Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs
aa r X i v : . [ m a t h . P R ] O c t Dirichlet forms and ultrametric Cantor sets associated tohigher-rank graphs
Jaeseong Heo, Sooran Kang and Yongdo LimOctober 29, 2019
Abstract
The aim of this paper is to study the heat kernel and the jump kernel of the Dirich-let form associated to ultrametric Cantor sets ∂ B Λ that is the infinite path space ofthe stationary k -Bratteli diagram B Λ , where Λ is a finite strongly connected k -graph.The Dirichlet form which we are interested in is induced by an even spectral triple( C Lip ( ∂ B Λ ) , π φ , H , D, Γ) and is given by Q s ( f, g ) = 12 Z Ξ Tr (cid:0) | D | − s [ D, π φ ( f )] ∗ [ D, π φ ( g )] (cid:1) dν ( φ ) , where Ξ is the space of choice functions on ∂ B Λ × ∂ B Λ . There are two ultrametrics, d ( s ) and d w δ , on ∂ B Λ which make the infinite path space ∂ B Λ an ultrametric Cantor set. Theformer d ( s ) is associated to the eigenvalues of Laplace-Beltrami operator ∆ s associatedto Q s , and the latter d w δ is associated to a weight function w δ on B Λ , where δ ∈ (0 , µ on ∂ B Λ has the volume doubling propertywith respect to both d ( s ) and d w δ and we study the asymptotic behavior of the heat kernelassociated to Q s . Moreover, we show that the Dirichlet form Q s coincides with a Dirichletform Q J s ,µ which is associated to a jump kernel J s and the measure µ on ∂ B Λ , and weinvestigate the asymptotic behavior and moments of displacements of the process. Contents k -graphs and ultrametric Cantor sets 4 k -graphs and k -Bratteli diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Weights and ultrametrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Introduction
Higher-rank graphs (or k -graphs) Λ and their C ∗ -algebras C ∗ (Λ) were introduced in [17] as ageneralization of directed graphs and Cuntz-Krieger algebras in order to give a combinatorialmodel and to study abstract properties of C ∗ (Λ) such as simplicity and ideal structure. Higher-rank graphs and the associated C ∗ -algebras have extensively been studied in last two decadeson the classification theory, K-theory of C ∗ (Λ), and the study of spectral triples and of KMSstates. (See [3, 20, 6, 10, 9, 13] and references therein). The C ∗ -theory of Cuntz algebras andCuntz-Krieger algebras associated to graphs (and k -graphs) not only have played an importantrole in operator algebras and noncommutative geometry, but also have been used in manyapplications in engineering and physics: branching laws for endomorphisms, Markov measuresand topological Markov chains, wavelets and multiresolution analysis, and iterated functionsystems (IFS) and fractals. (See [1, 7, 2, 18, 5] and references therein).In the recent papers [8, 9], authors have studied spectral triples and their relations to C ∗ -representations and wavelet decompositions of the Cuntz algebra O N , the Cuntz-Kriegeralgebra O A of an adjacency matrix A and C ∗ (Λ) based on the works of [19, 14]. In fact,Pearson-Bellissard [19] originally constructed a family of spectral triples to study geometryof ultrametric Cantor sets induced from weighted trees. Not only they studied ζ -function,Dixmier trace and their relations to (fractal) dimensions and (Hausdorff, probability) measureson ultrametric Cantor sets, but also they constructed a family of Dirichlet forms from thespectral triples and studied the eigenvalues, eigenspaces of the Laplace-Beltrami operatorsassociated to the Dirichlet forms, and investigated the associated diffusion on triadic Cantorsets. Using the framework developed in [19], Julien and Savinien [14] applied the theory tostudy substitution