅ . By Lemma 2.1, for every i = 1 , . . . , N, there exists a continuous function f i : I N → R such that f i ≤ x i = 0] and f i ≥ x i = 1] . Moreover, C i = { x ∈ I N : f i ( x ) = 0 } . The continuous vector field f → := ( f , . . . , f N ) : I N → R N satisfies the assumptions ofthe Poincar´e − Miranda theorem and therefore there exists ¯ x ∈ I N such that f i (¯ x ) = 0 , ∀ i = 1 , . . . , N. Hence ¯ x ∈ T Ni =1 C i . (cid:3) Remark 3.1.
A very special case in Definition 3.1 occurs when Z = R N , X = I N and h = Id R N . In order to avoid a cumbersome notation, we denote the pair ( I N , Id R N )simply by I N . This is, for example, the setting in the Hurewicz − Wallman lemma [16]and in the work by Kampen [17]. ⊳ As a first application of Theorem 3.2 we present a corollary which generalizes a resultdue to J. Kampen in [17], providing also a more direct proof.
Corollary 3.1. (see [17, Corollary 4, p.513] ) Let φ = ( φ , . . . , φ N ) : R N ⊇ I N → R N be a continuous map such that for every j ∈ { , . . . , N } one of the following conditionsholds: ( C ) φ j ([ x j = 0] ∪ [ x j = 1]) ⊆ [0 , E ) for every continuous path γ = ( γ , . . . , γ N ) : [0 , → I N such that γ j (0) = 0 and γ j (1) = 1 , it holds that φ j ( γ ) ⊇ [0 , . Then φ has at least a fixed point in I N . If condition ( C ) holds for some j ∈ { , . . . , N } , we say that j is a contractivedirection , while we say that j is an expansive direction when ( E ) is satisfied. Oncefor all, we point out that the term “contractive” has to be meant in a broad manneras it does not imply that the map is a contraction in the classical sense. UTTING SURFACES 13
Proof.
For every i ∈ { , . . . , N } we define S i := { x = ( x , . . . , x N ) ∈ I N : x i = φ i ( x ) } . Let j ∈ { , . . . , N } be fixed and let γ : [0 , → I N be a continuous map such that γ (0) ∈ [ x j = 0] and γ (1) ∈ [ x j = 1] . If j is a contractive direction, so that ( C ) holds, we have that φ j ( γ (0)) ≥ γ j (0)and φ j ( γ (1)) ≤ γ j (1) . Bolzano theorem ensures the existence of t ∗ ∈ [0 ,
1] suchthat φ j ( γ ( t ∗ )) = γ j ( t ∗ ) , that is γ ( t ∗ ) ∈ S j . On the other hand, if j is an expansive direction and thus ( E ) holds, there exist t , t ∈ [0 ,
1] such that φ j ( γ ( t )) = 0 ≤ γ j ( t ) as well as φ j ( γ ( t )) = 1 ≥ γ j ( t ) . Bolzano theorem ensures the existence of ˜ t ∈ [0 ,
1] (with t ≤ ˜ t ≤ t or t ≤ ˜ t ≤ t )such that φ j ( γ (˜ t )) = γ j (˜ t ) , that is γ (˜ t ) ∈ S j . The assumptions of Theorem 3.2 are thus satisfied with respect to X = I N and h = Id R N and so T Ni =1 S i = ∅ . By definition, any point ¯ x ∈ T Ni =1 S i is such that φ (¯ x ) = ¯ x. (cid:3) Corollary 3.1 extends [17, Corollary 4, p.513] where, instead of ( C ) , the strongercondition( C ′ ) φ j ( I N ) ⊆ [0 , Definition 3.2.
Assume we have two N − dimensional rectangles R := Q Ni =1 [ a i , b i ]and R := Q Ni =1 [ c i , d i ] and let ψ : R N ⊇ R → R N be a continuous map. Let1 ≤ i < i < · · · < i k ≤ N be a finite sequence of indexes. We say that R ψ − covers R in ( i , i , . . . , i k ) − direction if the following conditions hold: • for every j ∈ { i , i , . . . , i k } , [ c j , d j ] ⊆ h max x ∈R , x j = a j ψ j ( x ) , min x ∈R , x j = b j ψ j ( x ) i or [ c j , d j ] ⊆ h max x ∈R , x j = b j ψ j ( x ) , min x ∈R , x j = a j ψ j ( x ) i ; • for every j
6∈ { i , i , . . . , i k } , ψ j ( R ) ⊆ [ c j , d j ] . Corollary 3.2. (see [55] ) Let ψ : R N ⊇ R → R N be a continuous map, for R := Q Ni =1 [ a i , b i ] and suppose there exists a finite sequence of indexes ≤ i < i < · · ·
The homeomorphism h = ( h , . . . , h N ) : R N → R N , with h i ( x , . . . , x N ) := a i + ( b i − a i ) x i (3.1)maps the unitary cube I N onto R . It is straightforward to check that the map φ := h − ◦ ψ ◦ h satisfies assumption ( E ) along the components in { i , i , . . . , i k } , whilecondition ( C ′ ) holds along the remaining components. Hence Corollary 3.1 appliesensuring the existence of at least a fixed point ¯ x ∈ I N for the map φ. This impliesthat ¯ y := h (¯ x ) ∈ R is a fixed point for ψ. (cid:3) Corollary 3.2 is a trivial case of a widely more general result (by P. Zgliczy´nski in[55]) that we recall below as Theorem 3.3. Actually, the author considered a morerestrictive covering condition (i.e. covering with margin δ ) for maps defined on thewhole R N in order to apply his main result also to the case of small perturbations ofa given map. Theorem 3.3. (see [55, Theorem 1, p.1042] ) Suppose we have a family of N − dimensionalrectangles R l := Q Ni =1 [ a ( l ) i , b ( l ) i ] and a family of continuous maps ψ l : R l → R N , for l = 0 , . . . , m − . Assume there exists a finite sequence of indexes ≤ i
Definition 3.3.
Assume we have two N − dimensional rectangles R := Q Ni =1 [ a i , b i ]and R := Q Ni =1 [ c i , d i ] and let ψ : R N ⊇ R → R N be a continuous map. Let1 ≤ i < i < · · · < i k ≤ N be a finite sequence of indexes. We say that R ψ − covers R in ( i , i , . . . , i k ) − direction along the paths if the following conditions hold: • for every j ∈ J := { i , i , . . . , i k } and every continuous path γ = ( γ , . . . , γ N ) :[0 , → R satisfying γ j (0) = a j and γ j (1) = b j , it holds that ψ j ( γ ) ⊇ [ c j , d j ] ; • for every j J, ψ j ( R ) ⊆ [ c j , d j ] . Remark 3.2.
We observe that Definition 3.2 and Definition 3.3 coincide in theone − dimensional case, while they differ for N ≥ J = ∅ (that is, if at leastone expansive direction is present). To be more precise, it is straightforward to verifythat any map ψ satisfying Definition 3.2, with respect to a pair of N − dimensionalrectangles, fulfills also Definition 3.3. On the other hand, we give an example (seeExample 3.1 below) of a two − dimensional map which satisfies the latter definition,but not the former. This shows that, in principle, Corollary 3.1 is more general thanCorollary 3.2. ⊳ Example 3.1.
