Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space

and Applied Analysis 3 Definition 2.1 see 23 . Let X, d be a metric space, f : X → X a map, and S a set of X with at least two points. Then, S is called a scrambled set of f if, for any two distinct points x, y ∈ S, lim inf n→∞ d ( f x , f ( y )) 0, lim sup n→∞ d ( f x , f ( y )) > 0. 2.1 The map f is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set S of f . Definition 2.2 see 24 . Let X, d be a metric space. A map f : V ⊂ X → V is said to be chaotic on V in the sense of Devaney if i the periodic points of f in V are dense in V ; ii f is topologically transitive in V ; iii f has sensitive dependence on initial conditions in V . Now, we give the definition of hyperbolic fixed point. Definition 2.3 see 25 . Let X be a Banach space, U ⊂ X be a set, and F : U ⊂ X → X be a map. Assume that z is a fixed point of F, F is continuously differentiable in some neighborhood of z, and denote DF z the Fréchet derivative of F at z. The fixed point z is called hyperbolic if σ DF z ∩ {λ ∈ C : |λ| 1} ∅, where σ A denotes the spectrum of a linear operator A. The hyperbolic fixed point z is called a saddle point if σ DF z ∩ {λ ∈ C : |λ| > 1}/ ∅ and σ DF z ∩ {λ ∈ C : |λ| < 1}/ ∅. If F is invertible, then for any p0 ∈ X, the setO F p0 : {Fk p0 : k ≥ 0} is said to be the forward orbit of F from p0, the set O− F p0 : {Fk p0 : k ≤ 0} is said to be the backward orbit of F from p0. Since it is not required that F is invertible in this paper, a backward orbit of p0 is a set O− F p0 {pj : j ≤ 0} with pj 1 F pj , j ≤ −1, which may not exist, or exist but may not be unique. A whole orbit of p0 is the union O F p0 ∪ O− F p0 , denoted by OF p0 in the case that it has a backward orbit O− F p0 . The stable set W s z, F and the unstable set W z, F of a hyperbolic fixed point z of F are defined by W z, F : { p0 ∈ X: the forward orbit { pj } ∞ j 0 of F from p0 such that, pj −→ z as j −→ ∞ } , W z, F : { p0 ∈ X: there exists a backward orbit { p−j } ∞ j 0 of F from p0 such that p−j −→ z as j −→ ∞ } , 2.2 respectively. The local stable and unstable sets are defined by W loc z,U, F : W s z, F ∩U, W loc z,U, F : W z, F ∩U, 2.3 respectively, where U is some neighborhood of z. If U B z, r or B z, r for some r > 0, then the corresponding local stable and unstable sets of F are denoted by W loc z, r, F and W loc z, r, F , respectively. By the Stable Manifold Theorem 26 , if F is continuously 4 Abstract and Applied Analysis differentiable in some neighborhood of a saddle point z, then there exists a neighborhood U of z such that the corresponding local stable and unstable set of z are submanifolds of X, respectively. In the following, we first give the definition that two manifolds intersect transversally and then give the definition of transversal homoclinic orbit for continuous maps. Definition 2.4 see 25 . Two submanifolds V and W in a manifold M are transverse in M provided for any point q ∈ V ∩ W , we have that TqV TqW TqM, where TqV and TqW denote the tangent spaces of V and W at q, respectively, and “ ” means the sum of the two subspaces this allows for the possibility that V ∩W ∅ . Remark 2.5. If M R, then V and W in M are transverse in M provided for any point q ∈ V ∩ W , we have that TqV TqW R. Obviously, if dim TqV dim TqW n, then the sum of the two subspaces TqV and TqW is a direct one, denoted by ⊕. Definition 2.6 see 12 . LetX be a Banach space, F : X → X be a map, and z ∈ X be a saddle point of F. i An orbitOF p0 {pj} ∞ j −∞ is said to be a homoclinic orbit asymptotic to z if p0 / z and limj→ ∞pj limj→−∞pj z. ii A homoclinic orbit OF p0 {pj} ∞ j −∞ to z is said to be transversal if there exists an open neighborhood U of z such that p−i ∈ W loc z,U, F and pj ∈ W loc z,U, F for any sufficiently large integers i, j ≥ 0, and F j sends a disc in W loc z,U, F containing p−i diffeomorphically onto its image that is transversal to W loc z,U, F at pj . The following lemma is taken from Theorems 3.1 and 5.2, Corollary 6.1, and the result in Section 7 of 12 . Lemma 2.7. Let F : Z → Z be a map, where Z X × Y , and X and Y are Banach spaces. i Let A and B be linear continuous maps in X and Y , respectively, with the absolute values of the spectrum of A less than 1 and the absolute values of the spectrum of B larger than 1, and ‖A‖, ‖B−1‖ ≤ λ0 for some constant 0 < λ0 < 1. ii Assume that U is an open neighborhood of 0 in Z and f1 : U → X, f2 : U → Y are C k ≥ 1 maps with fi 0 0, Dfi 0 0, i 1, 2. Further, assume that Df1, Df2 are uniformly continuous in U, and satisfies that for some constants 0 < θ < 1 − λ0 and γ > 0, ‖Df1 x, y ‖, ‖Df2 