tilings on a weighted Bratteli diagram (which is a special kind of tree).Although it is known in [19, 14] that the basis of eigenspaces of the Laplace-Beltrami oper-ators are related to wavelets (for example, Haar wavelets for the case of the triadic Cantor set[19]), it was first noticed in [8, 9] that particular representations of C ∗ -algebras (Cuntz alge-bras, Cuntz Krieger algebras and C ∗ (Λ)) studied in [7] give “scaling and translation” operatorsthat relate the eigenspaces to wavelets/wavelet decompositions in general cases. In [9], authorsassociated a finite strongly connected k -graph Λ to a stationary k -Bratteli diagram B Λ andobtained Cantor sets from the infinite path space ∂ B Λ of B Λ and ultrametric d w δ associatedto weight functions w δ on ∂ B Λ , where δ ∈ (0 , Then they [9] were also able to associatePearson-Bellissard spectral triples to the k -graph Λ under a mild hypothesis and obtain similarresults to those of [19] for ζ -function, Dixmier trace and Laplace-Beltrami operators ∆ s andtheir eigenspaces. Note that the Dixmier trace induces a self-similar measure µ on ∂ B Λ whichagrees with the Perron-Frobenius measure M introduced in [13] under a mild hypothesis. (SeeCorollary 3.10 of [9]).For the spectral triple ( C Lip ( ∂ B Λ ) , π φ , H , D, Γ) , the Dirichlet form Q s on L ( ∂ B Λ , µ ) isgiven by Q s ( f, g ) = 12 Z Ξ Tr (cid:0) | D | − s [ D, π φ ( f )] ∗ [ D, π φ ( g )] (cid:1) dν ( φ ) , where Ξ is the space of choice functions on ∂ B Λ × ∂ B Λ and ν is a measure induced from µ .Then a classical theory implies that there exists a non-positive definite self-adjoint operator This number δ ∈ (0 ,
1) is the abscissa of convergence of the ζ δ -function associated to the spectral triples.See Theorem 3.8 of [9]. See the details in Section 3. s , called the Laplace-Beltrami operator, given by h− ∆ s ( f ) , g i = Q s ( f, g ) . (See Section 3 of this paper and Section 8.3 of [19]). With the self-similar measure µ on ∂ B Λ (given in (3)), we get a metric measure space ( ∂ B Λ , µ, d w δ ). Note that the Markovian semigroupor the heat kernel associated to the Dirichlet form Q s has not been investigated in [9].According to Section 13 of [16], Kigami showed that the Dirichlet form associated to Pearson-Bellissard spectral triple of an ultrametric Cantor set induced from a weighted tree is a specialkind of the Dirichlet forms on Cantor sets constructed in [16] without a spectral triple. Infact, the Cantor set in [16] is obtained from a random walk on a tree, and the measure andthe Dirichlet form on it are given in terms of effective resistances (see [16, Section 2, Sec-tion 4]). Moreover, Kigami described how his theory is related to noncommutative geometryby identifying a generalized jump kernel for the Dirichlet form of [19] and comparing his results( ζ -function, self-similar measure, asymptotic behavior of the heat kernel) to those of [19] whenthe Cantor set is given by the complete binary tree.We think that Kigami’s observation on the complete binary tree can be extended to a moregeneral tree, in particular, a stationary k -Bratteli diagram, associated to a finite strongly con-nected k -graph. In this paper, we study the asymptotic behavior of the associated heat kerneland identify the jump kernel J s for the process associated to the Dirichlet form Q s . It turnedout that the Dirichlet form Q s coincides with the Dirichlet form Q J s ,µ given in Definition 4.2,where J s is a generalized jump kernel given in (15) (see Proposition 4.3). Also, the asymptoticbehavior of the heat kernel p t associated to Q s describes the jump process across the gaps ofthe