Let φ = ( f, g ) : R → R be the continuous map defined by f ( x, y ) := 12 + c cos(2 π ( kx + ℓ ( y − ))) ,g ( x, y ) := 12 + d sin(2 π ( y + mx )) , where c, d ∈ R and k, l, m ∈ N are chosen in order to satisfy0 < d ≤ < c, ℓ > d , k ≥ ℓ + 1 , m ≥ . By the above positions, we see immediately that g ( x, y ) ∈ h − d,
12 + d i ⊆ [0 , , ∀ ( x, y ) ∈ I , UTTING SURFACES 15 so that φ is contractive in the second component. We prove now that φ is expansive(along the paths) with respect to the first component. More precisely, we check that,for J = { } , I φ − covers I in the x − direction along the paths.Let γ = ( γ , γ ) : [0 , → I be a continuous map such that γ (0) = 0 and γ (1) = 1 . We claim that for F ( t ) := f ( γ ( t ) , γ ( t )) ,F ([0 , ⊇ [0 , { πkγ ( t ) : t ∈ [0 , } coincides with the interval [0 , πk ] , while the set { πℓγ ( t ) − πℓ : t ∈ [0 , } is contained in the interval [ − πℓ, πℓ ] , so thatthe set { π ( kγ ( t ) + ℓ ( γ ( t ) − )) : t ∈ [0 , } contains the interval [ πℓ, πk − πℓ ]whose length is at least 2 π. Therefore, F ([0 , ⊇ (cid:2) − c, + c (cid:3) ⊇ [0 ,
1] and thus wehave proved that φ agrees with Definition 3.3.On the other hand, it is not true that I φ − covers I in the x − direction. In fact, for x = 0 or x = 1 , [0 , * h max (0 , y ) ∈ I f (0 , y ) , min (1 , y ) ∈ I f (1 , y ) i , [0 , * h max (1 , y ) ∈ I f (1 , y ) , min (0 , y ) ∈ I f (0 , y ) i . In fact, it is not difficult to prove even more, that is, for every ˆ x ∈ [0 ,
1] the set { f (ˆ x, y ) : y ∈ [ − d, + d ] } covers [0 ,
1] and therefore, it cannot lie at the left or atthe right of the interval [0 , . More precisely, for every ˆ x ∈ [0 ,
1] it holds that:min { f (ˆ x, y ) : y ∈ [ − d, + d ] } = − c < , max { f (ˆ x, y ) : y ∈ [ − d, + d ] } = + c > . This shows that there is no sub − rectangle of the form R := [ a, b ] × [ − d, + d ] suchthat R φ − covers I in the x − direction. ⊳ Next, we provide an improvement of Corollary 3.1 which will be our main tool in theproof of Theorem 3.4 below.
Corollary 3.3.
Let R := Q Ni =1 [ a i , b i ] be an N − dimensional rectangle with opposite i − faces R ℓi := { x ∈ R : x i = a i } , R ri := { x ∈ R : x i = b i } and let φ = ( φ , . . . , φ N ) : R N ⊇ R → R N be a continuous map. Suppose there exists J ⊆ { , . . . , N } such that the following conditions hold: ( C ) φ j ( R ℓj ∪ R rj ) ⊆ [ a j , b j ] , ∀ j J ;( E W ) for each j ∈ J there is a (nonempty) compact set W j ⊆ R such that for ev-ery continuous path γ = ( γ , . . . , γ N ) : [0 , → R satisfying γ j (0) = a j and γ j (1) = b j , there exists a sub − path σ with σ ⊆ W j and φ j ( σ ) ⊇ [ a j , b j ] . Then φ has at least a fixed point in W := T j ∈ J W j . Proof.
We follow the argument already described along the proof of Corollary 3.1. Wealso assume that there exists at least one j ∈ { , . . . , N } for which ( E W ) is satisfied The role of W in the theorem is meaningful only if ( E W ) holds for at least one component j. Otherwise, if ( C ) holds for every j = 1 , . . . , N, we read the theorem for W = R . (otherwise the result follows directly from Rothe fixed point theorem).For every j J, we define S j := { x = ( x , . . . , x N ) ∈ R : x j = φ j ( x ) } , while, for j ∈ J, we set S j := { x = ( x , . . . , x N ) ∈ W j : x j = φ j ( x ) } . Let j ∈ { , . . . , N } be fixed and let γ : [0 , → R be a continuous map such that γ (0) ∈ R ℓj and γ (1) ∈ R rj . If j is a contractive direction, so that ( C ) holds, by repeating exactly the sameargument as in the proof of Corollary 3.1, we find that there exists t ∗ ∈ [0 ,
1] suchthat φ j ( γ ( t ∗ )) = γ j ( t ∗ ) , that is γ ( t ∗ ) ∈ S j . On the other hand, if j is an expansive direction so that ( E W ) holds, there exist0 ≤ t < t ≤ γ ( t ) ∈ W j , for every t ∈ [ t , t ] and, moreover, φ j ( γ ( t )) = a j ≤ γ j ( t ) as well as φ j ( γ ( t )) = b j ≥ γ j ( t ) (or φ j ( γ ( t )) = b j ≥ γ j ( t ) as well as φ j ( γ ( t )) = a j ≤ γ j ( t )). Bolzano theorem ensures the existence of ˜ t ∈ [ t , t ] suchthat φ j ( γ (˜ t )) = γ j (˜ t ) and also γ (˜ t ) ∈ W j . Hence, γ (˜ t ) ∈ S j . The assumptions of Theorem 3.2 are thus satisfied with respect to X = R and h defined as in (3.1). Therefore, T Ni =1 S i = ∅ . By definition, any point ¯ x ∈ T Ni =1 S i issuch that φ (¯ x ) = ¯ x. Moreover ¯ x ∈ W. (cid:3) Remark 3.3.
Clearly, Corollary 3.3 is still true if we replace hypothesis ( C ) with( C ′ ) φ j ( R ) ⊆ [ a j , b j ] , ∀ j J. This assumption (which is slightly more restrictive than ( C ) ) is crucial when weconsider compositions of maps. ⊳ We are now in position to prove our extension of Theorem 3.3. Along the proof,we use the following notation:Let α, β ∈ R with α < β. We denote by η [ α,β ] the (continuous) projection of R ontothe interval [ α, β ] , defined by η [ α,β ] ( s ) := max { α, min { s, β } } . (3.2) Theorem 3.4.
Suppose we have a family of N − dimensional rectangles R l := Q Ni =1 [ a ( l ) i , b ( l ) i ] and a family of continuous maps ψ l : R l → R N , for l = 0 , . . . , m − . Assume thereexists a finite sequence of indexes ≤ i < i < · · · < i k ≤ N, such that for l = 0 , . . . , m − , R l ψ l − covers R l +1 ( mod m ) in ( i , i , . . . , i k ) − direction along thepaths. Then there exists w ∈ R such that ψ l ◦ ψ l − ◦ · · · ◦ ψ ( w ) ∈ R l +1 , for l = 0 , , . . . , m − ψ m − ◦ ψ m − ◦ · · · ◦ ψ ( w ) = w. (3.4) Proof.