x, y ‖ < θ for all x, y ∈ B 0, γ ⊂ U. iii Let F : U → Z be of the following form: F ( x, y ) ( Ax f1 ( x, y ) , By f2 ( x, y )) , 2.4 and have the local stable and unstable manifoldsW loc 0, U, F { x, y | x, y ∈ U, y 0} andW loc 0, U, F { x, y | x, y ∈ U, x 0}/ {0}. iv Assume that O p0 {pi}i −∞ is a homoclinic orbit of F with pi → 0 as i → ±∞, and there exists an integerN > 0 such that p−N ∈ W loc 0, U, F , pN ∈ W loc 0, U, F , and Abstract and Applied Analysis 5and Applied Analysis 5 iv1 F2N sends a disc O1 ∩ W loc 0, U, F centered at p−N diffeomorphically onto O2 F2N O1 containing pN ; iv2 O2 intersects W loc 0, U, F transversally at pN . Then O p0 is a transversal homoclinic orbit of F. Furthermore, there exists an integer k > 0 and a subset Λ in a neighborhood of O p0 such that F on Λ is topologically conjugate to the full shift map on the doubly infinite sequence of two symbols. Consequently, F is chaotic in the sense of both Li-Yorke and Devaney, and its topological entropy h F ≥ log 2/k. Note that it is not required that F is a diffeomorphism, even F may not be continuous on the whole space Z in Lemma 2.7. 3. Distribution of Maps with Transversal Homoclinic Orbits In this section, we first consider distribution of maps with transversal homoclinic orbits in a continuous self-map space, which consists of continuous maps that transform a closed, bounded, and convex set in a Banach space into itself. At the end of this section, we discuss distribution of chaotic maps in a continuous map space, in which a map may not transform its domain into itself. Without special illustration, we always assume that X, ‖ · ‖X and Y, ‖ · ‖Y are Banach spaces, andDX andDY are bounded, convex, and open sets in X and Y , respectively. It is evident that D DX × DY is a bounded, convex, and open set in Z X × Y , where the norm ‖ · ‖ on Z is defined by ‖ x, y ‖ max{‖x‖X, ‖y‖Y}, for any x, y ∈ Z, where x ∈ X, y ∈ Y . Introduce the following map space: C0 ( D,D ) : { f : D −→ D is continuous and has a fixed point in D } . 3.1 For any f ∈ C0 D,D , let ∥ ∥f ∥ ∥ : sup {∥ ∥f x ∥ ∥ : x ∈ D } , 3.2 and for any f, g ∈ C0 D,D , let d ( f, g ) : ∥f − g∥. 3.3 Then C0 D,D , d is a metric space. It may not be complete because a limit of a sequence of maps in C0 D,D is continuous and bounded, but may not have a fixed point in D. But in the special case that Z is finite-dimensional, C0 D,D , d is a complete metric space by the Schauder fixed point theorem. In this section, we first study distribution of maps with transversal homoclinic orbits in C0 D,D . 6 Abstract and Applied Analysis For convenience, by x, y ∈ Z denote x ∈ X and y ∈ Y , by Fix f denote the set of all the fixed points of f . For x1, y1 , x2, y2 ∈ Z with x1, y1 / x2, y2 , by l x1, y1 , x2, y2 denote the straight half-line connecting x1, y1 and x2, y2 : l (( x1, y1 ) , ( x2, y2 )) : { u ( x1, y1 ) t (( x2, y2 ) − x1, y1 )) : t ≥ 0. 3.4 Lemma 3.1 see 22, Lemma 3.1 . For every map f ∈ C0 D,D and any ε > 0, there exists a map g ∈ C0 D,D such that d f, g < ε, Fix g ∩ D/ ∅, and g is continuously differentiable in some neighborhood of some point x∗ ∈ Fix g ∩D. Lemma 3.2. For every map f ∈ C0 D,D and every ε > 0, there exists a map F ∈ C0 D,D with d f, F < ε such that F has a transversal homoclinic orbit in D. Proof. Fix any f ∈ C0 D,D . By Lemma 3.1, it suffices to consider the case that f has a fixed point z z1, z2 ∈ D with z1 ∈ X, z2 ∈ Y , and is continuously differentiable in some neighborhood of z. For any ε > 0, there exists a positive constant r0 < ε/4 with B z, r0 ⊂ D such that ∥ ∥f ( x, y ) − z∥ < ε 4 , ( x, y ) ∈ B z, r0 . 3.5 Let r r0/4, take two constants a, b with r < a < b < r0/3 and take two points p1 x0, z2 ∈ B z, r , p0 z1, y0 ∈ B z, b \ B z, a , where x0 ∈ X, y0 ∈ Y . The rest of the proof is divided into three steps. Step 1. Construct a map F that is locally controlled near z. Define F ( x, y ) ⎧ ⎨ ⎩ ( λ x − z1 z1, μ ( y − z2 ) z2 ) , ( x, y ) ∈ B z, r , ( x x0 − z1 , y − ( y0 − z2 )) , ( x, y ) ∈ B z, b \ B z, a , 3.6 where x ∈ X, y ∈ Y , and λ and μ are real parameters and satisfy |λ| < 1, b r < ∣μ ∣∣ < r0 r . 3.7 Note that |μ| > 1 since r < b. For any x, y ∈ B z, a \ B z, r , F x, y is defined as follows. Let x′ 1, y′ 1 and x′ 2, y ′ 2 be the intersection points of the straight line l z, x, y with ∂B z, r and ∂B z, a , respectively, see Figure 1 . Set F ( x, y ) F ( x′ 1, y ′ 1 ) t ( x, y )( F ( x′ 2, y ′ 2 ) − Fx′ 1, y′ 1 )) , 3.8 where t x, y ∈ 0, 1 is determined as follows; ( x, y ) ( x′ 1, y ′ 1 ) t ( x, y )(( x′ 2, y ′ 2 ) − x′ 1, y′ 1 )) . 3.9 Abstract and Applied Analysis 7and Applied Analysis 7


