- Email: [email protected]
On a property specific to the tent map
Chaos, Solitons and Fractals 29 (2006) 1256–1258 www.elsevier.com/locate/chaos
On a property specific to the tent map Akihiko Kitada *, Yoshihito Ogasawara Department of Materials Science and Engineering, Laboratory of Mathematical Design for Materials, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan Accepted 24 August 2005
Abstract Let a set {Xk; k 2 K} of subspaces of a topological space X be a cover of X. Mathematical conditions are proposed for each subspace Xk to define a map gX k : X k ! X which has the following property specific to the tent map known in the bakerÕs transformation. Namely, for any infinite sequence x0, x1, x2, . . . of Xk, k 2 K, we can find an initial point x0 2 x0 such that gx0 ðx0 Þ 2 x1 ; gx1 ðgx0 ðx0 ÞÞ 2 x2 ; . . .. The conditions are successfully applied to a closed cover of a weak self-similar set. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The map u : [0, 1] ! [0, 1], u(x) = 2x for x 2 [0, 1/2], u(x) = 2 2x for x 2 [1/2, 1] is called tent map. The tent map well known in the dynamical system theory is characterized by the property that for any infinite sequence x0, x1, x2, . . . each term xi of which is either [0, 1/2] or [1/2, 1], we can find an initial point x0 2 x0 such that u(x0) 2 x1, u(u(x0)) 2 x2, u(u(u(x0))) 2 x3, . . . The continuity and the ontoness of the map u causes this remarkable property. In the present short note, taking a weak self-similar set S instead of [0, 1], we will generalize the above property of the tent map. That is, the tent map is a special case of the map in the following proposition.
2. A generalization of a property specific to the tent map As a generalization of a property which characterizes the tent map, we have the following proposition. Proposition. Let (X, s) be a compact T2-space and fX k 2 I; k 2 Kg be a closed cover of X. If, for each k 2 K, there exists a continuous map gXk from the subspace (Xk, sXk) onto (X, s), for any infinite sequence x0, x1, x2, . . ., we can find an initial point x0 2 x0 such that gx0(x0) 2 x1, gx1(gx0(x0)) 2 x2, . . . Here, xi 2 {Xk; k 2 K}, i = 0, 1, 2, . . . Proof. The symbol gi(a) denotes gxi ðgxi1 ð gx1 ðgx0 ðaÞÞ ÞÞ. Now suppose that the sequence x0, x1, x2, . . . of Xk ends in failure in finite steps for any initial point x0 2 x0. Since gxi1 : xi1 ! X is an onto map, there exists a point p1 2 xi1 *
Corresponding author. E-mail address: [email protected] (A. Kitada).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.159
A. Kitada, Y. Ogasawara / Chaos, Solitons and Fractals 29 (2006) 1256–1258
1257
such that gxi1 ðp1 Þ 2 xi . Since the map gxi2 : xi2 ! X is also onto, there exists a point p2 2 x2 such that gxi2 ðp2 Þ ¼ p1 . Continuing in this manner, we have a point pi 2 x0 such that gx0 ðpi Þ ¼ pi1 2 x1 . Therefore, for any i, we have a point xi 2 x0 such that gi1(xi) 2 xi. Let ni > i be the first number such that gni 1 ðxi Þ 62 xni . It is evident that the sequence {xi} of the initial point is a set which consists of infinitely many different kinds of points. Since x0 is a compact set, we have an accumulation point n of {xi} in x0. Let us consider the sequence gx0 ðnÞ; gx1 ðgx0 ðnÞÞ; . . . which starts from the point n 2 x0. From the assumption that the sequence which starts from n ends in failure in finite steps, there exists a point nn such that gnn 1 ðnÞ 62 xnn . Since any compact T2-space (X, s) is a T3-space, for xnn 2 I, there exists a s-open set H(q0) containing the point q0 ¼ gnn 1 ðnÞ ¼ gxn 1 ðgnn 2 ðnÞÞ such that H ðq0 Þ \ xnn ¼ /. The continuity of the map gxn 1 : n n ðxnn 1 ; sxnn 1 Þ ! ðX ; sÞ guarantees the existence of a s-open set H(q1) containing the point q1 ¼ gnn 2 ðnÞ such that gxn 1 ðxnn 1 \ H ðq1 ÞÞ H ðq0 Þ. Here, sxnn 1 ¼ fG \ xnn 1 ; G 2 sg. Continuing in this manner, finally, we have a s-open n set H(n) containing the point n such that gx0 ðx0 \ H ðnÞÞ H ðqnn 1 Þ 2 s. Since the point n is an accumulation point in a T1-space ðx0 ; sx0 Þ, there exists a number i0 > nn such that xi0 2 x0 \ H ðnÞ. If gx0 ðxi0 Þ 62 x1 , the contradictory relation 1 ¼ ni0 6 nn must be valid. Therefore, gx0 ðxi0 Þ 2 x1 \ H ðqnn 1 Þ. In this manner, we attain the relation gnn 2 ðxi0 Þ 2 xnn 1 \ Hðq1 Þ and then, gnn 1 ðxi0 Þ 2 H ðq0 Þ. This means thatgnn 1 ðxi0 Þ 62 xnn . Since ni0 is the number of the step at which the sequence starting from xi0 fails for the first time, the relation ni0 ¼ nn contradicts the relation nn < ni0 .
3. An application to a closed cover of a weak self-similar set Now, let us recall the definition of the weak self-similar set [1,2]. A set S in a metric space X with a metric d is called a weakSself-similar set based on a system of weak contractions fj : X ! X, j = 1, . . ., m (2 6 m < 1) provided that the relation mj¼1 fj ðSÞ ¼ S holds. Namely, {f1(S), . . ., fm(S)} is a cover of S. Here, weak contraction means a map satisfying the estimate dðfj ðxÞ; fj ðyÞÞ 6 aj ðtÞdðx; yÞ for dðx; yÞ < t; 0 < aj ðtÞ < 1; inf aj ðtÞ > 0. t>0
Lemma. Let X be a compact metric space equipped with the metric d and fj : X ! X, j = 1, . . ., m be the weak contractions satisfying three conditions. (i) Each fj is S one to one. (ii) The set mj¼1 fx 2 X ; fj ðxÞ ¼ xg is notP degenerate. (iii) There exists a point t0 > 0 such that mj¼1 aj ðt0 Þ < 1. Then, there exists in X a weak self-similar set S for which each fj(S) is compact, perfect and zero-dimensional. Proof. We have already proved [3] the existence of a compact, perfect and zero-dimensional weak self-similar space (S, s) under the conditions (i), (ii), (iii). Here, s is a topology of S defined by d. Since fj(S) S, we can define a map gj : (S,s) ! (S, s) as gj(x) = fj(x) for x 2 S. The map gj is continuous and (S, s) is compact, then the set fj(S)(= gj(S)) must be a compact set in (S, s). If there exists a point x 2 fj(S) such that fxg 2 sfj ðSÞ ¼ fG \ fj ðSÞ; G 2 sg, from the continuity of the map gj, there exists a s-open u containing a point of the set g1 j ðxÞ such that gj(u) {x}. Since fj, therefore gj, is assumed to be one to one, open set u must be a singleton set. This contradicts the assumption that S is perfect. Therefore, the subspace fj(S) of S is perfect. Next, let x be a point of fj(S) and u be a sfj ðSÞ -open containing x. There exists U 2 s such that u = U \ fj(S). Since S is zero-dimensional, we can find in U a clopen set V (i.e. V 2 s \ I) containing x. The relation u V \ fj ðSÞ 2 sfj ðSÞ \ Ifj ðSÞ means that the subspace fj(S) is zero-dimensional. Here, Ifj ðSÞ ¼ fK \ fj ðSÞ; K 2 Ig. h Let (X, s) be a compact, perfect, zero-dimensional T1-space. It is known [3–5] that for any nonempty compact metric space Y, there exists a continuous map f from X onto Y. From this fact, Proposition and Lemma, we can immediately obtain the following statement. Statement. Let S be a weak self-similar set obtained in Lemma. Then, for any infinite sequence x0, x1, x2, . . . where xi 2 {f1(S), . . ., fm(S)}, we can find continuous onto maps gfj ðSÞ : fj ðSÞ ! S; j ¼ 1; . . . ; m and a point x0 2 x0 such that gx0 ðx0 Þ 2 x1 ; gx1 ðgx0 ðx0 ÞÞ 2 x2 ; . . .