Cantor set (see Theorem 4.8). Note that Chen and Kumagai [4] have investigated the heatkernel estimates for this kind of jump processes associated to a jump-type Dirichlet form suchas Q J s ,µ .We begin in Section 2 with definitions of k -graphs and k -Bratteli diagrams and discusshow we obtain ultrametric Cantor sets C using weighted k -Bratteli diagrams associated to k -graphs. In Section 3, we first review the construction of Pearson-Bellissard’s spectral triplesand Dirichlet forms Q s , s ∈ R on C . Then we show in Proposition 3.2 that the Laplace-Beltramioperator ∆ s associated to Q s is in fact unbounded when s < δ , where δ ∈ (0 , d ( s ) associated to the eigenvalues of ∆ s in Proposition 3.3.In Section 4, we first show that the intrinsic ultrametric d ( s ) has the volume doubling propertywith respect to the self-similar measure µ on the infinite path space ∂ B Λ . We prove our maintheorem, which states that both ultrametrics d ( s ) and d w δ are asymptotically equivalent inthe sense that d ( s ) ( x, y ) ≍ ( d w δ ( x, y )) δ − s , where δ ∈ (0 ,
1) and see (19) for ‘ ≍ ’. Finally, wediscuss the heat kernel and its asymptotic behavior and moments of displacement of the processassociated to the Dirichlet form Q s in Theorem 4.8. Acknowledgement
The first author J.H. and the third author Y.L. were supported by National Research Founda-tion of Korea (NRF) grant funded by the Korea government (MEST) No.NRF-2015R1A3A2031159.The second author S.K. was supported by Basic Science Research Program through NRF grantfunded by the Ministry of Education No.NRF-2017R1D1A1B03034697.3 k -graphs and ultrametric Cantor sets k -graphs and k -Bratteli diagrams Throughout this article, N denotes the non-negative integers and | S | denotes the number ofelements in a set S unless specified otherwise.A directed graph is given by a quadruple E = ( E , E , r, s ), where E is the set of verticesof the graph, E is the set of edges, and maps r, s : E → E denote the range and source ofeach edge. A vertex v in a directed graph E is a source if r − ( v ) = ∅ . Definition 2.1. [17, Definition 1.1] A higher-rank graph (or k-graph) is a small category Λequipped with a degree functor d : Λ → N k satisfying the factorization property : whenever λ is a morphism in Λ such that d ( λ ) = m + n , there are unique morphisms µ, ν ∈ Λ such that d ( µ ) = m, d ( ν ) = n , and λ = µν .We often consider k -graphs as a generalization of directed graphs. For n ∈ N k , we writeΛ n := { λ ∈ Λ : d ( λ ) = n } , and we call morphisms λ ∈ Λ n paths with degree n ∈ N k . If n = 0 ∈ N k , then Λ is the set ofobjects of Λ, which we also refer to as the vertices of Λ. There are maps r, s : Λ → Λ whichidentify the range and source of each morphism, respectively. For v, w ∈ Λ and n ∈ N k , wedefine two sets v Λ n := { λ ∈ Λ n : r ( λ ) = v } , v Λ w = { λ ∈ Λ : r ( λ ) = v, s ( λ ) = w } . We say that a k -graph Λ is finite if | Λ n | < ∞ for all n ∈ N k ; Λ has no sources (or is source-free )if | v Λ n | > v ∈ Λ and n ∈ N k ; Λ is strongly connected if, for all v, w ∈ Λ , v Λ w = ∅ .For each 1 ≤ i ≤ k , we write e i for the standard basis vector of N k , and define a matrix A i ∈ M Λ ( N ) by A i ( v, w ) = | v Λ e i w | . We call A i the i -th vertex matrix of Λ. Note that the factorization property implies that thematrices A i commute, i.e. A i A j = A j A i for 1 ≤ i, j ≤ k . In fact, for a pair of composableedges f ∈ v Λ e i u , g ∈ u Λ e j w , the factorization property implies that there exist unique edges˜ f ∈ Λ e i w , ˜ g ∈ v Λ e j such that f g = ˜ g ˜ f since d ( f g ) = e i + e j = e j + e i = d (˜ g ˜ f ). Example 2.2.
