On the Existence of a Self-Similar Coarse Graining of a Self-Similar Space
Akihiko Kitada, Tomoyuki Yamamoto, Tsuyoshi Yoshioka, Shousuke Ohmori
aa r X i v : . [ m a t h - ph ] J u l On the Existence of a Self-Similar CoarseGraining of a Self-Similar Space
Akihiko Kitada, Tomoyuki Yamamoto, Tsuyoshi Yoshioka and ShousukeOhmoriLaboratory of mathematical design for materials,Faculty of Science and Engineering, Waseda University,3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555
Abstract
A topological space homeomorphic to a self-similar space is demonstrated tobe self-similar. There exists a self-similar space S whose coarse graining ishomeomorphic to S . The coarse graining of S is, therefore, self-similar again.In the same way, the coarse graining of the self-similar coarse graining of S is, furthermore, self-similar. These situations succeed endlessly. Such a self-similar S is generated actually from an intense quadratic dynamics. Keywords : self-similar set, fractals, dynamical system, Cantor set, coarsegraining
In the fractal sciences, the fine structure of the self-similar space is charac-terized by the property that every details looks similar with the whole. Inthe present report, we are oppositely concerned with the coarse structuresof a self-similar space, that is, with the problem ”what self-similar space canhave a coarse graining of it with a self-similarity again?”. According to A.Fern´andez [1], the procedure of the coarse graining or the block construction[2] of a space in the statistical physics corresponds mathematically to thatof the construction of a quotient space which is defined by a classification ofall points in the space through the identification of the different points basedon an equivalence relation.At first, a sufficient condition for a given topological space to be metriz-able and self-similar with respect to the metric is investigated, and, second,the existence of a decomposition space [3] as a coarse graining of a self-similarspace S whose self-similarity is defined by a system of weak contractionswhich is topologically closely related to that defining the self-similarity of S
1s discussed in a quite elementary way. As a consequence, we are convincedthat there exists a sequence of self-similar coarse graining of a self-similarspace even for the quadratic dynamics known to be one of the simplest dy-namical system. Finally, it is noted that each step of the sequence canequally generate a topological space characteristic of condensed matter suchas dendrite [4].
An answer of the problem ”for what topological space, can we find a systemof weak contractions which makes the space self-similar?” is simply stated asfollows.
Proposition.
The existence of a self-similar space which is homeomorphic[5] to ( Y, τ ) is sufficient for a topological space ( Y, τ ) to be a metrizable spaceand self-similar with respect to the metric. Proof.
Let (
X, τ d ) be self-similar based on a system of weak contractions p j : ( X, τ d ) → ( X, τ d ), d ( p j ( x ) , p j ( x ′ )) ≤ α j ( η ) d ( x, x ′ ) for d ( x, x ′ ) < η, ≤ α j ( η ) < , j = 1 , . . . , m (2 ≤ m < ∞ ). That is, m [ j =1 p j ( X ) = X . Using ahomeomorphism h : ( X, τ d ) ≃ ( Y, τ ), we can define a metric ρ on Y as ρ ( y, y ′ ) = d ( h − ( y ) , h − ( y ′ )) , y, y ′ ∈ Y. The metric topology τ ρ is identical with the initial topology τ . From therelations 1) and 2) below, the metric space ( Y, τ ρ ) is confirmed to be self-similar by a system of weak contractions q j : ( Y, τ ρ ) → ( Y, τ ρ ) , j = 1 , . . . , m where q j is topologically conjugate to p j with the above homeomorphism h ,that is, q j = h ◦ p j ◦ h − .1) ρ ( q j ( y ) , q j ( y ′ )) = d ( h − ( q j ( y )) , h − ( q j ( y ′ )))= d ( p j ( h − ( y )) , p j ( h − ( y ′ ))) ≤ α j ( η ) d ( h − ( y ) , h − ( y ′ ))= α j ( η ) ρ ( y, y ′ ) for ρ ( y, y ′ ) < η .2) m [ j =1 q j ( Y ) = m [ j =1 q j ( h ( X )) = h ( m [ j =1 p j ( X )) = h ( X ) = Y. (cid:3) Existence of a self-similar decompositionspace
As an application of Proposition, we will show the existence of a self-similardecomposition space of a self-similar space.Let S be a self-similar, perfect [6], zero-dimensional (0-dim) [7], compactmetric space, and ( X, τ d ) be any compact metric space which is self-similar.Then, there exists a continuous map f from S onto X [8], and X is homeo-morphic to the decomposition space ( D f , τ ( D f )) of S with a homeomorphism h : ( X, τ d ) ≃ ( D f , τ ( D f )) , x f − ( x ) [9]. Here, D f = { f − ( x ) ⊂ S ; x ∈ X } and τ ( D f ) = {U ⊂ D f ; [ U (= [ D ∈U D ) is an open set of S } . The decom-position topology τ ( D f ) is identical with a metric topology τ ρ with a metric ρ ( y, y ′ ) = d ( h − ( y ) , h − ( y ′ )) , y, y ′ ∈ D f [10]. Since the metric space ( X, τ d )is assumed to be self-similar, from Proposition, the decomposition space( D f , τ ρ ) must be self-similar based on a system of weak contractions eachof which is topologically conjugate