CCyclic Cohomology Groups of Some Self-similar Sets
Takashi Maruyama
Abstract
We define a variant of the Young integration on some kinds of self-similar sets whichare called cellular self-similar sets . This variant is an analogue of the Young integrationdefined on the unit interval. We give the criteria of the variant on cellular self-similarsets, and also show that the variant is a cyclic 1-cocycle of the algebra of complex-valued α -H¨older continuous functions on the cellular self-similar sets. This suggests that thecocycle is a variant of currents. Fractal sets introduced by Mandelbrot [16] are complex-behaved spaces difficult to analyse.For instance, for the Cantor sets, their Hausdorff dimensions are different, although all ofthem are topologically isomorphic. Since the Hausdorff dimension is an invariant of fractalsets which is stable under bi-Lipschitz transformations, it is hard to say that the (co)homologytheories which are homotopy invariant can capture deeper topological quantities of fractalsets.Connes introduced cyclic cohomology theory [4], which turns out to be a generalisationof the de Rham homology theory. He proposed Quantised calculus in [5] and exploits theDixmier trace as a non-smooth analogue of the integration on manifolds. Namely, he appliedit to the Cantor sets and succeeded to recover their Minkowski contents as the value of acertain Dixmier trace. This result suggests that cyclic cohomology theory could be one ofsuitable methods to study fractal sets.Another approach to analyse some fractal sets was proposed by Moriyoshi and Natsume[18]. For the Sierpinski gasket SG , they exploit the algebra C ( SG ) of Lipschitz functionson SG to construct a cyclic 1-cocycle φ of C ( SG ) . They also show that, when Lipschitzfunctions are seen as 1-H¨older continuous functions, the regularity α = C ( SG ) for thewell-definedness of φ can be reduced to the half of the Hausdorff dimension dim H ( SG ) / φ , Moriyoshi and Natsume exploit the Youngintegration on the unit interval I . It has following properties, which explain a reason whyH¨older continuous functions are used and the Hausdorff dimension arises in the results of[18]: the Young integration on the unit interval was developed in [26]. This is a bilinearfunction from the product W α × W β of the Wiener classes such that α + β > = dim H ( I ) tocomplex numbers: Y : W α × W β → C . a r X i v : . [ m a t h . M G ] D ec specially, if we restrict the domain of Y to the algebra C α ( I ) of α -H¨older continuous func-tions on I , the map Y is well-defined for 2 α > = dim H ( I ) . We note that C α ( I ) contains thealgebra C ∞ ( I ) of smooth functions on I , and then, for f , g ∈ C ∞ ( I ) we further get Y ( f , g ) = (cid:90) I f dg . In this sense, the Young integration may be considered as a generalisation of the integrationof differential 1-forms, and does make it possible to integrate some non-smooth functions.Moreover, for a Jordan curve C composed of a finite number of unit intervals, the Youngintegration Y along C turns out to be a cyclic 1-cocycle if 2 α > = dim H ( C ) : Y : C α ( C ) × C α ( C ) → C . All those results above motivate us to extend the cyclic 1-cocycle φ of C ( SG ) onto a certainclass of fractal sets by exploiting the Young integration.In this paper, we extend the cocycle of the Sierpinski gasket defined in [18] to a certainclass of self-similar sets by exploiting the Young integration, and show that the cocycles canbe applied to some examples. More detailed and precise statements are given as follows.We first define cellular self-similar sets, the preliminary notions of which are given inSection 2.2 below. Cellular self-similar sets K | X | are self-similar sets that are based on linearcell complexes, and the unit interval is a prototype of cellular self-similar sets. The precisedefinition is as follows: Definition 1.1 (Definition 3.1) . Let | X | be a 2-dimensional finite convex linear cell complexand { F j } j ∈ S a set of similitudes F j : | X | → | X | indexed by a finite set S . We also let | X | = (cid:83) j ∈ S F j ( | X | ) . The triple ( | X | , S , { F j } j ∈ S ) is called a cellular self-similar structure if it satisfiesa) ∂ | X | ⊂ ∂ | X | , andb) int F i ( | X | ) ∩ int F j ( | X | ) = /0, for all i (cid:54) = j ∈ S .Let ( | X | , S , { F j } j ∈ S ) be a cellular self-similar structure. Then ( | X | , S , { F j } j ∈ S ) yields asequence {| X n |} n ∈ N of 2-dimensional cell complexes, and, by Theorem 2.5 below, the se-quence gives rise to the cellular self-similar set K | X | with respect to ( | X | , S , { F j } j ∈ S ) . Forevery n ∈ N , | X n | is subdivided into a simplicial complex | X sn | by Lemma 1 of Chapter 1 in[27]. From this simplicial complex, we get 1-chains b n , I n , o n and I n \ I n − ∈ ˜ S ( X sn ; C ) whosegeometric incarnations are subspaces of 1-skelton of | X sn | ; see Section 3.1 for the details.On the other hand, the algebra C α ( | X sn | ) of complex-valued α -H¨older continuous func-tions defined on | X sn | is a subspace of the function space F ( | X sn | ; C ) = { f : | X sn | → C } asa C -vector space. Therefore, C α ( | X sn | ) can generate a C -vector space C α , ( | X sn | ) with thedifferential and cup product of the Alexander-Spanier cochain complex [21], which is a sub-space of Hom C ( ˜ S ( | X sn | ) , C ) ; see Section 3.2 for the details. For f and g ∈ C α ( K | X | ) we havea cochain f (cid:94) δ ( g ) − g (cid:94) δ ( f ) ∈ C α , ( | X sn | ) for any n ∈ N , which is denoted by ω n ( f , g ) .Finally we set φ n ( f , g ) as ω n ( f , g )( I n ) and call the sequence { φ n ( f , g ) } n ∈ N the cyclic quasi-1-cocycle of f and g , the definition of which is given in Section 3.2.The first main theorem states that if 2 α > dim H ( K | X | ) , we can define a bilinear map φ : C α ( K | X | ) × C α ( K | X | ) → C by taking the limit of the cyclic quasi-1-cocycle { φ n ( f , g ) } n ∈ N .2his implies that the bilinear map φ may be seen as a generalisation of the classical Youngintegration of the unit interval. Theorem 1.2 (Theorem 3.8, Existence theorem) . Let ( | X | , S , { F j } j ∈ S ) be a cellular self-similar structure with S ≥ and K | X | the cellular self-similar set with respect to ( | X | , S , { F j } j ∈ S ) .We also let C α ( K | X | ) be the algebra of α -H¨older continuous functions on K | X | . If α > dim H ( K | X | ) , then the cyclic quasi-1-cocycle { φ n ( f , g ) } is a Cauchy sequence for any f ,g ∈ C α ( K | X | ) . The map φ is originally defined by Moriyoshi and Natsume [18] for the algebra C Lip ( SG ) of complex-valued Lipschitz functions on the Sierpinski gasket SG , and the construction isbased on the classical Young integration on the unit interval. The authors use the simplexes I n to prove the existence of the cyclic cocycle of the Sierpinski gasket. An obstacle to extendthe construction to cellular self-similar sets is that, for each n ∈ N , the lengths of 1-simplicesbelonging in | X n | are not equal. The key technical ingredient to overcome the difficulty is theexistence of 2-dimensional simplicial complex | K sn , n + | whose boundary is a disjoint union of ∂ ( | X n | ) and ∂ ( | X n + | ) . By properties of cellular self-similar sets, we can prove that lengthsof 1-simplices of | K n , n + | have an upper bound which tends to 0 as n → ∞ . This propertyplays a crucial role to prove that { φ n ( f , g ) } n ∈ N is a Cauchy sequence.The