We set J := { i , . . . , i k } . If J = ∅ , it follows that ψ l ( R l ) ⊆ R l +1 , for l = 0 , . . . , m − m ) , and theresult is an immediate consequence of the Brouwer fixed point theorem. Thus, forthe remainder of the proof, we assume J = ∅ . First of all, we extend the map ψ l = ( ψ ( l )1 , . . . , ψ ( l ) N ) to a continuous mapping e ψ l = ( e ψ ( l )1 , . . . , e ψ ( l ) N ) : R N → R N , UTTING SURFACES 17 defined by e ψ ( l ) i ( x ) := η [ a ( l +1) i , b ( l +1) i ] ( ψ ( l ) i ( P R l ( x ))) , ∀ i = 1 , . . . , N, where we have called P R l the projection of R N onto the rectangle R l , given by P R l ( x ) := (cid:0) η [ a ( l )1 , b ( l )1 ] ( x ) , . . . , η [ a ( l ) N , b ( l ) N ] ( x N ) (cid:1) . Then, for every j ∈ J, we define W j as the set of the points x ∈ R satisfying thefollowing conditions: ψ (0) j ( x ) ∈ [ a (1) j , b (1) j ] ,ψ (1) j ( ψ ( x )) ∈ [ a (2) j , b (2) j ] , ... ψ ( m − j ( ψ m − ◦ · · · ◦ ψ )( x ) ∈ [ a ( m − j , b ( m − j ] ,ψ ( m − j ( ψ m − ◦ ψ m − ◦ · · · ◦ ψ )( x ) ∈ [ a (0) j , b (0) j ] . We are going to apply Corollary 3.3 (in the version of Remark 3.3) to the compositemap φ = ( φ , . . . , φ N ) := e ψ m − ◦ e ψ m − ◦ · · · ◦ e ψ . Notice that φ is well defined as a continuous map on R . Indeed, by the property ofthe projections η [ a ( l ) i , b ( l ) i ] we have that e ψ l ( R l ) ⊆ R l +1 , ∀ l = 0 , . . . , m − . As a preliminary observation we note that any fixed point w for φ, with w ∈ W := T j ∈ J W j , satisfies the conditions (3.3) and (3.4). This follows from the fact that η [ a ( l ) i , b ( l ) i ] ( s ) = s, for s ∈ [ a ( l ) i , b ( l ) i ]and that, when j J, e ψ ( l ) j ( x ) = ψ ( l ) j ( x ) , ∀ x ∈ R l . Let j ∈ { , . . . , N } be a fixed index. We distinguish two cases:( a ) j J. In this situation, for every l = 0 , . . . , m − , we have e ψ ( l ) j ( x ) ∈ [ a ( l +1) j , b ( l +1) j ] . Therefore, φ satisfies ( C ′ ) . ( b ) j ∈ J. Let γ = ( γ , . . . , γ N ) : [0 , → R be a continuous path satisfying γ j (0) = a (0) j and γ j (1) = b (0) j . Our aim is to prove that there exists a sub-path σ of γ with σ ⊆ W j and such that φ j ( σ ) ⊇ [ a (0) j , b (0) j ] . By the assumptions, we know that ψ (0) j ( γ ) ⊇ [ a (1) j , b (1) j ] . If we consider the continuous real valued function g : [0 , → R , g ( t ) := ψ (0) j ( γ ( t )) , we can find 0 ≤ t < t ≤ g ( t ) ∈ [ a (1) j , b (1) j ] , for every t ∈ [ t , t ] and,either g ( t ) = a (1) j and g ( t ) = b (1) j , or g ( t ) = b (1) j and g ( t ) = a (1) j . The restriction γ := γ | [ t ,t ] is a sub-path of γ such that ψ (0) j ( γ ) = [ a (1) j , b (1) j ] . Moreover, we also havethat e ψ (0) j ( γ ( t )) = ψ (0) j ( γ ( t )) , ∀ t ∈ [ t , t ] . If we like, we can take a continuous path σ : [0 , → R , in the same equivalenceclass of γ , such that e ψ (0) j ( σ ( t )) = ψ (0) j ( σ ( t )) ∈ [ a (1) j , b (1) j ] , ∀ t ∈ [0 , ,ψ (0) j ( σ (0)) = a (1) j , ψ (0) j ( σ (1)) = b (1) j . Now, with respect to the path γ := e ψ ◦ σ : [0 , → R , we are exactly in the same situation like we were with respect to the path γ : [0 , →R . At this moment, we can proceed by induction, just repeating a finite number oftimes the previous argument, until we find a sub-path σ of γ, with σ : [0 , → R satisfying the following conditions: e ψ ( l ) j ( e ψ l − ◦ · · · ◦ e ψ )( σ ( t )) = ψ ( l ) j ( ψ l − ◦ · · · ◦ ψ )( σ ( t )) ∈ [ a ( l +1) j , b ( l +1) j ] , ∀ t ∈ [0 , , ∀ l = 0 , . . . , m − m ) ,ψ ( m − j ( ψ m − ◦ · · · ◦ ψ )( σ (0)) = a (0) j , ψ ( m − j ( ψ m − ◦ · · · ◦ ψ )( σ (1)) = b (0) j . Therefore σ ⊆ W j and φ j ( σ ) = [ a (0) j , b (0) j ] , that is, ( E W ) of Corollary 3.3 is satisfied.In this manner we have proved that for every j ∈ { , . . . , N } either ( C ′ ) or ( E W )is fulfilled with respect to φ. Hence, Corollary 3.3 ensures the existence of at leasta fixed point for ψ in W := T j ∈ J W j that, as already observed, satisfies conditions(3.3) and (3.4). The proof is complete. (cid:3) Continua of fixed points for maps depending on parameters
In this section we still consider the intersection of generalized surfaces which sep-arate the opposite edges of an N − dimensional cube, but in the case in which thenumber of the cutting surfaces is smaller than the dimension of the space. Here ourmain tool is a result by Fitzpatrick, Massab´o and Pejsachowicz (see [12, Theorem1.1]) on the covering dimension of the zero set of an operator depending on parame-ters. For the reader’s convenience, we recall the concept of covering dimension as canbe found in [10]. Definition 4.1. [10, p.54, p.208] Let Z be a metric space. We say that dim Z ≤ n if every finite open cover of the space Z has a finite open [closed] refinement oforder ≤ n. The object dim Z ∈ N ∪ {∞} is called the covering dimension or the ˇCech − Lebesgue dimension of the metric space Z. According to [12], if z ∈ Z, we alsosay that dim Z ≥ j at z if each neighborhood of z has dimension at least j. The order of a family A of subsets of Z (used in the above definition) is the largestinteger n such that the family A contains n + 1 sets with a non-empty intersection; ifno such integer exists we say that the family A has order infinity. By a classical resultfrom Topology (see [10, The coincidence theorem]) the covering dimension coincideswith the inductive dimension [10, p.3] for separable metric spaces.We keep the notation of the previous section. In particular, we recall that if R := Q Ni =1 [ a i , b i ] is an N − dimensional rectangle, we denote its opposite i − faces by R ℓi := { x ∈ R : x i = a i } , R ri := { x ∈ R : x i = b i } . UTTING SURFACES 19
Theorem 4.1.
Let R := Q Ni =1 [ a i , b i ] be an N − dimensional rectangle and let P =( p , . . . , p N ) be any interior point of R . Let n ∈ { , . . . , N − } be fixed. Supposethat F = ( F , . . . , F n ) : R → R n is a continuous mapping such that, for each i ∈{ , . . . , n } , F i ( x ) < , ∀ x ∈ R ℓi and F i ( x ) > , ∀ x ∈ R ri or F i ( x ) > , ∀ x ∈ R ℓi and F i ( x ) < , ∀ x ∈ R ri . Define also the affine map π : R N → R N − n , π j ( x , . . . , x N ) := x j − p j , j = N − n + 1 , . . . , N. (4.1) Then there exists a connected subset Z of F − (0) = { x ∈ R : F i ( x ) = 0 , ∀ i = 1 , . . . , n } whose dimension at each point is at least N − n. Moreover, dim(
Z ∩ ∂ R ) ≥ N − n − and π : Z ∩ ∂ R → R N − n \ { } is essential.Proof. We define the continuous mapping H = ( F, π ) :
R → R N . By the assumptions on F and π we havedeg( H, R o ,
0) = ( − d = 0 , where d is the number of components i ∈ { , . . . , n } such that F i ( x ) > x ∈ R ℓi (and also F i ( x ) < x ∈ R ri ). Hence π turns out to be a complementing map for F (according to [12]). A direct application of [12, Theorem 1.1] gives the thesis (notethat the dimension m in [12, Theorem 1.1] corresponds to our N − n ). (cid:3) Remark 4.1. If a i < < b i ( ∀ i = 1 , . . . , N ) we can take P = 0 , so that thecomplementing map is just the projection π : R N → R N − n . ⊳ For the reader’s convenience, we recall that (according to the definitions in [12]), givenan open bounded set
O ⊆ R N , a continuous map π : O → R N − n is a complement forthe continuous map F : O → R n if the topological degree deg(( π, F ) , O ,
0) is definedand nonzero. We also recall (see [14]) that a mapping f of a space X into a space Y is said to be inessential if f is homotopic to a constant; otherwise f is essential .A more elementary version of Lemma 4.1 can be given for the zero set of a vectorfield with range in R N − . In this case we can achieve our result by a direct use of theclassical Leray − Schauder continuation theorem [26], instead of the more sophisticatedtools in [12]. Namely, we have:
Theorem 4.2.
Let R := Q Ni =1 [ a i , b i ] be an N − dimensional rectangle and let F =( F , . . . , F N − ) : R → R N − be a continuous mapping such that, for each i ∈ { , . . . , N − } , F i ( x ) < , ∀ x ∈ R ℓi and F i ( x ) > , ∀ x ∈ R ri or F i ( x ) > , ∀ x ∈ R ℓi and F i ( x ) < , ∀ x ∈ R ri . Then there exists a closed connected subset Z of F − (0) = { x ∈ R : F i ( x ) = 0 , ∀ i = 1 , . . . , N − } such that Z ∩ R ℓN = ∅ , Z ∩ R rN = ∅ . Proof.