1258
A. Kitada, Y. Ogasawara / Chaos, Solitons and Fractals 29 (2006) 1256–1258
4. Conclusion Let X be a compact metric space and a set {Xk; k 2 K} of subspaces of X be a cover of X. If each Xk is perfect, zerodimensional compact set of X, we can obtain a set of maps gX k : X k ! X ; k 2 K which has the analogous property with the tent map. Namely, for any infinite sequence x0, x1, x2, . . . of Xk, k 2 K, there exists an initial point x0 2 x0 such that gx0 ðx0 Þ 2 x1 ; gx1 ðgx0 ðx0 ÞÞ 2 x2 ; . . . For example, the closed cover {f1(S), . . ., fm(S)} of a weak self-similar set S based on a system of weak contractions fj, j = 1, . . ., m characterized by the conditions (i), (ii), (iii) in Lemma can generate a set of maps gfj ðSÞ : fj ðSÞ ! S; j ¼ 1; . . . ; m which has the above property.
Acknowledgements The authors are grateful to Professors H. Fukaishi of Kagawa University and S. Tasaki of Waseda University, for useful discussions.
References [1] Kitada A, Konishi T, Watanabe T. An estimate of the Hausdorff dimension of a weak self-similar set. Chaos, Solitons & Fractals 2002;13:363. [2] Kitada A. On a topological embedding of a weak self-similar, zero-dimensional set. Chaos, Solitons & Fractals 2004;22:171. [3] Kitada A, Ogasawara Y. On a decomposition space of a weak self-similar set. Chaos, Solitons & Fractals 2005;24:785. [4] Nadler Jr SB. Continuum theory. New York: Marcel Dekker; 1992. p. 106. [5] Nadler Jr SB. Continuum theory. New York: Marcel Dekker; 1992. p. 109.
On a property specific to the tent map Akihiko Kitada *, Yoshihito Ogasawara Department of Materials Science and Engineering, Laboratory of Mathematical Design for Materials, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan Accepted 24 August 2005
Abstract Let a set {Xk; k 2 K} of subspaces of a topological space X be a cover of X. Mathematical conditions are proposed for each subspace Xk to define a map gX k : X k ! X which has the following property specific to the tent map known in the bakerÕs transformation. Namely, for any infinite sequence x0, x1, x2, . . . of Xk, k 2 K, we can find an initial point x0 2 x0 such that gx0 ðx0 Þ 2 x1 ; gx1 ðgx0 ðx0 ÞÞ 2 x2 ; . . .. The conditions are successfully applied to a closed cover of a weak self-similar set. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The map u : [0, 1] ! [0, 1], u(x) = 2x for x 2 [0, 1/2], u(x) = 2 2x for x 2 [1/2, 1] is called tent map. The tent map well known in the dynamical system theory is characterized by the property that for any infinite sequence x0, x1, x2, . . . each term xi of which is either [0, 1/2] or [1/2, 1], we can find an initial point x0 2 x0 such that u(x0) 2 x1, u(u(x0)) 2 x2, u(u(u(x0))) 2 x3, . . . The continuity and the ontoness of the map u causes this remarkable property. In the present short note, taking a weak self-similar set S instead of [0, 1], we will generalize the above property of the tent map. That is, the tent map is a special case of the map in the following proposition.