proof of the above theorem immediately yields the following corollary. This provesthat the bilinear map φ is a non-commutative representation of the Young integration. Corollary 1.3 (Corollary 3.10) . For any f , g ∈ C α ( K | X | ) with α > dim ( K | X | ) , we have φ ( f , g ) = (cid:90) Young ∂ | X | f dg = · ( Young integration of f and g along ∂ | X | ) . In particular, if | X | (cid:54) = | X | , for and x : = id ∈ C α ( K | X | ) , we get φ ( , x ) = (cid:90) Young ∂ | X | dx = · ( length of ∂ | X | ) . After we define φ of the cellular self-similar set K | X | , we prove that φ is a cyclic 1-cocycle of C α ( K | X | ) and represents a nontrivial element in the first cyclic cohomology group HC ( C α ( K | X | )) . This theorem shows that φ may be seen as a non-commutative generalisationof the integration on manifolds. Theorem 1.4 (Theorem 3.11) . Under the assumption of the existence theorem : a) The bilinear map φ is a cyclic -cocycle of C α ( K | X | ) .b) If | X | (cid:54) = | X | , the cocycle φ represents a non-trivial element [ φ ] in HC ( C α ( K | X | )) . For the proof of the first statement, we need to use the Leibniz rule of the cup productdefined on the Alexander-Spanier cochain complex. Corollary 1.3 immediately completes theproof of the second statement since 1 ⊗ x represents an element in the Hochschild homologygroup.By Theorem 1.4, we find that the cocycle φ has the following additional properties: φ candetect the Hausdorff dimensions of cellular self-similar sets and distinguish them by their di-mensions. For instance, we get the cocycles φ of the Sierpinski gasket SG and the Sierpinski3arpet SC , whose thresholds of the well-definedness are different. Namely, their thresholdsare dim H ( SG ) = log H ( SC ) = log
8. Since bi-Lipschitz transformations preservethe Hausdorff dimension, the cocycles can prove that SG and SC are not bi-Lipschitz home-omorphic. Moreover, if we have a bi-Lipschitz transformation between cellular self-similarsets K | X | and K | X (cid:48) | , the algebra C α ( K | X | ) is isomorphic to C α ( K | X (cid:48) | ) . Therefore, we further getthe following property: the cocycle φ is invariant under bi-Lipschitz transformations. Afterthe proof of the main results, we apply the results to some examples, and extend the cocycleto some variants of cellular self-similar sets. Conventions
We assume that algebras have unit unless otherwise stated, and all base rings of algebrasis the field C of complex numbers. The Euclidean space R n is endowed with the standardEuclidean metric. • ⊗ = ⊗ C . • Z ≥ = N ∪ { } . • ∂ X : the boundary of a topological space X . • C α ( X ) : the algebra of complex-valued α -H¨older continuous functions on a metricspace X whose sum and multiplication are given by the pointwise sum and multiplica-tion. • C Lip ( X ) : the algebra of complex-valued Lipschitz functions on a metric space X whosesum and multiplication are given by the pointwise sum and multiplication. Acknowledgements
The author thanks his advisor Hitoshi Moriyoshi for his constant advice and encouragements.The author also thanks Tatsuki Seto for many discussions.
In this section, we recall the key materials for the main theorems, the Young integral, self-similar sets.
We begin with a quick review of the Young integral basically following [26] except the slightchanges of the notation.Let I be the unit interval [ , ] and f , g complex-valued functions defined on I . We makea subdivision χ of I = x < x < · · · < x n − < x n = F ( χ ) = n ∑ i = f ( x i )( g ( x i ) − g ( x i − )) . Then F ( χ ) can be also written as F ( χ ) = ∑ < i ≤ j ≤ n δ ( f )( x i − , x i ) · δ ( g )( x j − , x j ) + f ( )( g ( ) − g ( )) . Here δ ( f )( x i − , x i ) denotes f ( x i ) − f ( x i − ) , and this notation is similar to the differential ofthe Alexander-Spanier cohomology theory; see also Chapter 6.4 in [21]. We also let α , β > S α , β [ , ] = S α , β [ , f , g ] the upper bound of (cid:18) ∑ i | δ ( f )( x i − , x i ) | α (cid:19) α (cid:18) ∑ i | δ ( g )( x i − , x i ) | β (cid:19) β for every subdivision of I . Following lemmas of [26], if α + β > ξ ∈ [ , ] is a divisionpoint of χ , we have (cid:12)(cid:12)(cid:12) F ( χ ) − f ( ξ )( g ( ) − g ( )) (cid:12)(cid:12)(cid:12) ≤ ( + ζ ( α + β )) · S α , β [ , ] , where ζ ( α + β ) denotes the zeta function of α + β .This inequality yields to a more general inequality for the sum associated to χ : for thegiven subdivision χ , let a point x i − ≤ ξ i ≤ x i for each i , and applying this inequality for eachinterval [ x i − , x i ] and summing up, we get (cid:12)(cid:12)(cid:12)(cid:12) F ( χ ) − n ∑ i = f ( ξ i )( g ( x i ) − g ( x i − )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ { + ζ ( α + β ) } · n ∑ i = S α , β [ x i − , x i ; f , g ] . Moreover if we have another subdivision χ (cid:48) of I and subdivision points x j − ≤ ξ (cid:48) j ≤ x j , then (cid:12)(cid:12)(cid:12)(cid:12) n ∑ i = f ( ξ i )( g ( x i ) − g ( x i − )) − m ∑ j = f ( ξ (cid:48) j )( g ( x (cid:48) j ) − g ( x (cid:48) j − )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ { + ζ ( α + β ) } · (cid:26) n ∑ i = S α , β [ x i − , x i ; f , g ] + m ∑ j = S α , β [ x (cid:48) j − , x (cid:48) j ; f , g ] (cid:27) . Definition 2.1.
We say that the
Stieltjes integral (cid:90) f dgexists in the Riemann sense with the value J , if there exist J ∈ C and a function ε δ > δ > ε δ → δ →
0, and if all the segments [ x i − , x i ] of asubdivision χ have lengths less than δ >
0, then (cid:12)(cid:12)(cid:12) J − ∑ i f ( ξ i )( g ( x i ) − g ( x i − )) (cid:12)(cid:12)(cid:12) < ε δ .
5e observe that, for the integrability in the Riemann sense, it is sufficient that the differ-ence of any of two sums of the formula ∑ i f ( ξ i )( g ( x i ) − g ( x i − )) of Definition 2.1, for eachof which the length of [ x i − , x i ] is less than δ , is less than ε δ . By the inequality just beforeDefinition 2.1, this is the case if for some α , β > α + β > n ∑ i = S α , β [ x i − , x i ; f , g ] < ε δ . For the existence of the integrability, we define W α ( δ ) to be the set of functions such that thevalue V ( δ ) α ( f ) defined below has an upper bound: V ( δ ) α ( f ) = sup | χ |≤ δ (cid:26)(cid:16) ∑ i | f ( x i ) − f ( x i − ) | α (cid:17) α (cid:27) < ∞ . Here | χ | denotes the maximum length of the intervals of χ , and the supremum runs over allsubdivisions χ such that | χ | is less than or equal to δ . Finally we define the Wiener classW α to be the set of functions f such that V ( δ ) α ( f ) with respect to the variable δ has an upperbound. Theorem 2.2 (Theorem on Stieltjes integrability) . If f ∈ W α and g ∈ W β where α , β > and α + β > , have no common discontinuities, their Stieltjes integral exists in the Riemannsense. The Wiener class W α is closed under the pointwise sum and scalar multiplication for 0 < α <
1. Therefore, if we regard the integration as a function from W α × W α to C , this functionturns out to be a bilinear function. On the other hand, it is clear from the definition that theset C α ( I ) of complex-valued α -H¨older continuous functions defined on I is a subset of W α .Moreover, C α ( I ) is closed under the pointwise multiplication in addition to the pointwisesum and scalar multiplication. The integration restricted to C α ( I ) is also referred to as the Young integration . Remark . The Young integration is a special case of the Riemann-Stieltjes integration.