If we let Λ E be the category of finite paths of a directed graph E , then Λ E is a -graph with the degree functor d : Λ E → N which takes a finite path λ to its length | λ | (thenumber of edges making up λ ).Another fundamental example of a k -graph is the following. For any k ≥ , let Ω k be thesmall category with Obj(Ω k ) = N k and Mor(Ω k ) = { ( p, q ) ∈ N k × N k : p ≤ q } where p ≤ q means that each entry p i is less than and equal to q i for ≤ i ≤ k . The range and source maps r, s : Mor(Ω k ) → Obj(Ω k ) are given by r ( p, q ) = p and s ( p, q ) = q . If we define d : Ω k → N k by d ( p, q ) = q − p , then one can show that (Ω k , d ) is a k -graph. k -graph Λ by a k -graph morphism(meaning degree-preserving functor) x : Ω k → Λ and we write Λ ∞ for the set of all infinitepaths in Λ. We endow Λ ∞ with the topology generated by the collection of cylinder sets { Z ( λ ) : λ ∈ Λ } , where the cylinder set of λ is defined by Z ( λ ) = { λx ∈ Λ ∞ | s ( λ ) = r ( x ) } = { y ∈ Λ ∞ | y (0 , d ( λ )) = λ } . We note that Z ( λ ) is compact open for all λ ∈ Λ. We also see that Λ is a finite k -graph if andonly if Λ ∞ is compact.Consider a family of commuting N × N matrices { A , . . . , A k } with non-negative entries.We say that the family { A , . . . , A k } is irreducible if for each nonzero matrix A i , there exists afinite subset F ⊂ N k such that A F ( s, t ) > s, t ∈ N , where for n = ( n , . . . , n k ) ∈ N k and a finite subset F of N k , we write A n := k Y i =1 A n i i , and A F := X n ∈ F A n . By [13, Lemma 4.1], a k -graph Λ is strongly connected if and only if the family of vertexmatrices { A , . . . , A k } for Λ is irreducible. So, if a finite k -graph Λ is strongly connected, thenProposition 3.1 of [13] implies that there is a unique positive vector κ Λ ∈ (0 , ∞ ) Λ such that P v ∈ Λ κ Λ v = 1 and A i κ Λ = ρ i κ Λ for all 1 ≤ i ≤ k where ρ i := ρ ( A i ) denotes the spectral radius of A i . Such an eigenvector κ Λ is called the (unimodular) Perron-Frobenius eigenvector of Λ. Proposition 8.1 of [13] shows that there is aunique Borel probability measure M on Λ ∞ that satisfies the self-similarity condition given by M ( Z ( λ )) = ρ (Λ) − d ( λ ) κ Λ s ( λ ) for all λ ∈ Λ , (1)where Z ( λ ) is a cylinder set of λ ∈ Λ and κ Λ is the unique Perron-Frobenius eigenvector of Λ.More precisely, for λ ∈ Λ M ( Z ( λ )) = ρ − d ( λ ) . . . ρ − d ( λ ) k k κ Λ s ( λ ) , (2)where ρ (Λ) = ( ρ , . . . , ρ k ) and d ( λ ) = ( d ( λ ) , . . . , d ( λ ) k ). The measure M is often called thePerron-Frobenius measure on Λ ∞ .Now we describe Bratteli diagrams and k -Bratteli diagrams introduced in [2, 9] as follows. Definition 2.3. [2, 9] A
Bratteli diagram denoted by B is a directed graph with a vertex set B = F n ∈ N V n , and an edge set B = F ∞ n =1 E n , where E n consists of edges whose source vertexlies in V n +1 and whose range vertex lies in V n . A finite path η = η · · · η ℓ is a finite sequence ofedges with s ( η n ) = r ( η n +1 ). Definition 2.4. [9, Definition 2.3] Given a Bratteli diagram B , the set of all infinite paths ∂ B is defined by ∂ B = { ( x n ) ∞ n =1 : x n ∈ E n and s ( x n ) = r ( x n +1 ) for all n ∈ N \ { }} . Since y ∈ Λ ∞ , we have y : Ω k → Λ by definition. So, (0 , d ( λ )) ∈ Mor(Ω k ) ⊂ N k × N k and the image of(0 , d ( λ )) under y should be an element of Λ by definition. Thus, y (0 , d ( λ )) = λ ∈ Λ means that the initial pathof the infinite path y with degree d ( λ ) is the same as λ . For λ ∈ Λ, M ( Z ( λ )) = ρ (Λ) − d ( λ ) M ( Z ( s ( λ ))). η = η η · · · η ℓ of B , we define the cylinder set [ η ] by[ η ] = { x = ( x n ) ∞ n =1 ∈ ∂ B : x i = η i for all 1 ≤ i ≤ ℓ } . We note the collection T of all cylinder sets forms a compact open sub-basis for a locallycompact Hausdorff topology on ∂ B ; we will always consider ∂ B with this topology. Definition 2.5. [9, Definition 2.5] Let A , . . . , A k be commuting N × N matrices with non-negative integer entries. The stationary k -Bratteli diagram associated to the matrices A , . . . , A k is given by a filtered set of vertices B = F n ∈ N V n and a filtered set of edges B = F ∞ n =1 E n ,where the edges in E n go from V n +1 to V n , such that(i) for each n ∈ N , each V n consists of N vertices, which we will label 1 , . . . , N ;(ii) when n ≡ i (mod k ), there are A i ( p, q ) number of edges whose range is the vertex p of V n and whose source is the vertex q of V n +1 . Remark . For a finite strongly connected k -graph Λ, let B Λ be the corresponding stationary k -Bratteli diagram. We first note that Proposition 2.10 of [9] implies that the infinite path spaceΛ ∞ of Λ is homeomorphic to the infinite path space ∂ B Λ of B Λ . Thus, we can obtain the uniqueprobability measure µ on ∂ B Λ satisfying self-similarity condition by transferring the Perron-Frobenius measure M on Λ ∞ given in (1) as follows. Since we have V n ∼ = Λ for all n ∈ N , wewrite ( κ v ) v ∈ Λ (or ( κ v ) v ∈ V n ) again for the corresponding Perron-Frobenius eigenvectors at eachlevel of B Λ . Then the induced unique probability measure µ on ∂ B Λ is given by µ [ η ] = (cid:16) ρ q ρ . . . ρ t (cid:17) κ Λ s ( η ) , (3)where ρ = ρ . . . ρ k and η is a finite path in B Λ with r ( η ) ∈ V and | η | = qk + t for some q, t ∈ N and 0 ≤ t ≤ k − Example 2.7.
Here we consider two examples of -graphs with one vertex. The first one is a -graph with one vertex, two blue edges and two red edges and the second one is a -graph withone vertex, two blue edges and one red edge.First consider a -colored graph with one vertex v , two blue edges e , e and two red edges f , f : ve e f f Then there is a -graph Λ with the above skeleton and factorization rules given by e f = f e e f = f e , e f = f e , e f = f e . (4)6 hen the associated vertex matrices are A = (2) and A = (2) and the associated stationaryBratteli diagram B Λ is given by v v v v v . . . . Since the spectral radii of both A and A are , Proposition 2.17 of [9] implies that the infinitepath space ∂ B Λ is a Cantor set. According to [21], Λ is periodic with minimal period (1 , − and the corresponding C ∗ -algebra can be identified with C ( T ) ⊗ A , where A is simple. (See thedetails in Theorem 3.1 and Section 5 of [21]).The Perron-Frobenius eigenvector at each level of B Λ is given by ( κ v ) v ∈ Λ = 1 since thereis only one vertex v in Λ . Also note that any finite path γ in the Bratteli diagram B Λ can berepresented as γ = e i f i e i f i . . . e i n , or γ = e i f i e i f i . . . e i m − f i m , where i k ∈ { , } . The self-similar measure µ on ∂ B Λ is given by µ ([ γ ]) = (cid:16) (cid:17) n · (cid:16) (cid:17) n · (cid:16) (cid:17) ( n + n ) , where γ is a finite path of B Λ with n blue edges and n red edges.Now let Λ ′ be the -graph with one vertex v , two blue edges e , e and one red edge f thatsatisfies the factorization property: e f = f e and f e = e f. Then the associated vertex matrices are A = (2) and A = (1) and the associated stationaryBratteli