Introduction
Distribution of a set of maps with some dynamical properties in some continuous map space is a very interesting topic.In the 1960s, Smale 1 studied density of hyperbolicity.Some scholars believed that hyperbolic systems are dense in spaces of all dimensions, but it was shown that the conjecture is false in the late 1960s for diffeomorphisms on manifolds of dimension ≥2.The problem whether hyperbolic systems are dense in the one-dimension case was studied by many scholars.It was solved in the C 1 topology by Jakobson 2 , a partial solution was given in the C 2 topology by Blokh and Misiurewicz 3 , and C 2 density was finally proved by Shen 4 .In 2007, Kozlovki et al. got the result in C k topology; that is, hyperbolic i.e., Axiom A maps are dense in the space of C k maps defined in a compact interval or circle, k 1, 2, . . ., ∞, ω 5 .At the same time, some other scholars considered the distribution of hyperbolic diffeomorphisms in Diff M , where M is a manifold.Just like the work of Smale, Palis 6, 7 gave the following conjecture: 1 any f ∈ Diff M can be approximated by a hyperbolic diffeomorphism or by a diffeomorphism exhibiting a homoclinic bifurcation tangency or cycle , 2 any diffeomorphism can be C r approximated by a Morse-Smale one or by one exhibiting transversal homoclinic orbit.Later, it was shown that the conjecture 1 holds for C 1 diffeomorphisms of surfaces 8 .And some good results Definition 2.1 see 23 .Let X, d be a metric space, f : X → X a map, and S a set of X with at least two points.Then, S is called a scrambled set of f if, for any two distinct points x, y ∈ S, The map f is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set S of f.Definition 2.2 see 24 .Let X, d be a metric space.A map f : V ⊂ X → V is said to be chaotic on V in the sense of Devaney if i the periodic points of f in V are dense in V ; ii f is topologically transitive in V ; iii f has sensitive dependence on initial conditions in V .Now, we give the definition of hyperbolic fixed point.
Definition 2.3 see 25 .Let X be a Banach space, U ⊂ X be a set, and F : U ⊂ X → X be a map.Assume that z is a fixed point of F, F is continuously differentiable in some neighborhood of z, and denote DF z the Fréchet derivative of If F is invertible, then for any p 0 ∈ X, the set O F p 0 : {F k p 0 : k ≥ 0} is said to be the forward orbit of F from p 0 , the set O − F p 0 : {F k p 0 : k ≤ 0} is said to be the backward orbit of F from p 0 .Since it is not required that F is invertible in this paper, a backward orbit of p 0 is a set O − F p 0 {p j : j ≤ 0} with p j 1 F p j , j ≤ −1, which may not exist, or exist but may not be unique.A whole orbit of p 0 is the union O F p 0 ∪ O − F p 0 , denoted by O F p 0 in the case that it has a backward orbit O − F p 0 .The stable set W s z, F and the unstable set W u z, F of a hyperbolic fixed point z of F are defined by W s z, F : p 0 ∈ X: the forward orbit p j ∞ j 0 of F from p 0 such that, respectively.The local stable and unstable sets are defined by respectively, where U is some neighborhood of z.If U B z, r or B z, r for some r > 0, then the corresponding local stable and unstable sets of F are denoted by W s loc z, r, F and W u loc z, r, F , respectively.By the Stable Manifold Theorem 26 , if F is continuously differentiable in some neighborhood of a saddle point z, then there exists a neighborhood U of z such that the corresponding local stable and unstable set of z are submanifolds of X, respectively.
In the following, we first give the definition that two manifolds intersect transversally and then give the definition of transversal homoclinic orbit for continuous maps.Definition 2.4 see 25 .Two submanifolds V and W in a manifold M are transverse in M provided for any point q ∈ V ∩ W, we have that T q V T q W T q M, where T q V and T q W denote the tangent spaces of V and W at q, respectively, and " " means the sum of the two subspaces this allows for the possibility that V ∩ W ∅ .
, then the sum of the two subspaces T q V and T q W is a direct one, denoted by ⊕.Definition 2.6 see 12 .Let X be a Banach space, F : X → X be a map, and z ∈ X be a saddle point of F.
loc z, U, F and p j ∈ W s loc z, U, F for any sufficiently large integers i, j ≥ 0, and The following lemma is taken from Theorems 3.1 and 5.2, Corollary 6.1, and the result in Section 7 of 12 .
Lemma 2.7.Let F : Z → Z be a map, where Z X × Y , and X and Y are Banach spaces.
i Let A and B be linear continuous maps in X and Y , respectively, with the absolute values of the spectrum of A less than 1 and the absolute values of the spectrum of B larger than 1, and A , B −1 ≤ λ 0 for some constant 0 < λ 0 < 1.
ii Assume that U is an open neighborhood of 0 in Z and f Further, assume that Df 1 , Df 2 are uniformly continuous in U, and satisfies that for some constants 0 < θ < 1 − λ 0 and γ > 0, Df 1 x, y , Df 2 x, y < θ for all x, y ∈ B 0, γ ⊂ U.
iii Let F : U → Z be of the following form: and have the local stable and unstable manifolds is a homoclinic orbit of F with p i → 0 as i → ±∞, and there exists an integer N > 0 such that p −N ∈ W u loc 0, U, F , p N ∈ W s loc 0, U, F , and Then O p 0 is a transversal homoclinic orbit of F. Furthermore, there exists an integer k > 0 and a subset Λ in a neighborhood of O p 0 such that F k on Λ is topologically conjugate to the full shift map on the doubly infinite sequence of two symbols.Consequently, F is chaotic in the sense of both Li-Yorke and Devaney, and its topological entropy h F ≥ log 2/k.
Note that it is not required that F is a diffeomorphism, even F may not be continuous on the whole space Z in Lemma 2.7.