In this subsection we briefly recall the definition of self-similar sets and the Hausdorff di-mension. This subsection basically follows [12]. At the end of this subsection, we give someexamples of self-similar sets. We first begin with the definition of some maps from a metricspace ( X , d ) to itself. Definition 2.4.
Let ( X , d ) be a metric space.a) A map F : X → X is a contraction if there exists 0 < r ≤ d ( F ( x ) , F ( y )) ≤ r · d ( x , y ) for any x , y ∈ X . The real number r is called the contraction ratio .b) A contraction F : X → X is a similitude if d ( F ( x ) , F ( y )) = r · d ( x , y ) for any x , y ∈ X .We call r the similarity ratio . 6or a finite set { F j } j ∈ S of contractions defined on a complete metric space, there exists aunique compact subspace that is characterised by { F j } j ∈ S . Here is the precise statement ofthe existence of self-similar sets: Theorem 2.5.
Let X be a complete metric space. We also let S be a finite set and F i : X → Xcontractions indexed by S. We call the triple ( X , S , { F j } j ∈ S ) an iterated function system or IFS . Then, there exists a unique non-empty compact subset K X of X that satisfiesK X = (cid:91) j ∈ S F j ( K X ) . The compact set K X is called the self-similar set with respect to ( X , S , { F j } j ∈ S ) . Remark . In some literature the terminology “self-similar set” is used in a restricted sense.For instance, Hutchinson introduces the notion of “self-similar set” for a finite set of simil-itudes [11], and self-similar sets defined in Theorem 2.5 are referred to as attractors or in-variant sets ; see Section 9.1 in [7]. We employ Hutchinson’s definition of self-similar sets inthe last section to define cellular self-similar sets, the definition of which is given in Section3 . Theorem 2.7 (Contraction principle) . Let ( X , d ) be a complete metric space and F : X → Xa contraction with respect to the metric. Then there exists a unique fixed point of F, in otherwords, there exists a unique solution to the equation F ( x ) = x. Moreover if x ∗ is the fixedpoint of F, then { F n ( a ) } n ≥ converges to x ∗ for all a ∈ X where F n is the n-th iteration of F. Let ( X , d ) be a metric space and K ( X ) the set of non-empty compact subsets of X . Wedefine the Hausdorff metric δ on K ( X ) by δ ( A , B ) = inf { r > U r ( A ) ⊂ B and U r ( B ) ⊂ A } , where U r ( A ) = { x ∈ X : d ( x , A ) ≤ r } . Lemma 2.8.
The pair ( K ( X ) , δ ) forms a metric space. Moreover, if X is complete, ( K ( X ) , δ ) is also complete. We now assume that the metric space ( X , d ) is complete. Define F ( A ) = (cid:83) j ∈ S F j ( A ) for A ⊂ X , and then F : K ( X ) → K ( X ) is a contraction with respect to the metric δ . Therefore,by applying Theorem 2.7 to ( K ( X ) , δ ) and F , we get the self-similar set K X with respect to ( X , S , { F j } j ∈ S ) .We next define the Hausdorff dimension, which plays a key role to define cyclic cocycleson cellular self-similar sets, the definition of which are given in Section 3 below. Definition 2.9.
Let ( X , d ) be a metric space. We also let s > δ >
0. For any boundedset A ⊂ X , we define H s δ ( A ) = inf (cid:26) ∑ i ≥ diam ( E i ) s : A ⊂ (cid:91) i ≥ E i , diam ( E i ) ≤ δ (cid:27) . { E i } of A , which consist of sets, and diam ( E i ) denotes the diameter of E i . Also we define H s ( A ) = limsup δ ↓ H s δ ( A ) , and we call H s the s-dimensional Hausdorff measure of ( X , d ) . Remark . The s -dimensional Hausdorff measure is a complete Borel measure.The measure detects a critical point of the given subset. Lemma 2.11.
For any subset E ⊂ X , we have sup { s ∈ R | H s ( E ) = ∞ } = inf { s ∈ R | H s ( E ) = } . Definition 2.12.
The real number which satisfies Lemma 2.11 is called the
Hausdorff dimen-sion of E , and it is denoted by dim H ( E ) .In general, it is difficult to calculate the Hausdorff dimension. Namely, the Hausdorffdimensions of a few self-similar sets have been computed. However, if we have a self-similarset K X with respect to an IFS ( X , S , { F j } j ∈ S ) such that contractions are similitudes and thesimilitudes have “small” enough intersections, then there exists a useful way to compute theHausdorff dimension of K X . Theorem 2.13. [17, Theorem II]
Let X be a compact subspace in R n and { F j : R n → R n } j ∈ S a finite set of similitudes indexed with a finite set S. Suppose that the self-similar set K X withrespect to the IFS ( X , S , { F j } j ∈ S ) satisfies the open set condition , i.e., there exists a boundednon-empty open set O ⊂ R n such that (cid:91) j ∈ S F j ( O ) ⊂ O and F i ( O ) ∩ F j ( O ) = /0 f or any i (cid:54) = j ∈ S . Then the Hausdorff dimension dim H ( K X ) of the self-similar set K X is the unique real number α such that the following relation holds ∑ j ∈ S r α j = . Here r j denotes the similarity ratio of F j . Example 2.14.
In this example we give examples of self-similar sets and their dimensions.For later use, we explain contractions of each self-similar set and give an IFS ( X , S , { F j } j ∈ S ) that gives rise to the self-similar set. We also provide figures for each self-similar set, thatcorrespond to X , F ( X )(= (cid:83) j ∈ S F j ( X )) and F ◦ F ( X ) . • Sierpinski gasket
The Sierpinski gasket SG is a well-known examplesof self-similar sets. Here are the first 3steps of the construction of the Sierpinski gasket:8he space of the left-hand side X is an equilateral triangle in R . In the centre we have3 equilateral triangles, the length of whose edges are a half of the ones of X . The simili-tudes F , F and F are defined by the 3 triangles, and the similarity ratios of F j are . Theright-hand side is the space F ◦ F ( X ) . Then, we get an IFS ( X , S = { , , } , { F j } j ∈ S ) , andit gives rise to SG . Moreover, SG satisfies the open set condition. Namely, we can choosean open set O = int ( X ) , and we find that (cid:83) j ∈ S F j ( O ) ⊂ O and F i ( O ) ∩ F j ( O ) = /0 for any i (cid:54) = j ∈ S . Therefore, the Hausdorff dimension of SG is the root α given by the equation ∑ j ∈ S ( ) α = · ( ) α =
1, i.e., dim H ( SG ) = log • Pinwheel fractal
The Pinwheel fractal PW is a self-similar set which is modeled by the pinwheel tiling of theplane. There exist uncountably many pinwheel tilings, and therefore we have self-similarsets following them.The figure is one of the pinwheel fractals based on the most well-known pinwheel tiling of R . The left triangle consists of 3 edges whose lengths are 1, 2 and √
5, and we have 4 simil-itudes whose similarity ratios are √ . Therefore, we get an IFS ( X , S = { , · · · , } , { F j } j ∈ S ) which gives rise to PW . Since PW satisfies the open set condition, the Hausdorff dimensionof the pinwheel fractal is given by the root of the equation ∑ j ∈ S ( √ ) α = · ( √ ) α =
1, i.e.,dim H ( PW ) = log √ • Infinite Sierpinski gasket
Here we give a non-connected self-similar set based on the Sierpinski gasket.9he row represents the first 3 iterations of an IFS that consists of 4 similitudes, one of whichhas the similarity ratio and the rest has . The resulting cellular self-similar set consists ofinfinitely many countable connected components. In this section we define cyclic cocycles on certain subclass of self-similar sets and prove themain theorems. From now on, self-similar sets are assumed to be in R . First we define the kinds of self-similar sets on which we define cyclic cocycles.