diagram B Λ ′ is given by v v v v v . . . . Since spectral radii of A and A are ρ = 2 and ρ = 1 , and ρ ρ = 2 > , Proposition 2.17of [9] implies that the infinite path space ∂ B Λ ′ is a Cantor set. As described in Example 4.10 of[10], ∂ B Λ ′ is homeomorphic to Q ∞ i =1 Z . Moreover, there is a Λ ′ -semibranching function systemassociated to this -graph which gives rise to a monic representation of C ∗ (Λ ′ ) . (See the detailsin Example 3.6 of [11] and Example 4.10 of [10]).The self-similar measure µ on ∂ B Λ ′ is given by µ ([ η ]) = (cid:16) (cid:17) m · (cid:16) m · (cid:16) (cid:17) m , where η is a finite path of B Λ with m blue edges and m red edges. .2 Weights and ultrametrics For a Bratteli diagram (or stationary k -Bratteli diagram) B and n ∈ N , we let F n B = { η ∈ B : r ( η ) ∈ V , | η | = n } , where | η | is the length of η , and let F B = ∪ n ∈ N F n B . Definition 2.8. [19, 14, 9] A weight on B is a function w : F B → R + such that(i) for any vertex v ∈ V , w ( v ) < n →∞ sup { w ( λ ) : λ ∈ F n B} = 0;(iii) if η is a sub-path of λ , then w ( λ ) ≤ w ( η ).For x, y ∈ ∂ B , we write x ∧ y for the longest common initial sub-path of x and y . If r ( x ) = r ( y ), then we define x ∧ y = ∅ .We say that a metric d on a Cantor set C is an ultrametric if d induces the Cantor settopology and satisfies d ( x, y ) ≤ max { d ( x, z ) , d ( y, z ) } for all x, y, z ∈ C . It is not hard to see that a weight w on a Bratteli diagram gives an ultrametric d w as follows. Proposition 2.9. [9, Proposition 2.15] (c.f. [19, 14, 8]) Let B be a Bratteli diagram with aweight w . Suppose that ∂ B is a Cantor set. The function d w : ∂ B × ∂ B → R + given by d w ( x, y ) = if x ∧ y = ∅ if x = yw ( x ∧ y ) otherwise (5) is an ultrametric on ∂ B . Moreover, d w metrizes the cylinder set topology on ∂ B . It is shown in [9] that we can associate a k -graph to an ultrametric Cantor set under a mildhypothesis as follows. Let Λ be a finite strongly connected k -graph with vertex matrices A i andsuppose that each spectral radius ρ i of A i is bigger than 1. Then Proposition 2.17 of [9] impliesthat the infinite path space Λ ∞ is a Cantor set, and hence the infinite path space ∂ B Λ of thecorresponding stationary k -Bratteli diagram B Λ is also a Cantor set since Λ ∞ is homeomorphicto ∂ B Λ . Also, Proposition 2.19 of [9] implies that there exists a weight ω δ on B Λ for δ ∈ (0 , λ ∈ F B with | λ | = n ∈ N , we write n = qk + t for some q, t ∈ N with0 ≤ t ≤ k −
1. For each δ ∈ (0 , , we define w δ : F B → R + by w δ ( η ) = (cid:0) ρ q +11 · · · ρ q +1 t ρ qt +1 · · · ρ qk (cid:1) − /δ κ Λ s ( η ) , (6)where κ Λ is the unimodular Perron-Frobenius eigenvector for Λ. Moreover, Proposition 2.9above implies that the weight ω δ induces an ultrametric d ω δ on ∂ B and the metric topologyinduced by d w δ agrees with the cylinder set topology of ∂ B Λ . Thus if we let B d wδ ( x, r ) be theclosed ball of center x and radius r >
0, then B d wδ ( x, r ) = [ x . . . x n ] for some n ∈ N , where x = ( x n ) ∞ n =1 ∈ ∂ B Λ . (See the details of the proof in Proposition 2.15 of [9]).8 xample 2.10. For the -graph Λ given in Example 2.7, there is a weight w δ on B Λ given by w δ ( γ ) = 2 − n n δ where δ ∈ (0 , and γ is a finite path of B Λ with n blue edges and n red edges.Similarly for the -graph Λ ′ given in Example 2.7, there is a weight w ′ δ on B Λ ′ given by w ′ δ ( η ) = 2 − m δ where δ ∈ (0 , and η is a finite path of B Λ ′ with m blue edges and m red edges. For