Distribution of Maps with Transversal Homoclinic Orbits
In this section, we first consider distribution of maps with transversal homoclinic orbits in a continuous self-map space, which consists of continuous maps that transform a closed, bounded, and convex set in a Banach space into itself.At the end of this section, we discuss distribution of chaotic maps in a continuous map space, in which a map may not transform its domain into itself.
Without special illustration, we always assume that X, In this section, we first study distribution of maps with transversal homoclinic orbits in C 0 D, D .
For convenience, by x, y ∈ Z denote x ∈ X and y ∈ Y , by Fix f denote the set of all the fixed points of f.For x 1 , y 1 , x 2 , y 2 ∈ Z with x 1 , y 1 / x 2 , y 2 , by l x 1 , y 1 , x 2 , y 2 denote the straight half-line connecting x 1 , y 1 and x 2 , y 2 : Proof.Fix any f ∈ C 0 D, D .By Lemma 3.1, it suffices to consider the case that f has a fixed point z z 1 , z 2 ∈ D with z 1 ∈ X, z 2 ∈ Y , and is continuously differentiable in some neighborhood of z.
For any ε > 0, there exists a positive constant r 0 < ε/4 with B z, r 0 ⊂ D such that Let r r 0 /4, take two constants a, b with r < a < b < r 0 /3 and take two points , where x 0 ∈ X, y 0 ∈ Y .The rest of the proof is divided into three steps.
Step 1. Construct a map F that is locally controlled near z. Define where x ∈ X, y ∈ Y , and λ and μ are real parameters and satisfy Note that |μ| > 1 since r < b.
For any x, y ∈ B z, a \ B z, r , F x, y is defined as follows.Let x 1 , y 1 and x 2 , y 2 be the intersection points of the straight line l z, x, y with ∂B z, r and ∂B z, a , respectively, see Figure 1 where t x, y ∈ 0, 1 is determined as follows; x, y x 1 , y 1 t x, y x 2 , y 2 − x 1 , y 1 . 3.9 It is noted that when x, y continuously varies in B z, a \B z, r , so do the intersection points x 1 , y 1 and x 2 , y 2 .Consequently, t x, y and then F x, y are continuous in B z, a \ B z, r .Next, define F x, y f x, y for x, y ∈ D \ B z, r 0 .Finally, for any x, y ∈ B z, r 0 \ B z, b , suppose that x 1 , y 1 and x 2 , y 2 are the intersection points of the straight line l z, x, y with ∂B z, b and ∂B z, r 0 , respectively.Define F x, y as that in 3.8 , where t x, y is determined by 3.9 with x 1 , y 1 and x 2 , y 2 replaced by x 1 , y 1 and x 2 , y 2 , respectively.Hence, F x, y is continuous in B z, r 0 \ B z, b .
Obviously, z is a saddle fixed point of F, and x, y : y z 2 , x − z 1 X < r ⊂ W s loc z, r, F .