Definition 3.1.
Let | X | be a 2-dimensional finite convex linear cell complex and { F j } j ∈ S aset of similitudes F j : | X | → | X | indexed by a finite set S . We also let | X | = (cid:83) j ∈ S F j ( | X | ) .The triple ( | X | , S , { F j } j ∈ S ) is called a cellular self-similar structure if it satisfiesa) ∂ | X | ⊂ ∂ | X | , andb) int F i ( | X | ) ∩ int F j ( | X | ) = /0, for all i (cid:54) = j ∈ S .By Theorem 2 . K | X | with respect to the cellular self-similar structure ( X , S , { F j } j ∈ S ) and we call K | X | the cellular self-similar set with respect to ( | X | , S , { F j } j ∈ S ) . By construction, K | X | is a compact subset of | X | ⊂ R . Lemma 3.2.
Any cellular self-similar structure ( | X | , S , { F j } j ∈ S ) satisfies the open set condi-tion.Proof. The lemma follows immediately from the definition of cellular self-similar structures.Let ( | X | , S , { F j } j ∈ S ) be a cellular self-similar structure. For any n ∈ N , we define a cellcomplex | X n | as follows: first, for ω = ( j , · · · , j n ) ∈ S × n , we write F ω = F j ◦ · · · ◦ F j n . We define | X n | by the following skelton filtration: • sk ( | X n | ) = (cid:83) ω ∈ S × n F ω ( sk ( | X | )) , • sk ( | X n | ) = (cid:83) ω ∈ S × n F ω ( sk ( | X | )) , • sk ( | X n | ) = (cid:83) ω ∈ S × n F ω ( sk ( | X | )) = (cid:83) ω ∈ S × n F ω ( | X | ) .A 1-cell in | X n | is defined to be the closure of a connected component in sk ( | X n | ) − sk ( | X n | ) .The definition of a cellular self-similar structure yields | X n + | = (cid:91) j ∈ S F j ( (cid:91) ω ∈ S × n F ω ( | X | )) = (cid:91) j ∈ S F j ( | X n | ) , i n , n + : | X n + | (cid:44) → | X n | for every n ∈ Z ≥ . Moreover K | X | is written as the inverse limit of inclusion maps { i n , n + : | X n + | (cid:44) → | X n |} , that is, K | X | = ∞ (cid:92) n = | X n | . Therefore we also have a canonical inclusion map i n : K | X | (cid:44) → | X n | for each n ∈ Z ≥ .For a n ∈ N and a 1-cell | σ | in ∂ | X n | , we define E n σ to be the set of 1-cells of | X n + | whichare subspaces of | σ | . Then, we have | σ | = (cid:91) | τ |∈ E n σ | τ | . Lemma 3.3.
There exists M ∈ N that satisfy the following condition: for any n ∈ N and a -cell | σ | in ∂ | X n | we have E n σ ≤ M.Proof.
For every 1-cell | σ | in ∂ | X n | , there exists a unique ω ∈ S × n and a unique 1-cell | ˜ σ | in F ω ( | X | ) such that | σ | ⊂ | ˜ σ | . Since | X n + | is obtained by replacing each 2-cell F ω ( | X | ) by F ω ( | X | ) = F ω ( (cid:83) j ∈ S F j ( | X | )) , | ˜ σ | is subdivided by at most S | X n | is a convex linear cell complex, we can associate an abstract simplicialcomplex X sn by employing a lemma in [27]: Lemma 3.4. [27, Chapter I, Lemma 1]
A convex linear cell complex can be subdivided intoa simplicial complex without introducing any more vertices.
For any simplicial complex | X sn | and p ≥ S p ( X sn ) to be a set of ( p + ) -tuplesof points of sk ( X sn ) such that ( p + ) vertices are contained in a simplex of X sn , that is, S p ( X sn ) = (cid:110) ( x , · · · , x p ) ∈ sk ( X sn ) × ( p + ) | there exists a p -simplex σ ∈ X sn s . t . x i ∈ σ for ∀ i (cid:111) . We also define face maps σ i : S p ( X sn ) → S p − ( X sn ) for 0 ≤ i ≤ p , and the pair ( S ∗ ( X sn ) , σ i ) forms a semi-simplicial set; see the definition [6]. We note that, for p ≥ S p ( X sn ) contains a degenerate simplex ( x , · · · , x p ) , that is, a simplex ( x , · · · , x p ) ∈ S p ( X sn ) such that there existdistinct indexes i and j such that x i = x j . Now, we define ˜ S p ( X sn ; C ) to be the free C -modulegenerated by S p ( X sn ) and a map ˜ ∂ p : ˜ S p ( X sn ; C ) → ˜ S p − ( X sn ; C ) by˜ ∂ p ( x , · · · , x p ) = p ∑ j = ( − ) j σ i ( x , · · · , x p ) = p ∑ j = ( − ) j ( x , · · · , ˆ x j , · · · , x p ) . Then we have a commutative diagram:˜ S p ( X sn ; C ) π (cid:15) (cid:15) ˜ ∂ p (cid:47) (cid:47) ˜ S p − ( X sn ; C ) π (cid:15) (cid:15) C p ( X sn ; C ) ∂ p (cid:47) (cid:47) C p − ( X sn ; C ) , C p ( X sn ; C ) is the p th simplicial chain group of X sn whose coefficient is C , ∂ p a simplicialboundary map and π the quotient map. Remark . The chain map π is a chain equivalence; see Theorem 8 in Chapter 4.3 of [21]for details.We now assign the counterclockwise orientation on each 2-simplex in every | X sn | , andchoose a basis B n = { [ σ ] } of C ( X sn ; C ) consisting of non-degenerate p -simplexes σ in X sn .We assume that each element [ σ ] of B n represents the counterclockwise orientation.Now, we define simplicial chains for every n ∈ Z ≥ : let c n = ∑ [ σ ] ∈ B n [ σ ] ∈ C ( X sn ; C ) . Then ∂ ( c n ) ∈ C ( X sn ; C ) is the sum of all 1-simplices which lie on ∂ | X n | , and we can choose s n ∈ π − ( c n ) so that s n has no degenerate simplexes and each summand of ˜ ∂ ( s n ) ∈ ˜ S ( X sn ; C ) lies on ∂ | X n | . Now we define a boundary chain b n ∈ ˜ S ( X sn ; C ) by • b n = ˜ ∂ ( s n ) .We first let ε ( b n ) be the subset of 1-simplices in S ( X sn ) which are direct summands of b n .Since any σ ∈ ε ( b n ) is non-degenerate, we can take the geometric realisation | σ | ⊂ ∂ | X sn | .We also define a subset ε ( o n ) ⊂ ε ( b n ) by ε ( o n ) = (cid:110) σ ∈ ε ( b n ) | | σ | ⊂ ∂ | X | (cid:111) . For each σ ∈ ε ( o n ) , we have the sign of σ in b n and denote it by sgn ( σ ) . Define • o n = ∑ σ ∈ ε ( o n ) sgn ( σ ) · σ ∈ ˜ S ( X sn ; C ) , • I n = b n − o n ∈ ˜ S ( X sn ; C ) .Let ε ( I n ) = ε ( b n ) \ ε ( o n ) . We also define | ε ( I n ) | = (cid:83) σ ∈ ε ( I n ) | σ | , and ε ( I n \ I n − ) in a mannersimilar to ε ( o n ) : ε ( I n \ I n − ) = (cid:110) σ ∈ ε ( b n ) | | σ | ⊂ | ε ( I n ) |\| ε ( I n − ) | (cid:111) . Finally we define a 1-chain by • I n \ I n − = ∑ σ ∈ ε ( I n \ I n − ) sgn ( σ ) · σ ∈ ˜ S ( X sn ; C ) . Example 3.6.