2. A generalization of a property specific to the tent map As a generalization of a property which characterizes the tent map, we have the following proposition. Proposition. Let (X, s) be a compact T2-space and fX k 2 I; k 2 Kg be a closed cover of X. If, for each k 2 K, there exists a continuous map gXk from the subspace (Xk, sXk) onto (X, s), for any infinite sequence x0, x1, x2, . . ., we can find an initial point x0 2 x0 such that gx0(x0) 2 x1, gx1(gx0(x0)) 2 x2, . . . Here, xi 2 {Xk; k 2 K}, i = 0, 1, 2, . . . Proof. The symbol gi(a) denotes gxi ðgxi1 ð gx1 ðgx0 ðaÞÞ ÞÞ. Now suppose that the sequence x0, x1, x2, . . . of Xk ends in failure in finite steps for any initial point x0 2 x0. Since gxi1 : xi1 ! X is an onto map, there exists a point p1 2 xi1 *
Corresponding author. E-mail address: [email protected] (A. Kitada).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.159
A. Kitada, Y. Ogasawara / Chaos, Solitons and Fractals 29 (2006) 1256–1258
1257
such that gxi1 ðp1 Þ 2 xi . Since the map gxi2 : xi2 ! X is also onto, there exists a point p2 2 x2 such that gxi2 ðp2 Þ ¼ p1 . Continuing in this manner, we have a point pi 2 x0 such that gx0 ðpi Þ ¼ pi1 2 x1 . Therefore, for any i, we have a point xi 2 x0 such that gi1(xi) 2 xi. Let ni > i be the first number such that gni 1 ðxi Þ 62 xni . It is evident that the sequence {xi} of the initial point is a set which consists of infinitely many different kinds of points. Since x0 is a compact set, we have an accumulation point n of {xi} in x0. Let us consider the sequence gx0 ðnÞ; gx1 ðgx0 ðnÞÞ; . . . which starts from the point n 2 x0. From the assumption that the sequence which starts from n ends in failure in finite steps, there exists a point nn such that gnn 1 ðnÞ 62 xnn . Since any compact T2-space (X, s) is a T3-space, for xnn 2 I, there exists a s-open set H(q0) containing the point q0 ¼ gnn 1 ðnÞ ¼ gxn 1 ðgnn 2 ðnÞÞ such that H ðq0 Þ \ xnn ¼ /. The continuity of the map gxn 1 : n n ðxnn 1 ; sxnn 1 Þ ! ðX ; sÞ guarantees the existence of a s-open set H(q1) containing the point q1 ¼ gnn 2 ðnÞ such that gxn 1 ðxnn 1 \ H ðq1 ÞÞ H ðq0 Þ. Here, sxnn 1 ¼ fG \ xnn 1 ; G 2 sg. Continuing in this manner, finally, we have a s-open n set H(n) containing the point n such that gx0 ðx0 \ H ðnÞÞ H ðqnn 1 Þ 2 s. Since the point n is an accumulation point in a T1-space ðx0 ; sx0 Þ, there exists a number i0 > nn such that xi0 2 x0 \ H ðnÞ. If gx0 ðxi0 Þ 62 x1 , the contradictory relation 1 ¼ ni0 6 nn must be valid. Therefore, gx0 ðxi0 Þ 2 x1 \ H ðqnn 1 Þ. In this manner, we attain the relation gnn 2 ðxi0 Þ 2 xnn 1 \ Hðq1 Þ and then, gnn 1 ðxi0 Þ 2 H ðq0 Þ. This means thatgnn 1 ðxi0 Þ 62 xnn . Since ni0 is the number of the step at which the sequence starting from xi0 fails for the first time, the relation ni0 ¼ nn contradicts the relation nn < ni0 .