to each weak contraction which definesthe self-similarity of X . According to the self-similarity of the selected space X , the decomposition space D f of S can have various types of self-similarity.Now, let us consider a special case where the system of contructions defin-ing the self-similarity of the decomposition space D f of S is topologically re-lated to that defining the self-similarity of S . Let { S , · · · , S n } be a partitionof S [3] such that each S i is a clopen (closed and open) set of S . (Concerningthe existence of such partition of S , see Appendix.) Since the metric space S is perfect, 0-dim, compact, it is homeomorphic to the Cantor’s MiddleThird Set (abbreviated to CMTS) [11] as well as the space S . Therefore, S and S are homeomorphic. Let f : S → S be a not one to one , contin-uous, onto map. For example, the map f : S → S defined as f ( x ) = x for x ∈ S , f ( x ) ≡ q ∈ S for x ∈ S , · · · , f ( x ) ≡ q n ∈ S for x ∈ S n is acontinuous, onto map. It must be noted that D f is not trivial decompositionspace {{ x } ⊂ S ; x ∈ S } because the map f is not one to one [12]. Since thedecomposition space D f of S is homeomorphic to S [9], S must be home-omorphic to D f . Therefore, from Proposition, D f is self-similar based on asystem of weak contractions each of which is topologically conjugate to eachweak contraction which defines the self-similarity of S .Since the metric space D f is perfect, 0-dim and compact, the same sit-uation as for the initial space S can take place for the decomposition space3 f of S . Therefore, continuing this process endlessly, we obtain an infinitesequence of self-similar decomposition spaces or self-similar coarse grainingstarting from the self-similar space S , namely, a hierarchic structure of self-similar spaces as shown in Fig.1. In Fig. 1, the above mentioned decompo-sition space D f of S is denoted by D . D is self-similar due to a system ofweak contractions { f j = h ◦ f j ◦ ( h ) − : D → D ; j = 1 , · · · , m } . Here, { f j : S → S ; j = 1 , · · · , m } is a system of weak contractions which defines theself-similarity of S , and h is a homeomorphism from S to D . The decom-position space D of D is self-similar based on a system of weak contractions { f j = h ◦ f j ◦ ( h ) − : D → D ; j = 1 , · · · , m } where h is a homeomorphismfrom D to D . We can continue the procedure in this manner. Statement. [14, 15, 16]
Let ( Z, τ d ) be a compact metric space. If the sytem { f j : ( Z, τ d ) → ( Z, τ d ) , j = 1 , . . . , m } of weak contractions d ( f j ( z ) , f j ( z ′ )) ≤ α j ( η ) d ( z, z ′ ) for d ( z, z ′ ) < η, < α j ( η ) < , inf η> α j ( η ) > , j = 1 , · · · , m satisfies three conditionsi) Each f j is one to one,ii) The set S mj =1 { z ∈ Z ; f j ( z ) = z } is not a singleton,iii) P mj =1 inf η> α j ( η ) < ,then, there exists a perfect, 0-dim, compact S ( ⊂ Z ) such that S mj =1 f j ( S ) = S . Concludingly, we are convinced of the existence of a sequence as shownin Fig. 1 of self-similar coarse graining of a self-similar space based on theabove quadratic dynamics F µ ( x ) with a sufficiently large rate constant µ > S, D , D , · · · Since all of the metric spaces S, D , D , · · · in Fig. 1 are perfect, 0-dimand compact, there exist continuous maps [8], k from S onto the dendrite δ as a compact metric space, k from D onto δ , k from D onto δ, · · · ,respectively [17]. The decomposition spaces δ S = { k − ( x ) ⊂ S ; x ∈ δ } S due to f , δ D = { ( k ) − ( x ) ⊂ D ; x ∈ δ } of D due to k , δ D = { ( k ) − ( x ) ⊂ D ; x ∈ δ } of D due to k , · · · are homeomorphic to thedendrite δ , and therefore, δ S , δ D , δ D , · · · must have the dendritic structurein common (Fig. 3). For example, the self-similar space S generated froma quadratic dynamics F µ ( x ) = µx (1 − x ) with a sufficiently large µ > S . Appendix
Let S be a perfect, 0-dim T -space. Then, for any n , there exist n non-empty clopen (closed and open) sets S , · · · , S n of S such that S i ∩ S i ′ = φ for i = i ′ and n [ i =1 S i = S . For any n , there exist n non-empty clopen sets S i , · · · , S i n of S such that S i j ∩ S i j ′ = φ for j = j ′ and n [ j =1 S i j = S i . We cancontinue in this manner endlessly. proof ) To use the mathematical induction, let the statement hold for n − S is perfect, the open set S n − has at least two distinct points a and b . Since S is a T -space, there exists an open set u containing a such that b / ∈ u without loss of generality. Since S is 0-dim, there exists a clopenset v which contains the point a and is contained in the open set u ∩ S n − .Since b ∈ S n − − v , the clopen set S n − − v is not empty. Thus, we obtain adesired n -partition { S , · · · , S n − , v, S n − − v } of S . Concerning the subspace S i , it suffices to remember that any non-empty open set in a perfect space isperfect again. (cid:3) Acknowledgment
The authors are grateful to Professor H. Fukaishi of Kagawa Universityfor useful discussions.