For the Sierpinski gasket and the pinwheel fractal introduced in Section 2.2, we give spacesthat represent ε ( b ) , ε ( b ) , ε ( b ) , and ε ( I ) , ε ( I ) , ε ( I ) . The first row corresponds to ε ( b i ) ,and the second corresponds to ε ( I i ) . The dots in spaces denote the vertices of 1-simplices,i.e., 0-simplices. 12 Sierpinski gasket • Pinwheel fractal 13ext, for every n ∈ Z ≥ , we define a 2-dimensional cell complex | K n , n + | . For every n ∈ Z ≥ we endow | X n , n + | = | X n | − | X n + | (= the closure of | X n | − | X n + | ) , with a cell complex structure, whose structure is defined by the following skelton filtration: • sk ( | X n , n + | ) = sk ( ∂ | X n + | ) ∩ | X n , n + |• sk ( | X n , n + | ) = ∂ | X n , n + |• sk ( | X n , n + | ) = | X n , n + | We also define a subspace | K n , n + | in R to be | K n , n + | = [ , ] × ∂ | X n + | ∪ { } × | X n , n + | . We use z as the variable of the first coordinate of | K n , n + | . We now endow | K n , n + | with a 2-dimensional cell complex structure as follows: let p : | K n , n + | → | K n , n + || z = be a projectiondefined by p ( t , x ) = ( , x ) . We define • sk ( | K n , n + | ) = { } × sk ( ∂ | X n + | ) ∪ { } × ( sk ( ∂ | X n | ) ∪ sk ( | X n , n + | )) • sk ( | K n , n + | ) = { } × sk ( ∂ | X n + | ) ∪ { } × ( sk ( ∂ | X n | ) ∪ sk ( ∂ | X n , n + | ) ∪ | E n , n + | ) • sk ( | K n , n + | ) = | K n , n + | .Here, | E n , n + | = (cid:110) ( x , y ) | x ∈ { } × sk ( ∂ | X n | ) or x ∈ { } × sk ( | X n , n + | ) , y ∈ { } × sk ( ∂ | X n + | ) such that p ( x ) = y (cid:111) . By construction of | K n , n + | , we have ∂ | K n , n + | = { } × ∂ | X n + | ∪ { } × ∂ | X n | as a cell complex in R . By employing Lemma 3.4 again, the cell complex | K n , n + | is subdi-vided into a 2-dimensional simplicial complex | K sn , n + | , and we may therefore choose chains s n , n + , ˜ s n , n + and ˜˜ s n , n + ∈ ˜ S ( K sn , n + ; C ) so that the chains consist of non-degenerate sim-plexes and ˜ ∂ ( s n , n + ) = b n − b n + , ˜ ∂ ( ˜ s n , n + ) = I n − I n + , ˜ ∂ ( ˜˜ s n , n + ) = I n + \ I n . We define the sets ε ( s n , n + ) , ε ( ˜ s n , n + ) and ε ( ˜˜ s n , n + ) in a manner similar to the definition of ε ( b n ) , and assume that ˜ s n , n + and ˜˜ s n , n + are summands of s n , n + , in other words, ε ( ˜ s n , n + ) , ε ( ˜˜ s n , n + ) ⊂ ε ( s n , n + ) . By a closed cycle z in I n + \ I n we mean a subset z of ε ( I n + \ I n ) such that (cid:83) σ ∈ z | σ | is ho-momorphic to S , and denote (cid:83) σ ∈ z | σ | by | z | . We also denote by cyc ( I n + \ I n ) the set of closed14ycles in I n + \ I n and define ˜ z = ∑ σ ∈ z sgn ( σ ) · σ ∈ ˜ S ( K sn , n + ; C ) for z ∈ cyc ( I n + \ I n ) . Then,for every closed cycle z in I n + \ I n , there exists a non-degenerate 2-chain ˜˜ s z ∈ ˜ S ( K sn , n + ; C ) such that ˜ ∂ ( ˜˜ s z ) = ˜ z .For n = | ˜ K , | = [ , ] × ∂ ( | X | − | X | ) ∪ { } × | X , | and then | ˜ K , | is written as | ˜ K , | = (cid:91) z ∈ cyc ( I \ I ) | ε ( ˜˜ s z ) | since ∂ ( | X | − | X | ) = | ε ( I \ I ) | . Moreover, since, for every ω ∈ S × n , we have an inclusionmap i ω : ∂ ( | X | − | X | ) (cid:44) → F ω ( ∂ | X | ) , there exists a family { ˜ i ω } ω ∈ S × n of inclusion maps ˜ i ω : | ˜ K , | (cid:44) → | K n , n + | such that ˜ i ω | z = = i ω . Finally we fix a subdivision of | ˜ K , | and assume that the subdivision of the images of theinclusion maps are given by the subdivision of | ˜ K , | . -cocycle In this subsection, we define a sequence of complex numbers for given H¨older continuousfunctions, that we call a cyclic quasi-1-cocycle. In order to define the sequence, we first recalla cochain complex which gives rise to one of the classical cohomology theories in algebraictopology, so called Alexander-Spanier cohomology theory; see Chapter 6 of [21] for details.Let R be a ring. We also let X be a set and X ( p + ) the ( p + ) -fold product of X . Wedefine F p ( X ; R ) to be the abelian group of functions from X ( p + ) to R , whose sum is givenby the pointwise sum. A coboundary homomorphism δ : F p ( X ; R ) → F p + ( X ; R ) is definedby ( δ φ )( x , · · · , x p + ) = p + ∑ j = ( − ) j φ ( x , · · · , ˆ x j , · · · , x p + ) . We also introduce the cup product on the complex ( F ∗ ( X ; R ) , δ ) : for φ ∈ F p ( X ; R ) and φ ∈ F q ( X ; R ) the cup product φ (cid:94) φ ∈ F p + q ( X ; R ) is defined by ( φ (cid:94) φ )( x , · · · , x p + q ) = φ ( x , · · · , x p ) φ ( x p , · · · , x p + q ) . The Leibniz rule holds for the cup product: for φ ∈ F p ( X ; R ) and φ ∈ F q ( X ; R ) , δ ( φ (cid:94) φ ) = δ φ (cid:94) φ + ( − ) p φ (cid:94) δ φ . Now, we define a cochain subcomplex of ( F ∗ ( X ; R ) , δ ) : we assume that X is a metricspace and R the field of complex numbers C . We also let C α ( X ) be the algebra of