3. An application to a closed cover of a weak self-similar set Now, let us recall the definition of the weak self-similar set [1,2]. A set S in a metric space X with a metric d is called a weakSself-similar set based on a system of weak contractions fj : X ! X, j = 1, . . ., m (2 6 m < 1) provided that the relation mj¼1 fj ðSÞ ¼ S holds. Namely, {f1(S), . . ., fm(S)} is a cover of S. Here, weak contraction means a map satisfying the estimate dðfj ðxÞ; fj ðyÞÞ 6 aj ðtÞdðx; yÞ for dðx; yÞ < t; 0 < aj ðtÞ < 1; inf aj ðtÞ > 0. t>0
Lemma. Let X be a compact metric space equipped with the metric d and fj : X ! X, j = 1, . . ., m be the weak contractions satisfying three conditions. (i) Each fj is S one to one. (ii) The set mj¼1 fx 2 X ; fj ðxÞ ¼ xg is notP degenerate. (iii) There exists a point t0 > 0 such that mj¼1 aj ðt0 Þ < 1. Then, there exists in X a weak self-similar set S for which each fj(S) is compact, perfect and zero-dimensional. Proof. We have already proved [3] the existence of a compact, perfect and zero-dimensional weak self-similar space (S, s) under the conditions (i), (ii), (iii). Here, s is a topology of S defined by d. Since fj(S) S, we can define a map gj : (S,s) ! (S, s) as gj(x) = fj(x) for x 2 S. The map gj is continuous and (S, s) is compact, then the set fj(S)(= gj(S)) must be a compact set in (S, s). If there exists a point x 2 fj(S) such that fxg 2 sfj ðSÞ ¼ fG \ fj ðSÞ; G 2 sg, from the continuity of the map gj, there exists a s-open u containing a point of the set g1 j ðxÞ such that gj(u) {x}. Since fj, therefore gj, is assumed to be one to one, open set u must be a singleton set. This contradicts the assumption that S is perfect. Therefore, the subspace fj(S) of S is perfect. Next, let x be a point of fj(S) and u be a sfj ðSÞ -open containing x. There exists U 2 s such that u = U \ fj(S). Since S is zero-dimensional, we can find in U a clopen set V (i.e. V 2 s \ I) containing x. The relation u V \ fj ðSÞ 2 sfj ðSÞ \ Ifj ðSÞ means that the subspace fj(S) is zero-dimensional. Here, Ifj ðSÞ ¼ fK \ fj ðSÞ; K 2 Ig. h Let (X, s) be a compact, perfect, zero-dimensional T1-space. It is known [3–5] that for any nonempty compact metric space Y, there exists a continuous map f from X onto Y. From this fact, Proposition and Lemma, we can immediately obtain the following statement. Statement. Let S be a weak self-similar set obtained in Lemma. Then, for any infinite sequence x0, x1, x2, . . . where xi 2 {f1(S), . . ., fm(S)}, we can find continuous onto maps gfj ðSÞ : fj ðSÞ ! S; j ¼ 1; . . . ; m and a point x0 2 x0 such that gx0 ðx0 Þ 2 x1 ; gx1 ðgx0 ðx0 ÞÞ 2 x2 ; . . .
1258
A. Kitada, Y. Ogasawara / Chaos, Solitons and Fractals 29 (2006) 1256–1258
4. Conclusion Let X be a compact metric space and a set {Xk; k 2 K} of subspaces of X be a cover of X. If each Xk is perfect, zerodimensional compact set of X, we can obtain a set of maps gX k : X k ! X ; k 2 K which has the analogous property with the tent map. Namely, for any infinite sequence x0, x1, x2, . . . of Xk, k 2 K, there exists an initial point x0 2 x0 such that gx0 ðx0 Þ 2 x1 ; gx1 ðgx0 ðx0 ÞÞ 2 x2 ; . . . For example, the closed cover {f1(S), . . ., fm(S)} of a weak self-similar set S based on a system of weak contractions fj, j = 1, . . ., m characterized by the conditions (i), (ii), (iii) in Lemma can generate a set of maps gfj ðSÞ : fj ðSÞ ! S; j ¼ 1; . . . ; m which has the above property.
Acknowledgements The authors are grateful to Professors H. Fukaishi of Kagawa University and S. Tasaki of Waseda University, for useful discussions.
References [1] Kitada A, Konishi T, Watanabe T. An estimate of the Hausdorff dimension of a weak self-similar set. Chaos, Solitons & Fractals 2002;13:363. [2] Kitada A. On a topological embedding of a weak self-similar, zero-dimensional set. Chaos, Solitons & Fractals 2004;22:171. [3] Kitada A, Ogasawara Y. On a decomposition space of a weak self-similar set. Chaos, Solitons & Fractals 2005;24:785. [4] Nadler Jr SB. Continuum theory. New York: Marcel Dekker; 1992. p. 106. [5] Nadler Jr SB. Continuum theory. New York: Marcel Dekker; 1992. p. 109.