the rest of this paper, we denote by Λ and B Λ a finite, strongly connected k -graph andits associated stationary k -Bratteli diagram with infinite path space ∂ B Λ , respectively, unlessspecified otherwise.According to [9], there exists a family of spectral triples for the ultrametric Cantor setassociated to Λ under mild hypotheses. The associated ζ -function is also regular and themeasure µ on ∂ B Λ induced by the Dixmier trace turned out to be equivalent to the Perron-Frobenius measure M on Λ ∞ . Moreover, the family of spectral triples induces Laplace-Beltramioperators ∆ s , s ∈ R given via the Dirichlet forms as below. In this section, we first reviewthem, and then investigate the asymptotic behavior of the eigenvalues of ∆ s and the associatedultrametric d ( s ) called the intrinsic metric. Definition 3.1. [19, 14, 9] An odd spectral triple is a triple ( A , H , D ) consisting of a Hilbertspace H , an involutive algebra A of (bounded) operators on H and a densely defined self-adjointoperator D that has compact resolvent such that [ D, π ( a )] is a bounded operator for all a ∈ A ,where π is a faithful bounded ∗ -representation of A on H . An even spectral triple is an oddspectral triple with a grading operator (meaning self adjoint and unitary) Γ on H such thatΓ D = − D Γ, and Γ π ( a ) = π ( a )Γ for all a ∈ A .We first suppose that ∂ B Λ is a Cantor set. For any δ ∈ (0 , w δ be the weight on B Λ given in (6) and we denote by d w δ the induced ultrametric on ∂ B Λ as mentioned in the previoussection. Let F B Λ be the set of finite paths in B Λ whose ranges are in the first vertex set V of B Λ (as in the beginning of section 2.2). Suppose, in addition, that for λ ∈ F B Λ , we havediam[ λ ] = w δ ( λ ) , (7)where diam[ λ ] = sup { d w δ ( x, y ) | x, y ∈ [ λ ] } . Then as in [9] (c.f. [19, 14]), we can construct afamily of spectral triples for the ultrametric Cantor set ( ∂ B Λ , d w δ ) as follows. A choice function for ( ∂ B Λ , d w δ ) is a map φ : F B Λ → ∂ B Λ × ∂ B Λ such that φ ( λ ) =( φ ( λ ) , φ ( λ )) ∈ [ λ ] × [ λ ] and d w δ ( φ ( λ ) , φ ( λ )) = diam[ λ ] . (8) Note that the representation π is often included in the notation for a spectral triple. The construction of the spectral triple for ( ∂ B Λ , d w δ ) works for a ultrametric Cantor set induced by anyweighted tree as in [19]. Also note that the family of the spectral triples is indexed by the space of choicefunctions. d w δ ( φ ( λ ) , φ ( λ )) = diam[ λ ] = w δ ( λ ) by (7)). Here φ ( λ ) and φ ( λ ) are infinitepaths in [ λ ] and the subscripts 1,2 imply that the choice function gives a pair of (distinct)infinite paths in [ λ ] satisfying (8). The condition in (8) means that φ ( λ ) and φ ( λ ) satisfy φ ( λ ) ∈ [ λe ] and φ ( λ ) ∈ [ λe ′ ] for two different edges e, e ′ . According to [19], the space of choicefunctions is the analogue of the sphere bundle of a Riemannian manifold.We denote by Ξ the space of choice functions for ( ∂ B Λ , d w δ ). Since ∂ B Λ is a Cantor set, Ξis nonempty since it implies that for every finite path λ of B Λ we can find two distinct infinitepaths x, y ∈ [ λ ] such that φ ( λ ) = x and φ ( λ ) = y . (See Proposition 2.4 of [9]).Let C Lip ( ∂ B Λ ) be the pre- C ∗ -algebra of Lipschitz continuous functions on ( ∂ B Λ , d w δ ) and let H = ℓ ( F B Λ ) ⊗ C . For φ ∈ Ξ, we define a ∗ -representation π φ of C Lip ( ∂ B Λ ) on H by π φ ( g ) = M λ ∈ F B Λ (cid:18) g ( φ ( λ )) 00 g ( φ ( λ )) (cid:19) . Then we define a