complex-valued α -H¨older continuous functions on X . Then, C α ( X ) is a subalgebra of F ( X ; C ) , andfor each p ∈ Z ≥ we define the submodule C α , p ( X ) of F p ( X ; C ) generated by C α ( X ) ⊂ F ( X ; C ) with the coboundary maps and the cup product.We now apply the construction for a cellular self-similar structure ( | X | , S , { F j } j ∈ S ) : let C α ( K | X | ) be the α -H¨older continuous functions defined on K | X | . For each n ∈ N , we endow15 k ( | X n | ) with the induced metric of R . Since we have an inclusion map j n : sk ( | X n | ) (cid:44) → K | X | for every n ∈ Z ≥ , we have a commutative diagram of cochain complexes F p ( K | X | ; C ) j ∗ n (cid:47) (cid:47) F p ( sk ( | X sn | ) ; C ) C α , p ( K | X | ) (cid:63)(cid:31) (cid:79) j ∗ n (cid:47) (cid:47) C α , p ( sk ( | X sn | )) (cid:63)(cid:31) (cid:79) The vector space F p ( sk ( | X sn | ) ; C ) is the set of complex-valued functions Func ( S p ( ∆ sk ( | X sn | ) ) , C ) defined on S p ( ∆ sk ( | X sn | ) ) : = sk ( | X sn | ) × p + . In a manner similar to the definition of the facemaps σ i of S p ( X sn ) , we define the face maps on S ∗ ( ∆ sk ( | X sn | ) ) , and then the pair ( S ∗ ( ∆ sk ( | X sn | ) ) , σ i ) turns out to be a semi-simplicial set, the definition of which is in [6], also known as the fun-damental ∞ -groupoid of sk ( | X sn | ) . Since the inclusion map S ∗ ( | X sn | ) (cid:44) → S ∗ ( ∆ sk ( | X sn | ) ) is amap of semi-simplicial sets, we therefore get the following commutative diagram: F p ( K | X | ; C ) j ∗ n (cid:47) (cid:47) F p ( sk ( | X sn | ) ; C ) extendlinearly (cid:47) (cid:47) Hom C ( ˜ S p ( ∆ sk ( | X n | ) ; C ) , C ) restrict (cid:15) (cid:15) C α , p ( K | X | ) (cid:63)(cid:31) (cid:79) j ∗ n (cid:47) (cid:47) C α , p ( sk ( | X sn | ) (cid:63)(cid:31) (cid:79) (cid:63)(cid:31) (cid:79) r (cid:47) (cid:47) Hom C ( ˜ S p ( | X sn | ; C ) , C ) . Now we define C α , p ( | X sn | ) = im ( r ) . For any f , g ∈ C α ( K | X | ) and p =
1, we have a 1-cochain ω n ( f , g ) = ( f (cid:94) δ g ) − ( g (cid:94) δ f ) in C α , ( | X sn | ) for every n ∈ N . As we define inSection 3.1, we also have I n ∈ ˜ S ( | X sn | ; C ) . For every n ∈ N , we have a complex number ω n ( f , g )( I n ) and denote it by φ n ( f , g ) . Definition 3.7.
Let f , g ∈ C α ( K | X | ) . We call the sequence { φ n ( f , g ) } n ∈ N the cyclic quasi-1-cocycle for f and g . -cocycles We prove the main results in this subsection. We refer the reader to [4, 5] for details of theHochschild cohomology groups and the cyclic cohomology groups.
Theorem 3.8 ( Existence theorem).
Let ( | X | , S , { F j } j ∈ S ) be a cellular self-similar structurewith S ≥ and K | X | the cellular self-similar set with respect to ( | X | , S , { F j } j ∈ S ) . We alsolet C α ( K | X | ) be the algebra of complex-valued α -H¨older continuous functions on K | X | . If α > dim H ( K | X | ) , then the cyclic quasi-1-cocycle { φ n ( f , g ) } n ∈ N is a Cauchy sequence forany f , g ∈ C α ( K | X | ) .Proof. We first endow | K n , n + | with a metric by d (( t , x ) , ( t (cid:48) , x (cid:48) )) = | x − x (cid:48) | R . Let f , g ∈ C α ( K | X | ) . Since we have an inclusion map sk ( | X n + | ) (cid:44) → K | X | for every n ∈ Z ≥ , we canextend f to f n ∈ C α ( sk ( | K n , n + | )) so that f n ( t , x ) = f ( x ) for ( t , x ) ∈ [ , ] × ∂ | X sn + | . Wealso let, for h , k ∈ C α ( sk ( | K n , n + | )) , ω n ( h , k ) = ( h (cid:94) δ k ) − ( k (cid:94) δ h ) be a 1-cochain in16 α , ( | K sn , n + | ) . Then, we have | φ n ( f , g ) − φ n + ( f , g ) | = | ω n ( f n , g n )( I n − I n + ) | = | ω n ( f n , g n )( ˜ ∂ ( ˜ s n , n + )) |≤ | ω n ( f n , g n )( ˜ ∂ ( ˜ s n , n + )) | + | ω n ( f n , g n )( ˜ ∂ ( s n , n + − ˜ s n , n + )) |≤ ∑ σ ∈ ε ( s n , n + ) | ω n ( f n , g n )( ˜ ∂ ( σ )) | = ∑ σ ∈ ε ( s n , n + ) | ( δ f n (cid:94) δ g n )( σ ) − ( δ g n (cid:94) δ f n )( σ ) | . (1)We note that every σ ∈ ε ( s n , n + ) is given by σ = ( x , y , z ) for some x , y , z ∈ sk ( | K n , n + | ) .Therefore, ( ) may be written as ( ) = ∑ ( x , y , z ) ∈ ε ( s n , n + ) | ( δ f n (cid:94) δ g n )( x , y , z ) − ( δ g n (cid:94) δ f n )( x , y , z ) | = ∑ ( x , y , z ) ∈ ε ( s n , n + ) (cid:12)(cid:12)(cid:12) ( f n ( y ) − f n ( x ))( g n ( z ) − g n ( y )) − ( g n ( y ) − g n ( x ))( f n ( z ) − f n ( y )) (cid:12)(cid:12)(cid:12) ≤ ∑ ( x , y , z ) ∈ ε ( s n , n + ) · c f · c g | y − x | α | z − y | α , (2)where c f and c g are the H¨older constants of f and g respectively.We now define a map to estimate the term ( ) . For any σ ∈ ε ( s n , n + ) \ ε ( ˜˜ s n , n + ) thereexists a unique ω = ( j , · · · , j n ) ∈ S × n such that p ( | σ | ) ⊂ ∂ F ω ( | X | ) . We therefore have amap ρ : ε ( s n , n + ) \ ε ( ˜˜ s n , n + ) → S × n , and define ˜ S × n to be im ( ρ ) . We