We split x = ( x , . . . , x N − , x N ) ∈ R ⊆ R N as x = ( y, λ ) with y = ( x , . . . , x N − ) ∈ M := N − Y i =1 [ a i , b i ] , λ = x N ∈ [ a N , b N ]and define f = f ( y, λ ) : M × [ a N , b N ] → R N − , f ( y, λ ) := F ( x , . . . , x N − , λ ) , treating, in this manner, the variable x N = λ as a parameter for the ( N − − dimensionalvector field f λ ( · ) = f ( · , λ ) . By the assumptions on F we havedeg( f λ , M o ,
0) = ( − d = 0 , ∀ λ ∈ [ a N , b N ] , where d is the number of the components i ∈ { , . . . , N − } such that F i ( x ) > x ∈ R ℓi (and also F i ( x ) < x ∈ R ri ). The Leray − Schauder continuationtheorem [26, Th´eor`eme Fondamental] (see also [27, 28]) ensures the existence of aclosed connected set
Z ⊆ { ( y, λ ) ∈ M × [ a N , b N ] : f ( y, λ ) = 0 ∈ R N − } whose projection onto the λ − component covers the interval [ a N , b N ] . By the abovepositions the thesis follows immediately. (cid:3)
Theorem 4.2 can be found also in [23] and it was then applied in [24].In the next lemma we take the unit cube I N := [0 , N as N − dimensional rectangleand choose the interior point P = ( , , . . . , ) . Lemma 4.1.
Let n ∈ { , . . . , N − } be fixed. Assume that, for each i ∈ { , . . . , n } , there exists a compact set S i ⊆ I N such that S i cuts the arcs between [ x i = 0] and [ x i = 1] in I N . Then there exists aconnected subset Z of n \ i =1 S i = ∅ , whose dimension at each point is at least N − n. Moreover, dim(
Z ∩ ∂I N ) ≥ N − n − and π : Z ∩ ∂I N → R N − n \ { } is essential (where π is defined as in (4.1) ).Proof. For any fixed index i ∗ ∈ { , . . . , n } we define the tunnel set T i ∗ := i ∗ − Y i =1 [0 , × R × N Y i = i ∗ +1 [0 , . UTTING SURFACES 21
It is immediate to check that S i ∗ cuts the arcs between [ x i ∗ = 0] and [ x i ∗ = 1] in T i ∗ . By Lemma 2.4 there exists a continuous function f i ∗ : T i ∗ → R such that f i ∗ ( x ) ≤ , ∀ x ∈ T i ∗ with x i ∗ ≤ f i ∗ ( x ) ≥ , ∀ x ∈ T i ∗ with x i ∗ ≥ . Moreover, S i ∗ = { x ∈ T i ∗ : f i ∗ ( x ) = 0 } . By this latter property and the fact that S i ∗ ⊆ I N it follows that f i ∗ ( x ) < , ∀ x ∈ T i ∗ with x i ∗ < f i ∗ ( x ) > , ∀ x ∈ T i ∗ with x i ∗ > . Now we define, for x = ( x , . . . , x i ∗ − , x i ∗ , x i ∗ +1 , . . . , x N ) ∈ R N , the continuous func-tion F i ∗ ( x ) := f i ∗ (cid:0) η [0 , ( x ) , . . . , η [0 , ( x i ∗ − ) , x i ∗ , η [0 , ( x i ∗ +1 ) , . . . , η [0 , ( x N ) (cid:1) , where η [0 , is the projection of R onto the interval [0 ,
1] defined as in (3.2). As aconsequence of the above positions we find that F i ∗ ( x ) < , ∀ x ∈ R N : x i ∗ < F i ∗ ( x ) > , ∀ x ∈ R N : x i ∗ > . We are ready to apply Theorem 4.1 to the map F = ( F , . . . , F n ) restricted to therectangle R := n Y i =1 [ − , × N Y i = n +1 [0 , . Clearly, (cid:0) F | R (cid:1) − (0) = n \ i =1 S i ⊆ I N and the thesis is achieved. (cid:3) Remark 4.2.
Both in Theorem 4.1 and in Lemma 4.1 the fact that we have privi-leged the first n components is purely conventional. It is evident that the results arestill true for any finite sequence of indexes i < i < · · · < i n in { , . . . , N } . Moreover,Lemma 4.1 is invariant under homeomorphisms in a sense that is described in Theo-rem 4.3 below. The same observation applies systematically to all the other results(preceding and subsequent) in which some directions are conventionally chosen. ⊳ In view of the next result we recall the concept of generalized rectangle b X := ( X, h )given in Definition 3.1, where h : I N → X ⊆ Z is a homeomorphism of the unit cube I N = [0 , N onto its image X. Theorem 4.3.
Let b X := ( X, h ) be a generalized rectangle of a metric space Z. Let afinite sequence of n indexes i < i < · · · < i n ( n ≥ ) be fixed in { , . . . , N } . Assumethat, for each j ∈ { i , . . . , i n } , there exists a compact set S j ⊆ X such that S j cuts the arcs between X ℓj and X rj in X. Then there exists a compactconnected subset Z of n \ k =1 S i k = ∅ , whose dimension at each point is at least N − n. Moreover, dim(
Z ∩ ϑX ) ≥ N − n − and π : h − ( Z ) ∩ ∂I N → R N − n \ { } is essential (where π is defined as in (4.1) ). Proof.
The result easy follows by moving to the setting of Lemma 4.1 through thehomeomorphism h − and repeating the arguments employed therein. (cid:3) We end this section by presenting a result (Corollary 4.1) which plays a crucial rolein the subsequent proofs. It concerns the case n = N − − Schauder principle and therefore,in some sense, is more elementary). Corollary 4.1 extends to an arbitrary dimensionsome results in [39, Appendix] which were proved only for N = 2 using [40]. Corollary 4.1.
Let b X := ( X, h ) be a generalized rectangle in a metric space Z. Let k ∈ { , . . . , N } be fixed. Assume that, for each j ∈ { , . . . , N } with j = k, there existsa compact set S j ⊆ X such that S j cuts the arcs between X ℓj and X rj in X. Then there exists a compactconnected subset C of N \ i =1 i = k S i = ∅ , such that C ∩ X ℓk = ∅ , C ∩ X rk = ∅ . Proof.
Without loss of generality (if necessary, by a permutation of the coordinates),we assume k = N. In this manner, using the homeomorphism h − : Z ⊇ X = h ( I N ) → I N , we can confine ourselves to the following situation:For each j ∈ { , . . . , N − } , there exists a compact set S ′ j := h − ( S j ) ⊆ I N that cuts the arcs between [ x i = 0] and [ x i = 1] in I N . Proceeding as in the proof of Lemma 4.1, for any fixed index i ∗ ∈ { , . . . , N − } we define the tunnel set T i ∗ := i ∗ − Y i =1 [0 , × R × N Y i = i ∗ +1 [0 , S ′ i ∗ cuts the arcs between [ x i ∗ = 0] and [ x i ∗ = 1] in T i ∗ . Hence, by Lemma 2.4 there exists a continuous function f i ∗ : T i ∗ → R such that f i ∗ ( x ) ≤ , ∀ x ∈ T i ∗ with x i ∗ ≤ f i ∗ ( x ) ≥ , ∀ x ∈ T i ∗ with x i ∗ ≥ . Moreover, S ′ i ∗ = { x ∈ T i ∗ : f i ∗ ( x ) = 0 } , as well as f i ∗ ( x ) < , ∀ x ∈ T i ∗ with x i ∗ < f i ∗ ( x ) > , ∀ x ∈ T i ∗ with x i ∗ > . We define, for x = ( x , . . . , x i ∗ − , x i ∗ , x i ∗ +1 , . . . , x N ) ∈ R N , the continuous function F i ∗ ( x ) := f i ∗ (cid:0) η [0 , ( x ) , . . . , η [0 , ( x i ∗ − ) , x i ∗ , η [0 , ( x i ∗ +1 ) , . . . , η [0 , ( x N ) (cid:1) , where η [0 , is the projection of R onto the interval [0 ,
1] defined as in (3.2). Then wehave F i ∗ ( x ) < , ∀ x ∈ R N : x i ∗ < F i ∗ ( x ) > , ∀ x ∈ R N : x i ∗ > . UTTING SURFACES 23
Now we consider the map F = ( F , . . . , F N − ) restricted to the rectangle R := N − Y i =1 [ − ε, ε ] × [0 , , for any fixed ε > . Since (cid:0) F | R (cid:1) − (0) = N − \ i =1 S ′ i ⊆ I N , the thesis follows by Theorem 4.2. In fact, we can define the set C := h ( Z ) , where Z ⊆ (cid:0) F | R (cid:1) − (0) comes from the statement of Theorem 4.2. (cid:3) Periodic points and chaotic dynamics for maps which expand thepaths
We provide now an extension to N − dimensional spaces of some results previouslyobtained in [35, 36] for the planar case. As in [35, 36] we are interested in the studyof maps which expand the arcs along a certain direction. To this aim, we reconsiderDefinition 3.1 in order to focus our attention on a generalized N − dimensional rectan-gle in which we have fixed (once for all) the left and right sides. In the applications,these opposite sides give an orientation (in a rough sense) of the generalized rectangleand they will be related to the expansive direction. Definition 5.1.