References [1] A. Fern´ a ndez: J. Phys. A 21 (1988) L295.[2] S.K. Ma : Modern theory of critical phenomena (Benjamin, 1976) p.246.[3] Let (
A, τ ) be a topological space. A partition D of A is a set { φ = D ⊂ A } of nonempty subsets of A such that D ∩ D ′ = φ for D = D ′ , D, D ′ ∈ D [ D (= [ D ∈D D ) = A . A decomposition space ( D , τ ( D )) of ( A, τ ) is atopological space whose topology τ ( D ) on a partition D of A is definedby τ ( D ) = {U ⊂ D ; [ U (= [ D ∈U D ) ∈ τ } . The space ( D , τ ( D )) is a kindof quotient space of ( A, τ ). See, for the detailed discussions, S.B. NadlerJr.,
Continuum theory (Marcel Dekker, 1992) p.36.[4] A dendrite is a metric space which is locally connected, connected andcompact. The reference in [3], p.165.[5] Topological spaces (
E, τ ) and (
F, τ ′ ) are said to be homeomorphic pro-vided that there exists a continuous, one to one, open (or closed) mapfrom E onto F . If E and F are homeomorphic, all of the topologicalproperties in E ( F ) are preserved in F ( E ). See, for example, A. Kitada: Isoukuukan to sono ouyou (Akakura Shoten, 2007) p.24 [in Japanese].[6] A topological space (
A, τ ) is said to be perfect provided that any set { x } composed of single point x ∈ A is not an open set, that is, { x } / ∈ τ .[7] A topological space ( A, τ ) is said to be 0-dim provided that at any point x ∈ A , and for any open set U containing x , there exists a closed andopen set (so-called a clopen set) u containing x such that u ⊂ U . See,for the detailed discussions, W. Hurewicz and H. Wallman, DimensionTheory (Princeton University Press, Princeton, 1941) p.10.[8] The reference in [3], p.106 and p.109.[9] The reference in [3], p.44. The decomposition space D f of S can havevarious types of topological structure. For example, if X is a dendrite(the reference in [3], p.165), also D f must be dendrite.[10] To topologize the set D f , we use the metric ρ defined by means of thehomeomorphism h rather than the Vietoris topology (see, for example,A. Illanes and S.B. Nadler Jr., Hyperspaces (Marcel Dekker, 1999) p.9)which has been customarily employed for the topological discussions ofthe phenomena in the Chaos-Fractal sciences (see, for example, J. Banks:Chaos, Solitons & Fractals (2005) 681). Since X and D f are home-omorphic, the employment of the metric topology τ ρ which is identicalwith the decomposition topology τ ( D f ) seems to be quite natural.611] The reference in [3], p.109.[12] One of the simplest example of such decomposition D f is as follows.Let the self-similar, perfect, 0-dim, compact metric space S be CMTSitself and let the partition { S , S } of S be the set { CMTS ∩ [0 , / , CMTS ∩ [2 / , } . Then, the set { f − ( x ) ⊂ CMTS , x ∈ S } where f − ( x ) = { x } ⊂ X for x ∈ S − { q } and f − ( q ) = { q } ∪ S , is a de-composition D f of CMTS.[13] R.L. Devaney: An introduction to chaotic dynamical systems (WestviewPress, 2003) 2nd ed., p.35.[14] A. Kitada and Y. Ogasawara: Chaos, Solitons & Fractals (2005) 785;A. Kitada and Y. Ogasawara: Chaos, Solitons & Fractals (2005) 1273.[15] A. Kitada, T. Konishi and T. Watanabe: Chaos, Solitons & Fractals 13(2002) 363.[16] S. Nakamura, T. Konishi and A. Kitada, J. Phys. Soc. Jpn. 64 (1995)731.[17] It is noted that continuous maps k, k , k , · · · must not be one to one.In fact, if they are one to one, they must be homeomorphisms between0-dim (i.e., totally disconnected) spaces S, D , D , · · · and a connetctedspace δ . It is impossible. 7 Figure 1: A hierarchic structure of self-similar spaces. h i , i = 1 , , · · · arehomeomorphisms. f j , f j , f j , · · · are weak contractions such that m [ j =1 f j ( S ) = S, m [ j =1 f j ( D ) = D , m [ j =1 f j ( D ) = D , · · · , respectively.8 f f a a ) b ) Figure 2: a) F µ ( x ) = µx (1 − x ) , µ > , x ∈ [0 , F µ ( x ) defines a system of contractions { f j : [0 , → [0 , , j = 1 , } which satisfies three conditions i), ii), iii) in Statement in the text. In fact, [ j =1 , { x ∈ [0 , f j ( x ) = x } = { , a } . 9 k kk dd dendrite (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) Figure 3: Generation of dendrites from each step of the sequence S, D , D , · · · . δ, δ S , δ D , δ D , · · ·· · ·