In this paper, we generalize the Cantor function to 2-dimensional cubes and construct a cyclic 2-cocycle on the Cantor dust. This cocycle is non-trivial on the pullback of the smooth functions on the 2-dimensional torus with the generalized Cantor function while it vanishes on the Lipschitz functions on the Cantor dust. The cocycle is calculated through the integration of 2-forms on the torus by using a combinatorial Fredholm module

Maruyama, Takashi

Seto, Tatsuki

English

Okayama University Scientific Achievement Repository

Math. J. Okayama Univ. 66 (2024), 115–124A COMBINATORIAL INTEGRATION ON THE CANTORDUSTTakashi MARUYAMA and Tatsuki SETOAbstract. In this paper, we generalize the Cantor function to 2-dimensionalcubes and construct a cyclic 2-cocycle on the Cantor dust. This co-cycle is non-trivial on the pullback of the smooth functions on the 2-dimensional torus with the generalized Cantor function while it vanisheson the Lipschitz functions on the Cantor dust. The cocycle is calculatedthrough the integration of 2-forms on the torus by using a combinatorialFredholm module.IntroductionCyclic cohomology for algebras [2] is a fundamental tool to study non-commutative geometry. One of its application is the study of fractal setsto which powerful tools such as de Rham homology (on smooth manifolds)cannot be applied. K-homology, one of other homotopy invariant theories,of fractal sets is also studied extensively. For some class of fractal sets suchas the Cantor set and the Cantor dust, the K0 homology groups are isomor-phic to∞∏Z, which is known as the Baer-Specker group [1]. One feature ofthe Baer-Specker group is that the group does not admit basis. This featuremakes the study of K0 groups for such fractal sets somewhat intractable be-cause local topological features of the spaces cannot induce local algebraicstructures. Cyclic cohomology on the one hand is expected to be favor-able in these cases because it can be characterized as a “linearization” ofK-homology through the Chern character.There is a study [4] by H. Moriyoshi and T. Natsume which presenteda variant of the Riemann-Stieltjes integration on the middle third Cantorset. Let CS =∞⋂n=0In be the middle third Cantor set, where I0 = [0, 1],I1 =[0,13], I2 =[0,132]∪[232,332]∪[632,732]∪[832, 1], . . . . We also denoteby (Hn, Fn) the Fredholm module on In which is defined by the direct sum ofConnes’ Fredholm module on intervals. In [4], a new class of algebra is intro-duced: an algebra P = c∗BV (S1) defined by the pull-back of the boundedvariation class on a unit circle BV (S1) with the canonical Cantor function c.Mathematics Subject Classification. Primary 46L87; Secondary 28A80.Key words and phrases. Fredholm module, Cantor dust, cyclic cocycle.115116 T. MARUYAMA AND T. SETOThen they define a functional ψ(f, g) = limn→∞ψn(f, g) = limn→∞Tr(ϵf [Fn, g]Fn)on P. The existence of the limit is proved by showing that the cocycle isreduced to the Riemann-Stieltjes integration. A key ingredient for the proofis the Cantor function (and its intermediate functions that converge into theCantor function); the function is used to map the iterated function systemof the Cantor set onto 2n-fold subdivisions of the unit interval on whichthe Riemann-Stieltjes integration is defined. Because the Cantor functionremains surjective