Let Z be a metric space and let b X := ( X, h ) be a generalized N − dimensional rectangle of Z. We set X ℓ := h ([ x N = 0]) , X r := h ([ x N = 1])and X − := X ℓ ∪ X r . The pair e X := ( X, X − )is called an oriented N − dimensional rectangle (or, simply, an oriented rectangle) of Z . For simplicity, the reference to the ambient space Z will be omitted when nopossibility of confusion may occur. Remark 5.1.
First of all we observe that, instead of the unit cube [0 , N , we couldhave chosen in the above definition any N − dimensional rectangle. In this case thesides X ℓ and X r would be defined (in a obvious manner) accordingly.A comparison between Definition 3.1 and Definition 5.1 shows that an orientedrectangle is just a generalized rectangle in which we have privileged the two sub-sets of its contour which correspond to the opposite faces for some fixed component(namely, the x N − component). The choice of the N − th component is purely conven-tional. For example, in some other papers (see [13, 38, 56]), the first component wasselected. Clearly, there is no substantial difference as the homeomorphism h couldbe composed with a permutation matrix (yielding to a new homeomorphism with thesame image set). From this point of view, our definition fits to the one of h − set of(1 , N − − type, given by Zgliczy´nski and Gidea in [56] for a subset of R N which isobtained as the counterimage of the unit cube through a homeomorphism of R N ontoitself. A similar concept is also considered by Gidea and Robinson in [13]: they call this object a (1 , N − − window and it is defined as a homeomorphic copy of the unitcube I N of R N through a homeomorphism whose domain is an open neighborhood of I N . ⊳ The next definition introduces the concept of stretching along the paths (alreadyconsidered in [35, 36, 38]) for maps between oriented rectangles.
Definition 5.2.
Let Z be a metric space and let e X := ( X, X − ) and e Y := ( Y, Y − )be oriented N − dimensional rectangles of Z. Let ψ : Z ⊇ D ψ → Z be a map (notnecessarily continuous on its whole domain D ψ ) and let D ⊆ X ∩ D ψ . We say that ( D , ψ ) stretches e X to e Y along the paths and write( D , ψ ) : e X ≎ −→ e Y if there exists a compact set K ⊆ D such that ψ is continuous on K and for everypath γ with γ ⊆ X and γ ∩ X ℓ = ∅ , γ ∩ X r = ∅ , there is a sub-path σ of γ such that σ ⊆ K and ψ ( σ ) ⊆ Y, with ψ ( σ ) ∩ Y ℓ = ∅ , ψ ( σ ) ∩ Y r = ∅ . We also write ( D , K , ψ ) : e X ≎ −→ e Y when we wish to put in evidence the role of the set K . In some applications, we take
K ⊆ D such that ψ ( K ) ⊆ Y. In this case, the condition ψ ( σ ) ⊆ Y is automaticallysatisfied. Remark 5.2.
Let e X = ( X, X − ) and e Y = ( Y, Y − ) be oriented N − dimensional rect-angles of a metric space Z and assume that( D , ψ ) : e X ≎ −→ e Y for some ψ : Z ⊇ D ψ → Z and D ⊆ X ∩ D ψ . From the above definition it turns outthat ( D ′ , ψ ) : e X ≎ −→ e Y , ∀ D ′ : ψ − ( Y ) ∩ D ⊆ D ′ ⊆ X ∩ D ψ . We also note that if D is closed and ψ is continuous on D , we can take K = D in thedefinition.Clearly, there are situations where there is no need to invoke the set K because theknowledge of D gives the required information. A particular case in which it is notnecessary to specify such a compact K occurs when X ⊆ D ψ : indeed, an easy criterionto verify the stretching condition in this particular context is checking that ψ ( X ) ⊆ Y and ψ ( X ℓ ) ⊆ Y ℓ as well as ψ ( X r ) ⊆ Y r , or ψ ( X ℓ ) ⊆ Y r as well as ψ ( X r ) ⊆ Y ℓ . On the other hand, in some cases, it may be useful to emphasize the existence ofa special set K . For instance, we could be interested in examples where ( D , K i , ψ ) : e X ≎ −→ e Y for different (even disjoint) sets K i ’s (see, e.g., Theorem 5.3) and also insituations in which either ψ is not defined on the whole set X or ψ is defined on X but ψ ( X ) Y. ⊳ UTTING SURFACES 25
As a consequence of Corollary 4.1 we obtain the following result which extends to N − dimensional rectangles a fixed point theorem (see [36, Th. 3.1]), originally provedfor N = 2 . Theorem 5.1.
Let e X := ( X, X − ) be an oriented N − dimensional rectangle of ametric space Z and let ψ : Z ⊇ D ψ → Z and D ⊆ X ∩ D ψ be such that ( D , K , ψ ) : e X ≎ −→ e X, (5.1) for some compact set K ⊆ D . Then there exists e w ∈ K such that ψ ( e w ) = e w. Proof.
Let h : I N = [0 , N → h ( I N ) = X ⊆ Z be a homeomorphism such that X ℓ = h ([ x N = 0]) and X r = h ([ x N = 1]) and consider the compact set of I N W := h − ( K ∩ ψ − ( X ))and the continuous mapping φ = ( φ , . . . , φ N ) : W → I N defined by φ ( x ) := h − (cid:0) ψ ( h ( x )) (cid:1) , ∀ x ∈ W . By the Tietze − Urysohn theorem [10, p.87] there exists a continuous map ϕ = ( ϕ , . . . , ϕ N ) : I N → I N , ϕ | W = φ. Let use define, for every i = 1 , . . . , N − , the closed sets S i := { x = ( x , . . . , x N − , x N ) ∈ I N : x i = ϕ i ( x ) } ⊆ I N . Since ϕ ( I N ) ⊆ I N , by the continuity of the ϕ i ’s, it is straightforward to check that S i cuts the arcs between [ x i = 0] and [ x i = 1] in I N (for each i = 1 , . . . , N − γ : [0 , → I N is a path with γ i (0) = 0 and γ i (1) = 1 , then, for the auxiliaryfunction g : [0 , ∋ t γ i ( t ) − ϕ i ( γ ( t )) , we have g (0) ≤ ≤ g (1) and therefore thereexists s ∈ [0 ,
1] such that γ i ( s ) = ϕ i ( γ ( s )) (Bolzano theorem), that is γ ∩ S i = ∅ . Thus the cutting property is proved.Now Corollary 4.1 guarantees the existence of a continuum
C ⊆ N − \ i =1 S i (5.2)such that C ∩ [ x N = 0] = ∅ , C ∩ [ x N = 1] = ∅ . By Lemma 2.5 we have that, for every ε > , there exists a path γ ε : [0 , → I N suchthat γ ε (0) ∈ [ x N = 0] , γ ε (1) ∈ [ x N = 1] and γ ε ( t ) ∈ B ( C , ε ) ∩ I N , ∀ t ∈ [0 , . By (5.1) and the definition of W and φ, there exists a sub-path σ ε of γ ε such that σ ε ⊆ W and φ ( σ ε ) ⊆ I N , with φ ( σ ε ) ∩ [ x N = 0] = ∅ , φ ( σ ε ) ∩ [ x N = 1] = ∅ . The Bolzano theorem applied this time to the continuous mapping x x N − ϕ N ( x )on σ ε implies the existence of a point˜ x ε = (˜ x ε , . . . , ˜ x εN ) ∈ σ ε ⊆ W such that ˜ x εN = ϕ N (˜ x ε ) . Taking ε = n and letting n → ∞ , by a standard compactness argument we find apoint ˜ x = (˜ x , . . . , ˜ x N ) ∈ C ∩ W such that ˜ x N = ϕ N (˜ x ) . By (5.2), recalling also the definition of the S i ’s, we find˜ x = ϕ (˜ x ) ∈ W . Then, since ϕ | W = φ, by the relation h ( φ ( x )) = ψ ( h ( x )) , ∀ x ∈ W , we have that h (˜ x ) = ψ ( h (˜ x )) ∈ h ( W ) and therefore e w := h (˜ x ) ∈ K ∩ ψ − ( X )is the desired fixed point for ψ. (cid:3) Having proved Theorem 4.3 and Theorem 5.1, we have now available the tools forextending to any dimension the results about periodic points and chaotic dynamicspreviously obtained for the two − dimensional case in [35, 36]. For sake of concisenesswe focus our attention only on some of them (selected from [35, 36]), that we presentbelow in the more general setting. Theorem 5.2.