note that, by Lemma3.3, there exists M ∈ N such that ρ − ( ω ) < M for any ω ∈ ˜ S × n . Moreover, since p ( | σ | ) ⊂ ∂ F ω ( | X | ) we have an inequalitydiam ( | σ | ) = diam ( p ( | σ | )) ≤ r j · · · · · r j n · d K | X | , where ( j , · · · , j n ) = ω ∈ ˜ S × n , r j are the similarity ratios of F j and d K | X | is the diameter of K | X | .On the other hand, we let L = ( I \ I ) be the number of closed cycles in I \ I . At the ( n + ) -step, for every ω ∈ S × n , there exist L closed cycles in F ω ( (cid:83) j ∈ S F j ( | X | )) = F ω ( | X | ) .We recall that for every closed cycle z in I n + \ I n there is a 2-chain ˜˜ s z ∈ ˜ S ( K sn , n + ) such that ε ( ˜˜ s z ) ⊂ ε ( ˜˜ s n , n + ) and ˜ ∂ ( ˜˜ s z ) = ˜ z ; see also Section 3.1. Therefore, ˜˜ s n , n + may be written as˜˜ s n , n + = ∑ ω ∈ S × n ∑ ≤ i ≤ L ˜˜ s ω , z i . We also recall from Section 3 . ω ∈ S × n we have an inclusion map ˜ i ω : | ˜ K s , | (cid:44) →| K n , n + | and im ( ˜ i ω ) = (cid:91) z ∈ cyc ( I n + \ I n ) s . t . | z |⊂ F ω ( | X | ) | ε ( ˜˜ s z ) | . Therefore, since the subdivision of the images im ( ˜ i ω ) are induced by the subdivision of | K s , | ,17e may define M = sup z ∈ cyc ( I n + \ I n ) { ε ( ˜˜ s z ) } = sup z ∈ cyc ( I \ I ) { ε ( ˜˜ s z ) } . From these arguments, ( ) is now decomposed into two parts: ( ) = ∑ ( x , y , z ) ∈ ε ( s n , n + ) \ ε ( ˜˜ s n , n + ) · c f · c g | y − x | α | z − y | α + ∑ ( x , y , z ) ∈ ε ( ˜˜ s n , n + ) · c f · c g | y − x | α | z − y | α ≤ ∑ ( j , ··· , j n ) ∈ ˜ S × n · c f · c g · ρ − ( ω ) · ( r α j · · · r α j n · d α K | X | )+ ∑ ( j , ··· , j n ) ∈ S × n ∑ ≤ i ≤ L · c f · c g · ε ( ˜˜ s z i ) · ( r α j · · · r α j n · d α K | X | ) ≤ ∑ ( j , ··· , j n ) ∈ S × n · c f · c g · M · ( r α j · · · r α j n · d α K | X | )+ ∑ ( j , ··· , j n ) ∈ S × n · c f · c g · L · M · ( r α j · · · r α j n · d α K | X | )= · c f · c g · d α K | X | · ( M + L · M ) · ( ∑ j ∈ S r α j ) n . We denote 2 · c f · c g · d α K | X | · ( M + L · M ) by K , and then we have | φ n + k ( f , g ) − φ n ( f , g ) | ≤ ∑ ≤ i ≤ k | φ n + i ( f , g ) − φ n + i − ( f , g ) |≤ ∑ ≤ i ≤ k K · ( ∑ j ∈ S r α j ) n + i − = K · ( ∑ j ∈ S r α j ) n · ∑ ≤ i ≤ k ( ∑ j ∈ S r α j ) i − . (3)Since we assume that 2 α > dim H ( K | X | ) and dim H ( K | X | ) is computed by the formula in The-orem 2.13, and therefore the term ( ∑ j ∈ S r α j ) is less than 1, and the term ∑ ≤ i ≤ k ( ∑ j ∈ S r α j ) i − converges to a finite value as k tends to ∞ . Therefore, we have ( ) ≤ K · ∞ ∑ i = ( ∑ j ∈ S r α j ) i − · ( ∑ j ∈ S r α j ) n , and the right hand side also converges to 0 as n tends to ∞ . This completes the proof of thetheorem.From now on, we assume that 2 α > dim H ( K | X | ) and define a bilinear map φ : C α ( K | X | ) × C α ( K | X | ) → C by φ ( f , g ) = lim n → ∞ φ n ( f , g ) . 18 emma 3.9. The map φ is independent of the choice of I n .Proof. In order to check the mentioned property of the bilinear map φ : C α ( K | X | ) × C α ( K | X | ) → C , we have to show that the cyclic quasi-1-cocycle converges to the same value regard-less of the choice of I n which represents the given orientation. Let I n , I (cid:48) n ∈ π − ([ I n ]) suchthat | ε ( I n ) | = | ε ( I (cid:48) n ) | and φ (cid:48) n ( f , g ) = ( f (cid:94) δ g )( I (cid:48) n ) − ( g (cid:94) δ f )( I (cid:48) n ) . Then there exists a2-dimensional simplicial complex J n such that | J n | = | ε ( I n ) | × [ , ] , and we may chooseˆ s n ∈ ˜ S ( J n ; C ) such that ˜ ∂ ( ˆ s n ) = I n − I (cid:48) n . We endow | J n | with a metric similar to the metricon | K n , n + | , and then we have | φ n ( f , g ) − φ (cid:48) n ( f , g ) | = | δ ω n ( f n , g n )( ˆ s n ) |≤ ∑ ( x , y , z ) ∈ ε ( ˆ s n ) c f · c g · | y − x | α · | z − y | α ≤ ∑ ( x , y ) ∈ ε ( I n ) · c f · c g · | y − x | α ≤ ∑ ( j , ··· , j n ) ∈ ˜ S × n · c f · c g · d α K | X | · r α j · · · · · r α j n ≤ · c f · c g · d α K | X | · ( ∑ j ∈ S r α j ) n → , as n → ∞ . This completes the proof of the well-definedness of φ .Based on the proof of Theorem 3.8, we can prove the following corollary. Corollary 3.10.
For any f , g ∈ C α ( K | X | ) , we have φ ( f , g ) = (cid:90) ∂ | X | f dg = · ( Young integral along ∂ | X | ) . In particular, for and x : = id ∈ C α ( K | X | ) , φ ( , x ) = (cid:90) ∂ | X | dx = · ( length of ∂ | X | ) . Proof.
By the construction of the cyclic quasi-1-cocycle of f , g ∈ C α ( K | X | ) , we have φ n ( f , g ) = ω n ( f , g )( I n ) = − ω n ( f , g )( o n ) + ω n ( f , g )( b n ) . The proof of Theorem 3.8 yields directly that the sequence { ω n ( f , g )( b n ) } n ∈ Z ≥ converges to0 if 2 α > dim H ( K | X | ) . Since { ω n ( f , g )( o n ) } n ∈ Z ≥ provides the Young integration along ∂ | X | ,which is the finite union of closed segments, we get the mentioned equalities. Theorem 3.11.