onto the unit interval when the function is restricted tothe Cantor set, the iterated function system for the Cantor set may be seenas a variant of subdivisions on I.In the present paper, by following the idea mentioned above, we constructa new cyclic 2-cocycle on the Cantor dust CD and show that the cocycleis not trivial. In order to construct the cocycle, we define a sequence offunctionals ϕn (see Definition 1), that is a generalization of a functional ψnon CS, by using a combinatorial Fredholm module on squares defined by theauthors [3] and the product of the Cantor function. The function inducesthe map c between the Cantor dust and 2-torus T2:Theorem (see Theorem 3.3). Let C∞(T2) be the smooth functions on thetorus T2. For f = c∗(f̃), g = c∗(g̃), h = c∗(h̃) ∈ c∗C∞(T2), we havelimn→∞ϕn(f, g, h) = 2∫T2f̃dg̃ ∧ dh̃.Moreover, the limit only depends on f, g, h ∈ c∗C∞(T2). Therefore, thefunctional ϕ defined by the limit is a cyclic 2-cocycle on c∗C∞(T2).We can take the value ϕn(f, g, h) for Lipschitz functions f, g, h ∈ CLip(CD)by the definition of ϕn. However, we have limn→∞ϕn(f, g, h) = 0 for anyf, g, h ∈ CLip(CD) (see Proposition 3.2). Thus the algebra c∗C∞(T2) is fardifferent from CLip(CD).All the main results to be shown also hold for the Sierpinski carpet. T2can be also replaced with a 2-dimensional sphere. In order to generalize ourmain theorem to general dimension, we need a general method to prove theexistence of a cyclic cocycle. We will leave the study for future work.1. A combinatorial Fredholm module on squaresIn this section, we review a combinatorial Fredholm module on squaresconstructed by the authors [3]. Let γ ⊂ R2 be a square of dimension 2 andV = {v0, v1, v2, v3} the set of vertices of γ; see the following figure for thenumbering of the vertices.Set V0 = {v0, v2} and V1 = {v1, v3}, so we have V = V0 ∪ V1. SetH+ = ℓ2(V0) = ℓ2(v0)⊕ ℓ2(v2), H− = ℓ2(V1) = ℓ2(v1)⊕ ℓ2(v3)COMBINATORIAL INTEGRATION 117v3 v2v0 v1Figure 1. Numbering of the verticesand H = H+ ⊕H−. The vector space H(∼= C4) is a Hilbert space of dimen-sion 4 with an inner product⟨f, g⟩ =3∑i=0f(vi)g(vi).We assume that H is Z2-graded with the grading ϵ = ±1 on H±, respec-tively. The C∗-algebra C(V ) of continuous functions on V acts on H bymultiplication:ρ(f) = (f(v0)⊕ f(v2))⊕ (f(v1)⊕ f(v3)).Set U =1√2[1 −11 1]and F =[U∗U]=1√21 1−1 11 −11 1. ThenF is a bounded operator on H and we have Fϵ+ ϵF = O. So (H, F ) is theFredholm module on C(V ).Set I = [0, 1] × [0, 1] and let fs : I → I (s = 1, . . . , N) be similitudes.Denote byrs =∥fs(x)− fs(y)∥Rn∥x− y∥Rn(< 1) (x ̸= y)the similarity ratio of fs. An iterated function system (IFS) (I, S = {1, . . . , N}, {fs}s∈S)defines the unique non-empty compact setK = K(γn, S = {1, . . . , N}, {fs}s∈S)called the self-similar set such that K =⋃Ns=1 fs(K). Denote by dimS(K)the similarity dimension of K, that is, the number s that satisfiesN∑s=1rss = 1.If an IFS (I, S, {fs}s∈S) satisfies the open set condition, we have dimH(K) =dimS(K), where we denote by dimH(K) the Hausdorff dimension of K.Set fs = fs1 ◦ · · · ◦ fsj for s = (s1, . . . , sj) ∈ S∞ =⋃∞j=0 S×j and f∅ = id.For the simplicity, we will denote by i the vertex fs(vi) of a square fs(I).We also denote by Vs the vertices of a square fs(I). Denote by es thelength of edge of fs(I), which equals∏jk=1 rsk . As