Assume there is a double sequence of oriented N − dimensional rectan-gles ( e X k ) k ∈ Z (with e X k = ( X k , X − k ) ) of a metric space Z and a sequence (( D k , ψ k )) k ∈ Z , with D k ⊆ X k , such that ( D k , ψ k ) : e X k ≎ −→ e X k +1 , ∀ k ∈ Z . Let us denote by X kℓ and X kr the two parts of X − k . Then the following conclusionshold: ( a ) There is a sequence ( w k ) k ∈ Z such that w k ∈ D k and ψ k ( w k ) = w k +1 , for all k ∈ Z ;( a ) For each j ∈ Z there exists a compact connected set C j ⊆ D j which cuts thearcs between X jℓ and X jr in X j and such that, for every w ∈ C j , there is asequence ( y i ) i ≥ j with y j = w and y i ∈ D i , ψ i ( y i ) = y i +1 , ∀ i ≥ j. The dimension of C j at each point is at least N − . Moreover, dim( C j ∩ ϑX j ) ≥ N − and π : h − ( C j ) ∩ ∂I N → R N − \ { } is essential (where π is defined asin (4.1) for p i = , ∀ i ); ( a ) If there are integers k and l, with k < l, such that e X k = e X l , then thereexists a finite sequence ( z i ) k ≤ i ≤ l , with z i ∈ D i and ψ i ( z i ) = z i +1 for each i = k, . . . , l − , such that z l = z k , that is, z k is a fixed point of ψ l − ◦ · · · ◦ ψ k . Proof.
We prove the conclusions of the theorem in the reverse order. So, let’s startwith the verification of ( a ) . By the assumptions and the definition of the “stretchingalong the paths” property, it is easy to check that( D , ψ l − ◦ · · · ◦ ψ k ) : e X k ≎ −→ e X l , (5.3)where D := { z ∈ D k : ψ j ◦ · · · ◦ ψ k ( z ) ∈ D j +1 , ∀ j = k, . . . , l − } . UTTING SURFACES 27
With the positions e X = e X k = e X l and ψ = ψ l − ◦ · · · ◦ ψ k , we read condition (5.3)as ( D , ψ ) : e X ≎ −→ e X and therefore the thesis follows immediately by Theorem 5.1.More precisely, if we like to put in evidence the role of the compact sets K k ’s, for( D k , K k , ψ k ) : e X k ≎ −→ e X k +1 , we have that( D , K , ψ ) : e X ≎ −→ e X, where we have set K := { z ∈ K k : ψ j ◦ · · · ◦ ψ k ( z ) ∈ K j +1 , ∀ j = k, . . . , l − } . As regards ( a ) , without loss of generality, we can assume j = 0 . Recall that byDefinition 5.2, since ( D k , ψ k ) : e X k ≎ −→ e X k +1 , ∀ k ∈ Z , it follows that for any k thereexists a compact set K k ⊆ X k such that ψ k is continuous on K k and for every path γ with γ ⊆ X k and γ ∩ X kℓ = ∅ , γ ∩ X kr = ∅ , there is a sub-path σ of γ such that σ ⊆ K k and ψ ( σ ) ⊆ X k +1 , with ψ ( σ ) ∩ X k +1 ℓ = ∅ , ψ ( σ ) ∩ X k +1 r = ∅ . Let us define theclosed set S := { z ∈ K : ψ j ◦ · · · ◦ ψ ( z ) ∈ K j +1 , ∀ j ≥ } (5.4)and fix a path γ such that γ ⊆ X and γ ∩ X ℓ = ∅ , γ ∩ X r = ∅ . Then, since( D , ψ ) : e X ≎ −→ e X , there exists a sub-path γ of γ with γ ⊆ K ⊆ X such that ψ ( γ ) ⊆ X and ψ ( γ ) ∩ X ℓ = ∅ , ψ ( γ ) ∩ X r = ∅ . Similarly, there exists a sub-path σ of σ := ψ ( γ ) with σ ⊆ K ⊆ D and such that ψ ( σ ) ⊆ X , ψ ( σ ) ∩ X ℓ = ∅ , ψ ( σ ) ∩ X r = ∅ . DefiningΓ := { x ∈ γ : ψ ( x ) ∈ σ } ⊆ { z ∈ K : ψ ( z ) ∈ K } and proceeding by induction, we can find a decreasing sequence of nonempty compactsets Γ := γ ⊇ Γ := γ ⊇ Γ ⊇ · · · ⊇ Γ n ⊇ Γ n +1 ⊇ . . . such that ψ j ◦ · · ·◦ ψ (Γ j +1 ) ⊆ X j +1 , ψ j ◦ · · ·◦ ψ (Γ j +1 ) ∩ X j +1 ℓ = ∅ , ψ j ◦ · · ·◦ ψ (Γ j +1 ) ∩ X j +1 r = ∅ , for j ≥ . Moreover, for every i ≥ , we have thatΓ i +1 ⊆ { z ∈ K : ψ j − ◦ · · · ◦ ψ ( z ) ∈ K j , ∀ j : 1 ≤ j ≤ i } . It is easy to see that ∩ + ∞ j =0 Γ j = ∅ and for any z ∈ ∩ + ∞ j =0 Γ j it holds that ψ n ◦· · ·◦ ψ ( z ) ∈K n +1 , ∀ n ∈ N . In this way we have shown that any path γ , with γ joining the twosides of X − , intersects S , i.e. S cuts the arcs between X ℓ and X r in X . Obviouslyany point belonging to the intersection of γ with S generates a sequence with theproperties required in ( a ) . The existence of the connected compact set C ⊆ D asin ( a ) follows from Theorem 4.3, setting n = 1 and i = 0 . Conclusion ( a ) follows now from ( a ) by a standard diagonal argument already em-ployed in previous works (see, e.g., [18, Proposition 5], [35, Theorem 2.2]). (cid:3) Remark 5.3.
An apparently more general version of Theorem 5.2 can be obtainedby assuming the X k ’s to be contained in possibly different metric spaces Z k ’s.If, at any step k ∈ Z , we have the further information that ( D k , K k , ψ k ) : e X k ≎ −→ e X k +1 , then, in each of the corresponding conclusions ( a ) , ( a ) , ( a ) we can be more preciseand add that w k ∈ K k , y k ∈ K k , or z k ∈ K k , respectively. ⊳ We end this paper with a few consequences of Theorem 5.2.Our applications deal with discrete dynamical systems exhibiting a chaotic behav-ior. Due to the many different definitions of chaos available in the literature, we state in a precise manner the one we use. The same concept of chaos has been alreadyconsidered in [35, 36, 37, 38] as well as in previous works by other authors (see, forinstance, [53]).
Definition 5.3.
Let Z be a metric space, ψ : Z ⊇ D ψ → Z be a map and let D ⊆ D ψ . Assume also that m ≥ ψ induces chaotic dynamics on m symbols in the set D if there exist m nonempty pairwise disjoint compact sets K , K , . . . , K m − ⊆ D , such that, for each two − sided sequence ( s i ) i ∈ Z ∈ { , . . . , m − } Z , there exists acorresponding sequence ( w i ) i ∈ Z ∈ D Z such that w i ∈ K s i and w i +1 = ψ ( w i ) , ∀ i ∈ Z (5.5)and, whenever ( s i ) i ∈ Z is a k − periodic sequence (that is, s i + k = s i , ∀ i ∈ Z ) for some k ≥ , there exists a k − periodic sequence ( w i ) i ∈ Z ∈ D Z satisfying (5.5). When wewant to stress the role of the K j ’s, we also say that ψ induces chaotic dynamics on m symbols in the set D relatively to ( K , . . . , K m − ). Remark 5.4.
We recall that the property expressed in (5.5) corresponds (in the caseof two symbols) to the definition of chaos in the sense of coin − tossing considered byKirchgraber and Stoffer in [21]. The same kind of chaotic behavior is also obtained byKennedy, Ko¸cak e Yorke in [18, Proposition 5]. As a further addition with respect to[18] and [21], our definition takes account also of the presence of periodic itinerariesgenerated by periodic points. ⊳ Theorem 5.3.