Under the assumption of Theorem 3.8 : a) The bilinear map φ gives rise to a cyclic -cocycle of C α ( K | X | ) .b) If | X | (cid:54) = | X | , the cocycle φ represents a non-trivial element [ φ ] in HC ( C α ( K | X | )) . roof. We have a linear map φ : C α ( K | X | ) ⊗ C α ( K | X | ) → C . It follows immediately that thecocycle satisfies the cyclic condition since φ n ( f , g ) satisfies the cyclic condition for any n ∈ Z ≥ . Accordingly, it remains to show that φ is a Hochschild cocycle. For f , g , h ∈ C α ( K | X | ) ,we may write b φ ( f , g , h ) as b φ ( f , g , h ) = φ ( f g , h ) − φ ( f , gh ) + φ ( h f , g )= lim n → ∞ φ n ( f g , h ) − lim n → ∞ φ n ( f , gh ) + lim n → ∞ φ n ( h f , g )= lim n → ∞ (cid:16) φ n ( f g , h ) − φ n ( f , gh ) + φ n ( h f , g ) (cid:17) = lim n → ∞ b φ n ( f , g , h ) . Therefore, to prove that b φ ( f , g , h ) = n → ∞ b φ n ( f , g , h ) = δ ( η (cid:94) τ ) = δ η (cid:94) τ + ( − ) deg ( η ) η (cid:94) δ τ , we have φ n ( f g , h ) = (cid:16) f g (cid:94) δ h − h (cid:94) δ ( f g ) (cid:17) ( I n )= (cid:16) ( f (cid:94) g (cid:94) δ h ) − ( h (cid:94) δ f (cid:94) g ) − ( h (cid:94) f (cid:94) δ g ) (cid:17) ( I n ) . Similarly, φ n ( f , gh ) = (cid:16) ( f (cid:94) δ g (cid:94) h ) + ( f (cid:94) g (cid:94) δ h ) − ( g (cid:94) h (cid:94) δ f ) (cid:17) ( I n ) , φ n ( h f , g ) = (cid:16) ( h (cid:94) f (cid:94) δ g ) − ( g (cid:94) δ h (cid:94) f ) − ( g (cid:94) h (cid:94) δ f ) (cid:17) ( I n ) . Therefore, b φ n ( f , g , h ) = − (cid:16) ( h (cid:94) δ f (cid:94) g ) + ( f (cid:94) δ g (cid:94) h ) + ( g (cid:94) δ h (cid:94) f ) (cid:17) ( I n ) . (4)Since − ( h (cid:94) δ f (cid:94) g )( I n ) = (cid:16) ( δ h (cid:94) f (cid:94) g ) + ( h (cid:94) f (cid:94) δ g ) − ( δ ( h f g )) (cid:17) ( I n )= (cid:16) ( δ h (cid:94) f (cid:94) g ) + ( h (cid:94) f (cid:94) δ g ) (cid:17) ( I n ) , we have ( ) = (cid:16) ( δ h (cid:94) f (cid:94) g ) + ( h (cid:94) f (cid:94) δ g ) − ( f (cid:94) δ g (cid:94) h ) − ( g (cid:94) δ h (cid:94) f ) (cid:17) ( I n )= ∑ ( x , y ) ∈ ε ( I n ) ± (cid:16) ( δ h (cid:94) f (cid:94) g ) + ( h (cid:94) f (cid:94) δ g ) − ( f (cid:94) δ g (cid:94) h ) − ( g (cid:94) δ h (cid:94) f ) (cid:17) ( x , y )= ∑ ( x , y ) ∈ ε ( I n ) ± ( h ( y ) − h ( x ))( g ( y ) − g ( x ))( f ( y ) − f ( x )) . | b φ n ( f , g , h ) | ≤ ∑ ( x , y ) ∈ ε ( I n ) | h ( y ) − h ( x ) | · | g ( y ) − g ( x ) | · | f ( y ) − f ( x ) | = ∑ ( x , y ) ∈ ε ( I n ) c f · c g · c h · | x − y | α ≤ c f · c g · c h · d α K | X | · M ∑ ( j , ··· , j n ) ∈ S × n r α j · · · · · r α j n = c f · c g · c h · d α K | X | · M · ( ∑ j ∈ S r α j ) n → , as n → ∞ . This completes the proof of (a).We now prove (b). We note that we have the pairing HH ( C α ( K | X | )) × HH ( C α ( K | X | )) → C . As seen in Theorem 3.10, we know that φ ( ⊗ x ) (cid:54) =
0, and this completes the proof of (b).
Remark . The algebra of α -H¨older continuous functions on a compact metric spaceadmits a Banach topology and it turns out to be a Banach algebra. However, we do not knowwhether or not the cocycle of Theorem 3.11 is continuous in the sense of a map betweenBanach algebras. We examine the cyclic cocycle on some cellular self-similar sets. The spaces on which thecocycles are examined are the examples given in Section 2 . • Sierpinski gasket
The theorems in the Section 3.3 may be applied to the Sierpinski gasket SG , and the cyclic1-cocycle φ on C α ( SG ) is well-defined for 2 α > dim H ( SG ) = log
3. Moreover, the cocycleis non-trivial since | X | (cid:54) = | X | . • pinwheel fractal Pinwheel fractal PF may also be seen as a cellular self-similar set. The self-similar structureconsists of 4 similitudes whose ratios are √ , see also section 2. The cyclic cocycle is well-defined if 2 α > dim H ( PF ) = log √ HC . • Infinite isolated Sierpinski gaskets
The second example in Section 2 . ISG the resulting cellular self-similar set. The cyclic 1-cocycle may be defined on the space, andthe cocycle is non-trivial. From now, we also discuss the structure of HC ( C Lip ( ISG )) .By the self-similar structure of ISG , π ( ISG ) = (cid:76) p ∈ N Z , each of whose summands cor-responds to a connected component Y p of ISG . Therefore
ISG may be written as
ISG = (cid:71) p ∈ N Y p . in p : Y p → (cid:71) p ∈ N Y p = ISG for any p ∈ N . We now fix a base point y p ∈ Y p for each p ∈ N , and define a cyclic 0-cocycle ψ p of C Lip ( Y p ) by taking the value of y p for any f ∈ C Lip ( Y p ) . Therefore, the canonicalinclusion map in p induces the map of cyclic cohomology groups: ( in p ) ∗ : HC ( C Lip ( Y p )) → HC ( C lip ( ISG )) . We now let P be a finite subset of N and assume that Ψ P = ∑ p ∈ P α p ( in p ) ∗ ([ ψ p ]) =
0. Wealso define c p ∈ C Lip ( ISG ) by c p ( y ) = (cid:40) , y ∈ Y p , otherwise . Then, for any ˜ p ∈ P , we have a pairing of the Hochschild homology group and the Hochschildcohomology group of C Lip ( ISG ) :0 = (cid:104) Ψ P , c p (cid:105) = ∑ p ∈ P α p ( in p ) ∗ ([ ψ p ])( c ˜ p ) = α ˜ p , and which means that the set { ( in p ) ∗ ([ ψ p ]) } p ∈ P is a linearly independent set. Since this argu-ment also works for any finite set P of N , we can conclude that { ( in p ) ∗ ([ ψ p ]) } p ∈ N forms a lin-early independent set of HC ( C Lip ( ISG )) , and therefore HC ( C Lip ( ISG )) contains (cid:76) p ∈ N C as a C -vector space. Strichartz introduces the notion of “fractafold” [22, 23], and on which he examines frac-tal versions of the classical theories, for example, Hodge-de Rham theory, spectral theory,homotopy theory. In particular, the Laplacian on some kinds of self-similar sets has beenextensively studied, and it is applied to various fields [2, 12, 22, 23]. Here, we will give someexamples of finite unions of cellular self-similar sets.The first example is the wedge sum of Sierpinski gasket and Sierpinski carpet with base pointsat their corners. Then the space is neither a cellular self-similar set nor a fractafold. However,22he theorem may be applied to the space. Namely, the space is seen as the projective limit ofthe following spaces:The figure is obtained by taking the wedge sum of the sequences which give rise to Sierpinskigasket and Sierpinski carpet. Similarly, we have sequences of boundary chains b , b , b andinner chains I , I , I respectively:We therefore have a cyclic quasi-1-cocycle, and the quasi-cocycle can be written by theelement-wise sum of cyclic quasi-1-cocycles of SG and SC . In order that that cyclic quasi-1-cocycle is a Cauchy sequence, it is enough that the H¨older index α satisfies the inequality2 α > dim H ( SC ) .From the point of this view, SG can be seen as a union of 3 Sierpinski gaskets, andtherefore SG may be seen as a “fractafold with boundary”, see [22, 23] for details. As definedin the previous subsection, we have a cyclic cocycle on SG .Finally, we will define a cyclic cocycle of the algebra of Lipschitz functions defined on a“fractafold” based on the Sierpinski gasket: 23he space is a union of four copies of Sierpinski gasket in R obtained by gluing the pointsat corners of a copy with each corner of the other Sierpinski gaskets. This space is one ofthe examples of what Strichartz calls “fractafolds without boundaries”, and we denote it by FSG . The space
FSG can be seen as the projective limit of a sequence of the spaces that isobtained by gluing copies of the sequence which gives rise to SG .We therefore get, by applying the theorem to each Sierpinski gasket, a cyclic 1-cocycle on C α ( FSG ) when 2 α > log Remark . Strichartz introduces the Hodge-de Rham theory for fractal graphs [1]. In thispaper, Laplacian on some fractal sets are defined by exploiting the Alexander-Spanier cochaincomplexes. However, we do not know whether or not there exist any relation between thecyclic 1-cocycle defined in the present paper and the Laplacian of [1].
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