introduced above, we set118 T. MARUYAMA AND T. SETOthe Hilbert space Hs = ℓ2(Vs) on fs(I) of the length es, which splits thepositive part H+s and the negative part H−s . Taking direct sum on squares,we defineHn =n⊕j=1⊕s∈S×jHs and Fn =n⊕j=1⊕s∈S×jF.The C∗-algebra C(K) acts on Hn by multiplication:ρn(f) =n⊕j=1⊕s∈S×jρ(f).2. Review on the Cantor dustWe introduce an IFS of the Cantor dust and 2-dimensional analogue ofthe Cantor function defined on a 1-dimensional interval. The IFS of theCantor dust is our main interest in the paper. The analogue of the Cantorfunction plays a crucial role in construction of cyclic cocycle in Section 3.Let I = [0, 1] × [0, 1]. The IFS of the Cantor dust is defined as a set offunctions {fs : I → R2}s=1,2,3,4 described as follows:f1(x) =13x, f2(x) =13x+13[02], f3(x) =13x+13[20], f4(x) =13x+13[22].This IFS induces a unique non-empty compact set CD in I such that CD =4⋃i=1fs(CD) holds; CD is called the Cantor dust.Let c0 = x : [0, 1] → R. We define in an inductive manner a sequenceof R-valued continuous functions defined on the unit interval {cn}n∈N asfollows:cn+1(x) =12cn(3x), 0 ≤ x ≤13 ,12 ,13 ≤ x ≤23 ,12cn(3x− 2) +12 ,23 ≤ x ≤ 1.The limit of {cn}n∈N exists and is called the Cantor function. We thengeneralize the construction to 2-dimensional case by taking the pair of thetwo identical sequences:cn = (cn, cn) : I → R2.The limit of the sequence {cn} also exists and equals the product of the Can-tor functions. We call the limit limn→∞cn 2-dimensional Cantor dust function.We note that the construction of the 2-dimensional Cantor dust functioncan be generalized to higher dimensional case.COMBINATORIAL INTEGRATION 1193. A combinatorial integration3.1. Approximating combinatorial integration. We apply the construc-tion of the combinatorial Fredholm module to the IFS of the Cantor dust(I, S = {1, 2, 3, 4}, {fs}s∈S) defined in Section 2. In order to construct ourcyclic cocycle, we need the multiplication operator ρ(f) and the commu-tator [F, f ] on Hs for f ∈ C(CD). Since ρ(f) is even, we can expressρ(f) =[f+f−]. We can write down the commutator [F, f ] as follows:[F, f ] =1√2− (f(0)− f(1)) − (f(0)− f(3))f(2)− f(1) − (f(2)− f(3))f(0)− f(1) − (f(2)− f(1))f(0)− f(3) f(2)− f(3) .Here, for the simplicity, we denote i = vi the vertices on the squares Vs.We denote the upper right 2× 2 block of [F, f ] by d−f and the lower left byd+f .We now construct a sequence of operators on⊕s∈S×nHs that will give riseto a cyclic cocycle. For f, g, h ∈ C(K), we havef [F, g][F, h] =[f+f−] [d−gd+g] [d−hd+h]=[f+d−gf−d+g] [d−hd+h]=[f+d−gd+hf−d+gd−h].By setting fi,j = f(j) − f(i), the two diagonal 2 × 2 blocks of f [F, g][F, h]can be expressed asf+d−gd+h =12[f(0)f(2)] [g0,1 g0,3−g2,1 g2,3] [−h0,1 h2,1−h0,3 −h2,3]=12[f(0)g0,1 f(0)g0,3−f(2)g2,1 f(2)g2,3] [−h0,1 h2,1−h0,3 −h2,3]=12[−f(0)(g0,1h0,1 + g0,3h0,3) f(0)(g0,1h2,1 − g0,3h2,3)f(2)(g2,1h0,1 − g2,3h0,3) −f(2)(g2,1h2,1 + g2,3h2,3)]=12[f(0)(g0,1h1,0 + g0,3h3,0) −f(0)(g0,1h1,2 − g0,3h3,2)f(2)(g2,3h3,0 − g2,1h1,0) f(2)(g2,1h1,2 + g2,3h3,2)]andf−d+gd−h =12[f(1)f(3)] [−g0,1 g2,1−g0,3 −g2,3] [h0,1 h0,3−h2,1 h2,3]=12[−f(1)g0,1 f(1)g2,1−f(3)g0,3 −f(3)g2,3] [h0,1 h0,3−h2,1 h2,3]120 T. MARUYAMA AND T. SETO=12[f(1)(−g0,1h0,1 − g2,1h2,1) f(1)(−g0,1h0,3 + g2,1h2,3)f(3)(−g0,3h0,1 + g2,3h2,1) f(3)(−g0,3h0,3 − g2,3h2,3)]=12[f(1)(g1,0h0,1 + g1,2h2,1) f(1)(g1,0h0,3 − g1,2h2,3)−f(3)(g3,2h2,1 − g3,0h0,1) f(3)(g3,0h0,3 + g3,2h2,3)].Here, we defineM = − 2(es)2[F, x][F, y] =1−11−1 .M can be expressed as N ⊕ N by denoting the off-diagonal 2 × 2 matri-ces by N . The following lemma indicates that the trace of an operatorf [F, g][F, h]M gives rise to a discretized version of integration on a square.Lemma 3.1. For any n ∈ N and s ∈ S×n, we have2 · Tr(f [F, g][F, h]M) =f(0)(g0,1h1,2 − g0,3h3,2) + f(2)(g2,3h3,0 − g2,1h1,0)− f(1)(g1,0h0,3 − g1,2h2,3)− f(3)(g3,2h2,1 − g3,0h0,1).Proof. The proof follows straightforward calculation. By multiplying Mwith f [F, g][F, h], we havef [F, g][F, h]M =[f+d−gd+h−f−d+gd−h] [NN]=[f+d−gd+hNf−d+gd−h N].Then we havef+d−gd+hN =12[f(0)(g0,1h1,0 + g0,3h3,0) −f(0)(g0,1h1,2 − g0,3h3,2)f(2)(g2,3h3,0 − g2,1h1,0) f(2)(g2,1h1,2 + g2,3h3,2)] [1−1]=12[f(0)(g0,1h1,2 − g0,3h3,2) f(0)(g0,1h1,0 + g0,3h3,0)−f(2)(g2,1h1,2 + g2,3h3,2) f(2)(g2,3h3,0 − g2,1h1,0)]andf−d+gd−h N =12[f(1)(g1,0h0,1 + g1,2h2,1) f(1)(g1,0h0,3 − g1,2h2,3)−f(3)(g3,2h2,1 − g3,0h0,1) f(3)(g3,0h0,3 + g3,2h2,3)] [1−1]=12[−f(1)(g1,0h0,3 − g1,2h2,3) f(1)(g1,0h0,1 + g1,2h2,1)−f(3)(g3,0h0,3 + g3,2h2,3) −f(3)(g3,2h2,1 − g3,0h0,1)].Therefore, the trace of f [F, g][F, h]M is given by the lemma. □When g = x and h = y are the coordinate functions of R2, respectively,each term of the right hand side of Lemma 3.1 is nothing but the summandof the Riemannian sum of f . So the sum of the trace in the left hand sidecan be regard as an analogue of the Riemannian sum on the Cantor dust.COMBINATORIAL INTEGRATION 121Definition 1. For any n ∈ N and f ⊗ g ⊗ h ∈ C(CD)⊗ C(CD)⊗ C(CD),define a mapϕn(f, g, h) =∑s∈S×nTr(f [F, g][F, h]M).We call the sequence of the maps {ϕn} the approximating combinatorialintegration.Proposition 3.2. For any functions f ∈ C(CD) and g, h ∈ CLip(CD), thesequence {ϕn(f, g, h)} converges to 0.Proof. By Lemma 3.1, we have∣∣ϕn(f, g, h)∣∣ ≤ ∑s∈S×n(∣∣f(0)(g0,1h1,2 − g0,3h3,2)∣∣+ ∣∣f(2)(g2,3h3,0 − g2,1h1,0)∣∣+∣∣f(1)(g1,0h0,3 − g1,2h2,3)∣∣+ ∣∣f(3)(g3,2h2,1 − g3,0h0,1)∣∣)≤ 2Lip(g)Lip(h) 19n∑s∈S×n(|f(0)|+ |f(1)|+ |f(2)|+ |f(3)|)≤ 8∥f∥Lip(g)Lip(h)4n9nfor any n ∈ N, f ∈ C(CD), and g, h ∈ CLip(CD). Here, ∥f∥ means thesupnorm of f and Lip(g) means the Lipschitz constant of g ∈ CLip(CD).Therefore, we have ϕn(f, g, h) → 0 (n→ ∞). □By Proposition 3.2, our approximating combinatorial integration does notrestore rich structure on the Lipschitz functions. So we need anothor classof functions on the Cantor dust.3.2. Non-triviality of a combinatorial integration. Let CDn =⋃s∈S×n fs(I)be a space obtained by applying the n-times composition of4⋃s=1fs to I. CDnhas 4n distinct squares {γni }1≤i≤4n with length of edge13n. For every n ∈ N,the restriction of cn to CDn is still surjective. The image of {γni }1≤i≤4n bycn is then a subdivision of I each of whose cell is a square with length12n.Therefore, the maximum length across all the cells tends to 0 as n→ ∞.We introduce an equivalence relation ∼ on ∂I: (a, 0) ∼ (a, 1) and (0, b) ∼(1, b). The quotient space turns out to be the torus T2 = I/ ∼. The map cninduces a surjective map CDn → T2. We denote the map by the same lettercn. Similarly, we can also construct a continuous function c : CD → T2 bythe restriction of the 2-dimensional Cantor dust function to CD.122 T. MARUYAMA AND T. SETOThe following theorem shows