3.10
Step 2. F ∈ C 0 D, D and satisfies that d f, F < ε.
From the definition of F, it is easy to know that F is continuous on D and has a fixed Next, we will prove that d f, F < ε.For x, y ∈ D\B z, r 0 , F x, y −f x, y 0 < ε.For x, y ∈ B z, r , it follows from 3.5 and 3.6 that For x, y ∈ B z, a \ B z, r , it follows from 3.5 , 3.6 , and 3.8 that

3.12
For x, y ∈ B z, b \ B z, a , from 3.5 and 3.6 , one has For x, y ∈ B z, r 0 \ B z, b , from 3.5 , 3.6 , and 3.8 , one has

3.14
Therefore, from the above discussion, d f, F F − f < ε.
Step 3. F has a transversal homoclinic orbit in D.
It follows from 3.6 that F p −1 p 0 and F p 0 p 1 , where Then 0 < δ < r by 3.7 , and consequently it follows from 3.10 that the disc Further, set the discs 3.17 2 F has a transversal homoclinic orbit in D; 3 F is chaotic in the sense of both Li-Yorke and Devaney; 4 the topological entropy h F > 0.

3.18
Hence, F satisfies assumptions ii and iii in Lemma 2.7 with γ r.By the discussions in Step 3 in the proof of Lemma 3.2, F satisfies assumption iv in Lemma 2.7, where O p 0 , O −1 , O 0 , and O 1 are the same as those in the proof of Lemma 3.2 and N 1.So, all the assumptions in Lemma 2.7 are satisfied.Consequently, 3 and 4 hold by Lemma 2.7.The proof is complete.
When it is not required that a map transforms its domain D into itself, the convexity of domain D can be removed and all the corresponding results to Lemmas 3.1 and 3.