Assume there is an oriented N − dimensional rectangle e X = ( X, X − ) of a metric space Z and a map ψ : Z ⊇ D ψ → Z. Let
D ⊆ X ∩ D ψ and suppose thereexist m ≥ nonempty and pairwise disjoint compact sets K , K , . . . , K m − ⊆ D such that ( D , K i , ψ ) : e X ≎ −→ e X , for i = 0 , . . . , m − . Then the following conclusions hold: ( b ) The map ψ induces chaotic dynamics on m symbols in the set D relatively to ( K , . . . , K m − );( b ) For each sequence of m symbols s = ( s n ) n ∈ { , , . . . , m − } N , there exists acompact connected set C s ⊆ K s which cuts the arcs between X ℓ and X r in X and such that, for every w ∈ C s , there is a sequence ( y n ) n with y = w and y n ∈ K s n , ψ ( y n ) = y n +1 , ∀ n ≥ . The dimension of C s at each point is at least N − . Moreover, dim( C s ∩ ϑX ) ≥ N − and π : h − ( C s ) ∩ ∂I N → R N − \ { } is essential (where π is defined asin (4.1) for p i = , ∀ i ).Proof. The result easy follows by applying Theorem 5.2 with the positions X k = X and ψ k = ψ, ∀ k ∈ Z , and noting that, in view of Remark 5.3, conclusion ( b ) is justa restatement of conclusion ( a ) in Theorem 5.2, while conclusion ( b ) comes fromconclusions ( a ) and ( a ) in Theorem 5.2 and by Definition 5.3. (cid:3) Several definitions of chaotic dynamics relate the behavior of the iterates of themap ψ to a particular operator (the Bernoulli shift) acting on the set of sequencesof m symbols. Our Definition 5.3 and the corresponding conclusion ( b ) achieved in UTTING SURFACES 29
Theorem 5.3 allow us to derive some facts in such a direction as well. To this aim,we first recall some basic notions, following [48].Let m ≥ m = { , . . . , m − } Z the set of the two − sided sequences of m symbols. The set Σ m can be endowed witha standard distance d ( s ′ , s ′′ ) := X i ∈ Z | s ′ i − s ′′ i | m | i | +1 , where s ′ = ( s ′ i ) i ∈ Z , s ′′ = ( s ′′ i ) i ∈ Z ∈ Σ m , (5.6)so that (Σ m , d ) is a compact metric space. The Bernoulli shift σ is the homeomorphismon Σ m defined by σ (( s i ) i ) := ( s i +1 ) i (5.7)and it represents one of the paradigms for chaotic dynamical systems in a (compact)metric space. In particular (as shown in [46, Theorem 7.12]), σ has positive topologicalentropy, expressed by h top ( σ ) = log( m )(see [46] for the pertinent definitions and more details).Let Λ be a compact metric space and let ψ : Λ → Λ be a continuous map. Wesay that ψ is semiconjugate to the two − sided m − shift if there exists a continuoussurjective mapping g : Λ → Σ m such that g ◦ ψ = σ ◦ g. (5.8)In a similar manner, if we denote byΣ + m = { , . . . , m − } N the set of the one − sided sequences of m symbols, endowed with a distance analogousto the one defined in (5.6), we say that ψ is semiconjugate to the one − sided m − shift if there exists a continuous surjective mapping g : Λ → Σ + m such that (5.8) holds.The following result (which is substantially a standard fact) connects the conceptof semiconjugation with the Bernoulli shift to the one of chaotic dynamics expressedin Definition 5.3. Its proof could be easily adapted from similar arguments previouslyappeared in the literature (see, for instance [18, 20] for semidynamical systems inducedby continuous maps of metric spaces), but, for sake of completeness, we provide hereall the details. Lemma 5.1.
Let Z be a metric space, ψ : Z ⊇ D ψ → Z be a map which is continuouson a set D ⊆ D ψ and induces therein chaotic dynamics on m ≥ symbols (relativelyto ( K , . . . , K m − ) ). Then, there exists a nonempty compact set Λ ⊆ m − [ j =0 K j , which is invariant for ψ and such that ψ | Λ is semiconjugate to the two − sided m − shift,so that the topological entropy h top ( ψ ) satisfies h top ( ψ ) ≥ log( m ) . Moreover, the subset P of Λ made by the periodic points of ψ is dense in Λ andif we denote by g : Λ → Σ m the continuous surjection in (5.8) , it holds also that the counterimage through g of any k − periodic sequence in Σ m contains at least one k − periodic point.Proof. Setting K := S m − j =0 K j , we defineΛ := { w ∈ K : ψ i ( w ) ∈ K , ∀ i ∈ N } = + ∞ \ i =0 ψ − i ( K )and P := { x ∈ Λ : ∃ k ≥ ψ k ( x ) = x } . Since K is compact and ψ is continuous on K , it follows immediately that also Λ is compact and that ψ (Λ ) ⊆ Λ (that is, Λ is invariant for ψ ). Let us now define g : Λ → Σ + m , as g ( w ) := ( s i ) i ∈ N ⇔ ψ i ( w ) ∈ K s i , ∀ i ∈ N . By Definition 5.3, the map g turns out to be surjective and the counterimage through g of any k − periodic sequence in Σ + m contains at least one k − periodic point (belongingto P ). The continuity of g comes from the continuity of ψ on Λ , the choice of thedistance d in (5.6) and the fact that the sets K j are compact and pairwise disjoint.Actually, g turns out to be uniformly continuous as it is defined on a compact metricspace. A direct inspection shows that the relation in (5.8) is satisfied and thereforethe map g induces a semiconjugation between ψ | Λ and the one − sided m − shift.Let Σ perm ⊆ Σ m be the set of the periodic two − sided sequences of m symbols. Since every two − sidedperiodic sequence of m symbols determines a one − sided periodic sequence of m sym-bols (and viceversa), we have that the map g | P may be considered as a function withvalues in Σ perm . In fact, for every w ∈ P , we have g ( w ) = ( s i ) i ∈ Z ∈ Σ perm ⇔ ψ i ( w ) ∈ K s i , ∀ i ∈ N . Thus, we can define a uniformly continuous and surjective map g : P → Σ perm , by setting, for each w ∈ P : g ( w ) := ( s i ) i ∈ Z ∈ Σ perm ⇔ ψ i ( w ) ∈ K s i , ∀ i ∈ Z . (5.9)Notice that g ◦ ψ ( w ) = σ ◦ g ( w ) , ∀ w ∈ P , where σ is the two − sided Bernoulli shift on m symbols defined in (5.7).Now, setting Λ := P ⊆ Λ , it holds that ψ (Λ) ⊆ Λ, so that Λ is compact and invariant for ψ. At last, we extendthe uniformly continuous surjective mapping g : P → Σ perm ⊆ Σ m to a continuous surjective function g : Λ → Σ m , such that g ◦ ψ ( x ) = σ ◦ g ( x ) , ∀ x ∈ Λ . UTTING SURFACES 31
From the above proved semiconjugacy condition and by [46, Theorem 7.2] it followsthat h top ( ψ ) ≥ h top ( σ ) = log( m ) . Hence we see that all the properties listed in the statement of the lemma are satisfied.The proof is complete. (cid:3)
Clearly, in view of the above lemma, conclusion ( b ) in Theorem 5.3 can be refor-mulated in terms of a semiconjugation between ψ and a Bernoulli shift.The next consequence of Theorem 5.2 deals with a situation which occurs in someODE models (see, e.g., [6, 33, 35]) where there are generalized rectangles linked eachother by a stretching map. We confine ourselves to the simpler case in which onlytwo objects are involved. More general examples could be considered as well. Corollary 5.1.