that the limit of approximating combina-torial integration with nontrivial value exists for some class of function onCD.Theorem 3.3. Let C∞(T2) be the smooth functions on the torus T2. Forf = c∗(f̃), g = c∗(g̃), h = c∗(h̃) ∈ c∗C∞(T2), we havelimn→∞ϕn(f, g, h) = 2∫T2f̃dg̃ ∧ dh̃.Moreover, the limit only depends on f, g, h ∈ c∗C∞(T2). Therefore, thefunctional ϕ(f, g, h) = limn→∞ϕn(f, g, h) is a cyclic 2-cocycle on c∗C∞(T2).Proof. Let⊞n be the set of equilateral cells that consist of 2n-fold subdivisionof I. By the first-order approximation, for g̃, h̃ ∈ C∞(T2) we haveg̃0,1 = g̃(1)− g̃(0) = 2−ng̃x(0) + o(2−n) (n→ ∞),h̃1,2 = h̃(2)− h̃(1) = 2−nh̃y(0) + o(2−n) (n→ ∞),g̃0,3 = g̃(3)− g̃(0) = 2−ng̃y(0) + o(2−n) (n→ ∞),h̃3,2 = h̃(2)− h̃(3) = 2−nh̃x(0) + o(2−n) (n→ ∞).Apply the above equations to f̃(0)(g̃0,1h̃1,2− g̃0,3h̃3,2) in Lemma 3.1, and wegetf̃(0)(g̃0,1h̃1,2−g̃0,3h̃3,2) = 4−nf̃(0)(g̃x(0)h̃y(0)−g̃y(0)h̃x(0))+o(4−n) (n→ ∞).Therefore, we havelimn→∞∑□∈⊞nf̃(0)(g̃0,1h̃1,2 − g̃0,3h̃3,2)= limn→∞4−n∑□∈⊞nf̃(0)(g̃x(0)h̃y(0)− g̃y(0)h̃x(0)) + 4no(4−n)=∫T2f̃(g̃xh̃y − g̃yh̃x)dx ∧ dy=∫T2f̃dg̃ ∧ dh̃.Similary, we obtainlimn→∞∑□∈⊞nf̃(2)(g̃2,3h̃3,0 − g̃2,1h̃1,0) = − limn→∞∑□∈⊞nf̃(1)(g̃1,0h̃0,3 − g̃1,2h̃2,3)= − limn→∞∑□∈⊞nf̃(3)(g̃3,2h̃2,1 − g̃3,0h̃0,1) =∫T2f̃dg̃ ∧ dh̃.COMBINATORIAL INTEGRATION 123Hence, by Lemma 3.1, we getlimn→∞ϕn(f, g, h) =12limn→∞∑s∈S×nTr(f̃ ◦ c[F, g̃ ◦ c] [F, h̃ ◦ c]M)=12limn→∞∑□∈⊞nTr(f̃ [F, g̃][F, h̃]M)=12limn→∞∑□∈⊞n(f̃(0)(g̃0,1h̃1,2 − g̃0,3h̃3,2)+ f̃(2)(g̃2,3h̃3,0 − g̃2,1h̃1,0)− f̃(1)(g̃1,0h̃0,3 − g̃1,2h̃2,3)−f̃(3)(g̃3,2h̃2,1 − g̃3,0h̃0,1))= 2∫T2f̃dg̃ ∧ dh̃.Assume that f = c∗(f̃) = c∗(f̃ ′), g = c∗(g̃) = c∗(g̃′) and h = c∗(h̃) =c∗(h̃′). In general, if c∗(p̃) = c∗(q̃) for p̃, q̃ ∈ C(T2), then c∗n(p̃) = c∗n(q̃) onthe boundary of CDn for any n ∈ N. Therefore, for every n ∈ N, we have∑s∈S×nTr(f̃ ◦ c[F, g̃ ◦ c] [F, h̃ ◦ c]M)=∑s∈S×nTr(f̃ ′ ◦ c[F, g̃′ ◦ c] [F, h̃′ ◦ c]M)Thus as n→ ∞, the value ϕ(f, g, h) depends only on f, g, h ∈ c∗C∞(T2). □Corollary 3.4. Given [p] ∈ K0(c∗(C∞(T2))) written as p = e ◦ c : CD →MN (C), the Connes’ pairing of the cyclic 2-cocycle [ϕ] with [p] is expressedas follows:⟨[ϕ], [p]⟩ = 1πi∫T2Tr(e(de)2).AcknowledgementSeto was supported by JSPS KAKENHI Grant Number 21K13795.References[1] R. Baer. Abelian groups without elements of finite order. Duke Math. J., 3(1):68–122,1937.[2] A. Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ.Math., 62:257–360, 1985.[3] T. Maruyama and T. Seto. A combinatorial fredholm module on self-similar sets builton n-cubes, 2019. arXiv:1912.05832 (to appear in J. Fractal Geom.).[4] H. Moriyoshi. On a cyclic volume cocycle in fractal geometry. talk on “Further devel-opement of Atiyah-Singer Index Theory”, 2013 (based on joint work with T. Natsume).124 T. MARUYAMA AND T. SETODepartment of EngineeringStanford University353 Jane Stanford Way Stanford CA 94305, USAe-mail address: takashi279@cs.stanford.edu, 49takashi@gmail.comGeneral Education and Research CenterMeiji Pharmaceutical University2-522-1 Noshio, Kiyose-shi, Tokyo, 204-8588 Japane-mail address: tatsukis@my-pharm.ac.jp(Received December 7, 2022 )(Accepted May 13. 2023 )

A combinatorial integration on the Cantor dust

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A combinatorial integration on the Cantor dust

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