3.20
Remark 3.6. 1 As we all know, under C 1 perturbation, the hyperbolicity of a map is preserved.But it is obvious that the conclusion does not hold in the C 0 sense.2 In the C 0 topology, Theorems 3.3 and 3.4 show the density of distributions of maps with transversal homoclinic orbits, and consequently in the sense of both Li-Yorke and Devaney.However, it is not true in the C 1 topology.For example, consider the map f x, y x 2 , y 2 , x, y ∈ I 2 0, 1 × 0, 1 .

3.21
Clearly, f ∈ C 0 I 2 , I 2 and z 0 is a globally asymptotically stable fixed point of f in I 2 .By Theorem 3.3, for each ε > 0, there exists a map F ∈ C 0 I 2 , I 2 with F − f < ε such that F is chaotic in the sense of both Li-Yorke and Devaney.But, in the C 1 topology, for each positive constant ε < 1/2 and for every map F ∈ C 1 I 2 , I 2 with F is globally asymptotically stable in I 2 , and so is not chaotic in any sense.

1 For 3 Then
X and Y, • Y are Banach spaces, and D X and D Y are bounded, convex, and open sets in X and Y , respectively.It is evident that D D X × D Y is a bounded, convex, and open set in Z X × Y , where the norm • on Z is defined by x, y max{ x X , y Y }, for any x, y ∈ Z, where x ∈ X, y ∈ Y .Introduce the following map space: C 0 D, D : f : D −→ D is continuous and has a fixed point in D .3.any f ∈ C 0 D, D , let f : sup f x : x ∈ D , 3.2 and for any f, g ∈ C 0 D, D , let d f, g : f − g .3.C 0 D, D , d is a metric space.It may not be complete because a limit of a sequence of maps in C 0 D, D is continuous and bounded, but may not have a fixed point in D. But in the special case that Z is finite-dimensional, C 0 D, D , d is a complete metric space by the Schauder fixed point theorem.

3 . 4 Lemma 3 . 1 Lemma 3 . 2 .
see 22, Lemma 3.1 .For every map f ∈ C 0 D, D and any ε > 0, there exists a map g ∈ C 0 D, D such that d f, g < ε, Fix g ∩ D / ∅, and g is continuously differentiable in some neighborhood of some point x * ∈ Fix g ∩ D. For every map f ∈ C 0 D, D and every ε > 0, there exists a map F ∈ C 0 D, D with d f, F < ε such that F has a transversal homoclinic orbit in D.

Then O − 1 ⊂Theorem 3 . 3 .
W u loc z, r, F , O 0 F O −1 , and O 1 F O 0 see Figure 2 .It is evident that O 1 intersects W s loc z, r, F transversally at point p 1 .In addition, by the definition of F in B z, b , one can get that F 2 : O −1 → O 1 is a diffeomorphism.Therefore, O p 0 is a transversal homoclinic orbit asymptotic to z of F by Definition 2.6, where N 1.The entire proof is complete.Let X and Y be Banach spaces, Z X × Y , and D be a bounded, convex, and open set in Z.Then, for every map f ∈ C 0 D, D and for any ε > 0, there exists a map F ∈ C 0 D, D satisfying

Theorem 3 . 4 . 2 F 3 F
2 and Theorem 3.3 still hold.In detail, let S be a bounded open set in Z and C 0 S, Z : f : S −→ Z is continuous and bounded, and has a fixed point in S .3.19 Then C 0 S, Z , d is a metric space, where d is defined the same as that in 3.3 .The results of Lemma 3.2 and Theorem 3.3 hold, where C 0 D, D is replaced by C 0 S, Z .Their proofs are similar.Now, we only present the detailed result corresponding to Theorem 3.3.Let X and Y be Banach spaces, Z X × Y , and S be a bounded open set in Z.Then, for every map f ∈ C 0 S, Z and for any ε > 0, there exists a map F ∈ C 0 S, Z satisfying the following 1 d f, F < ε; has a transversal homoclinic orbit in S; is chaotic in the sense of both Li-Yorke and Devaney; 4 the topological entropy h F > 0. Remark 3.5.A general Banach space Z may not be discomposed into a product of two Banach spaces with dimension greater than or equal to 1.However, it is true for Z R n with n ≥ 2. So Theorem 3.3 holds for each n-dimensional space R n with n ≥ 2. In addition, if D is a bounded and convex set in R n , every continuous map f : D → D has a fixed point in D by the Schauder fixed point theorem.In this case one has that C 0 D, D C D, D f : D −→ D is continuous .