Let e A and e A be oriented N − dimensional rectangles of a metricspace Z, with A ∩ A = ∅ , and let ψ : Z ⊇ D ψ → Z be a map. Assume there existcompact sets K i,j for i, j ∈ { , } , with K i,j ⊆ A i ∩ D ψ , ∀ i, j = 0 , such that ( K i,j , ψ ) : e A i ≎ −→ e A j , ∀ i, j = 0 , . Then the following conclusions hold: ( c ) For any two − sided sequence of two symbols s = ( s k ) k ∈ Z ∈ { , } Z , there existsa sequence ( w k ) k ∈ Z such that w k ∈ K s k ,s k +1 ⊆ A s k and ψ ( w k ) = w k +1 , for all k ∈ Z ;( c ) For any one − sided sequence of two symbols s = ( s n ) n ∈ N ∈ { , } N , there existsa compact connected set C s ⊆ K s ,s ⊆ A s which cuts the arcs between A s ℓ and A s r in A s and such that, for every w ∈ C s , there is a sequence ( y n ) n with y = w and y n ∈ K s n ,s n +1 , ψ ( y n ) = y n +1 , ∀ n ≥ . The dimension of C s at each point is at least N − . Moreover, dim( C s ∩ ϑ A s ) ≥ N − and π : h − ( C s ) ∩ ∂I N → R N − \ { } is essential (where π is defined asin (4.1) for p i = , ∀ i ); ( c ) For any two − sided sequence of two symbols s = ( s k ) k ∈ Z ∈ { , } Z which is m -periodic ( m ≥ ), there exists a m − periodic sequence ( w k ) k ∈ Z such that w k ∈ K s k ,s k +1 ⊆ A s k and ψ ( w k ) = w k +1 , for all k ∈ Z . Proof.
Recalling Remark 5.3, the result easy follows by applying Theorem 5.2 withthe position ψ k = ψ, ∀ k ∈ Z , and setting X k = A or X k = A , according to thevalue of s k in the considered sequence of two symbols. (cid:3) We end this paper with a result which applies Theorem 5.3 to a framework whichfits for possible applications to the detection of chaos via computer assisted proofs .In view of it we preface the following definition adapted from [36, 37, 38]. Indeed, it is not difficult to take advantage of the computations already performed in articleslike [15, 50, 51, 52] (regarding topological horseshoes in the sense of Kennedy and Yorke) and addto their conclusions also the existence of infinitely many periodic solutions.
Definition 5.4.
Let f M and e N be two oriented N − dimensional rectangles of thesame metric space Z. We say that f M is a vertical slab of e N and write f M ⊆ v e N if M ⊆ N and, either M ℓ ⊆ N ℓ and M r ⊆ N r , or M ℓ ⊆ N r and M r ⊆ N ℓ , so that any path in M joining the two sides of M − is also a path in N and joins thetwo opposite sides of N − . We say that f M is a horizontal slab of e N and write f M ⊆ h e N if M ⊆ N and every path in N joining the two sides of N − admits a sub-path in M that joins the two opposite sides of M − . Given three oriented N − dimensional rectangles e A , e B and e E of the same metric space Z, with E ⊆ A ∩ B , we say that e B crosses e A in e E and write e E ∈ { e A ⋔ e B} , if e E ⊆ v e A and e E ⊆ h e B . The above definitions, which imitate the classical terminology in [48, Ch.2.3], aretopological in nature and therefore do not necessitate any metric assumption (likesmoothness, lipschitzeanity, or similar ones often required in the literature). Wealso notice that the terms “vertical” and “horizontal” are employed in a purely con-ventional manner; the vertical is the expansive direction and the horizontal is thecontractive one (in a quite broad sense).Our next and final result (Theorem 5.4) depicts a situation when the domain andthe codomain of the mapping ψ are two intersecting oriented N − dimensional rectan-gles. A graphical illustration of it can be found in Figure 3, which is inspired by theSmale solenoid. Theorem 5.4.
Let e A and e B be oriented N − dimensional rectangles of a metric space Z and let D ⊆ A ∩ D ψ be a closed set such that ( D , ψ ) : e A ≎ −→ e B . Assume there exist m ≥ oriented N − dimensional rectangles e E , . . . , e E m − ∈ { e A ⋔ e B } . Then, ψ has at least a fixed point in each of the sets D ∩ E i ( i = 0 , . . . , m − ).Moreover, if m ≥ and D ∩ E i ∩ E j = ∅ , for i = j ( ∀ i, j ) , the following conclusions hold: ( d ) The map ψ induces chaotic dynamics on m symbols in the set D relatively to ( D ∩ E , . . . , D ∩ E m − ); UTTING SURFACES 33 ( d ) For each sequence of m symbols s = ( s n ) n ∈ { , , . . . , m − } N , there exists acompact connected set C s ⊆ D ∩ E s which cuts the arcs between E s ℓ and E s r in E s and such that, for every w ∈ C s , there is a sequence ( y n ) n with y = w and y n ∈ E s n , ψ ( y n ) = y n +1 , ∀ n ≥ . The dimension of C s at each point is at least N − . Moreover, dim( C s ∩ ϑ A ) ≥ N − and π : h − ( C s ) ∩ ∂I N → R N − \ { } is essential (where π is defined asin (4.1) for p i = , ∀ i ).Proof. First of all we show that(
D ∩ E i , ψ ) : e B ≎ −→ e B , ∀ i = 0 , . . . , m − . (5.10)Indeed, let γ be a path with γ ⊆ B and γ ∩ B ℓ = ∅ , γ ∩ B r = ∅ . Then, since e E i ⊆ h e B , ∀ i = 0 , . . . , m − , there exists a sub-path σ (= σ i ) of γ such that σ ⊆ E i and σ ∩ E iℓ = ∅ , σ ∩ E ir = ∅ . Recalling now that e E i ⊆ v e A , ∀ i = 0 , . . . , m − , it holds that σ ⊆ E i ⊆ A and σ ∩ A ℓ = ∅ , σ ∩ A r = ∅ . Finally, since ( D , ψ ) : e A ≎ −→ e B , there is a sub-path η (= η i ) of σ such that η ⊆ D∩E i , ψ ( η ) ⊆ B , with ψ ( η ) ∩B ℓ = ∅ , ψ ( η ) ∩B r = ∅ . Inthis way we have proved that any path γ with γ ⊆ B and γ ∩B ℓ = ∅ , γ ∩B r = ∅ admitsa sub-path η such that η ⊆ D ∩ E i and ψ ( η ) ⊆ B with ψ ( η ) ∩ B ℓ = ∅ , ψ ( η ) ∩ B r = ∅ . Therefore the condition in (5.10) has been checked and the existence of at least afixed point for ψ in D ∩ E i ( ∀ i = 0 , . . . , m −
1) follows by Theorem 5.1. To prove theremaining part of the statement, we apply Theorem 5.3 with the positions X = B and K i = D ∩ E i , for i = 0 , . . . , m − . (cid:3) Figure 3. The ellipsoidal body A is stretched by a continuous mapping ψ to a subset of the spiral − like set B . Both A and B are 3 − dimensionalgeneralized rectangles that we orientate as follows: the compact sets A ℓ and A r are the closure of the two components of ϑ A = ∂ A which are obtainedafter removing the darker part of the “lateral” surface; the compact sets B ℓ and B r are the two discs at the ends of the spiral body B (the order in whichwe label the two parts of A − and B − can be chosen arbitrarily). Accordingto Remark 5.2, the stretching condition ψ : e A ≎ −→ e B is fulfilled if we assumethat ψ ( A ) ⊆ B and that ψ ( A ℓ ) ⊆ B ℓ , as well as ψ ( A r ) ⊆ B r . Note thatwe do not require ψ to be a homeomorphism, nor ψ ( A ) = B . It is not evennecessary that the end sets B ℓ and B r of B lie outside A . Among the fiveintersections between A and B , only two (namely, the ones visible as a fullcrossing of the spiral − like set across the ellipsoidal body, that we call E and E ) correspond to a crossing in the sense of Definition 5.4. Therefore,Theorem 5.4 ensures the existence of at least a fixed point for ψ both in E and E and, moreover, ψ induces chaotic dynamics on two symbols in A (relatively to E and E ). Even if the drawn figures look smooth, there isno need of any regularity assumption neither for the sets (except of beinghomeomorphic to a cube) nor for their intersections. To conclude, we stress the fact that the definition of oriented N − dimensional rectan-gle can be slightly modified in order to take into account suitable perturbations ofthe domain and of the map. A similar setting has been analyzed, for instance, in [5],[19], [49] and [54]. A development of these topics (which have a relevant interest fromthe point of view of the applications) will be studied in a subsequent work, using ourapproach. Acknowledgment . The authors thank professor Gianluca Gorni for his help in theuse of Mathematica software.
UTTING SURFACES 35
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UTTING SURFACES 37
Fabio Zanolin
University of Udine, Department of Mathematics and Computer Science,via delle Scienze 206, 33100 Udine, Italy.mailto: [email protected]
Marina Pireddu