Dirac-type operator D on H by D = M λ ∈ F B Λ w δ ( λ ) (cid:18) (cid:19) . The grading operator
Γ is given byΓ = 1 ℓ ( F B Λ ) ⊗ (cid:18) − (cid:19) . Then one can show that π φ is a faithful ∗ -representation, and the unbounded operator D isself-adjoint with compact resolvent and the commutator [ D, π φ ( g )] is a bounded operator forall g ∈ C Lip ( ∂ B Λ ). Moreover, one can check that[Γ , π φ ( g )] = 0 for all g ∈ C Lip ( ∂ B Λ ) , and Γ D = − D Γ. Hence, we obtain an even spectral triple ( C Lip ( ∂ B Λ ) , π φ , H , D, Γ) for eachchoice function φ ∈ Ξ. (See the details in [19, 14, 9]).As defined in [9], the associated ζ δ -function is given by ζ δ ( s ) = 12 Tr( | D | − s )where s ∈ C . Since we have assumed the equation (7), the abscissa s of convergence of the ζ δ -function is δ by [9, Theorem 3.8]. Moreover, the associated Dixmier trace induces a measure µ w δ on ∂ B Λ by the formula µ w δ [ γ ] = µ w δ ( χ γ ) = lim s → s Tr( | D | − s π φ ( χ γ ))Tr( | D | − s )where χ γ is the characteristic function on a cylinder set [ γ ].According to Corollary 3.10 of [9], the induced Dixmier trace measure µ w δ is finite andindependent of δ under a mild hypothesis , so we denote by µ without subscript. Moreover, The product of the vertex matrices A = A . . . A k is irreducible µ agrees with the probability measure on ∂ B Λ given in (3). Hence, for λ ∈ F B Λ with | λ | = qk + t (0 ≤ t ≤ k − µ is given by µ [ λ ] = ( ρ . . . ρ t ) − ( q +1) ( ρ t +1 . . . ρ k ) − q κ s ( λ ) . (See Theorem 3.9 and Corollary 3.10 of [9] for the details).Now we describe the associated Dirichlet form and Laplace-Beltrami operator as follows.Recall that F B Λ is the set of finite paths in B Λ whose ranges are in the first vertex set V of B Λ . Let µ be the induced measure on ∂ B Λ as above and let χ λ be the characteristic functionof the cylinder set [ λ ] for λ ∈ F B Λ . For each s ∈ R , we define a sesquilinear form Q s on thedense subspace Dom( Q s ) := span { χ λ : λ ∈ F B Λ } of L ( ∂ B Λ , µ ) by Q s ( f, g ) = 12 Z Ξ Tr (cid:0) | D | − s [ D, π φ ( f )] ∗ [ D, π φ ( g )] (cid:1) dν ( φ ) (9)for f, g ∈ Dom( Q s ), where ν is the normalized measure on the set Ξ of choice functions givenby the measure µ . In particular, we have that for λ ∈ F B Λ , ν λ = µ × µ P ( e,e ′ ) ∈ ext ( λ ) µ [ λe ] µ [ λe ′ ] (10)where ext ( λ ) is the set of ordered pairs of distinct edges ( e, e ′ ) which extend γ one generationfurther, i.e., e = e ′ , r ( e ) = r ( e ′ ) = s ( λ ), and | e | = | e ′ | = 1.Using the similar argument given in Section 8 of [19], one can verify that the formula of Q s given in (9) gives rise to a closable Dirichlet form on L ( ∂ B Λ , µ ) as follows. Note first that for λ ∈ F B Λ , π ψ ( χ λ ) = M γ ∈ F B Λ (cid:18) χ λ ( ψ ( γ )) 00 χ λ ( ψ ( γ )) (cid:19) = π ψ ( χ λ ) ⊕ π ψ ( χ λ ) ⊕ π ψ ( χ λ ) , where π ψ ( χ λ ) = M γ ∈ F B Λ [ λ ] ⊆ [ γ ] (cid:18) χ λ ( ψ ( γ )) 00 χ λ ( ψ ( γ )) (cid:19) , π ψ ( χ λ ) = M γ ∈ F B Λ [ γ ] [ λ ] (cid:18) χ λ ( ψ ( γ )) 00 χ λ ( ψ ( γ )) (cid:19) ,π ψ ( χ λ ) = M γ ∈ F B Λ [ λ ] * [ γ ] and [ γ ] * [ λ ] (cid:18) χ λ ( ψ ( γ )) 00 χ λ ( ψ ( γ )) (cid:19) . Also note that D = M γ ∈ F B Λ w δ ( γ ) (cid:18) (cid:19) = D ⊕ D ⊕ D , where D = M γ ∈ F B Λ [ λ ] ⊆ [ γ ] w δ ( γ ) (cid:18) (cid:19) , D = M γ ∈ F B Λ [ γ ] [ λ ] w δ ( γ ) (cid:18) (cid:19) , D = M γ ∈ F B Λ [ λ ] * [ γ ] and [ γ ] * [ λ ] w δ ( γ ) (cid:18) (cid:19) . D , π ψ ( χ λ ) ] = 0. Also D π ψ ( χ λ ) = π ψ ( χ λ ) D , andhence [ D , π ψ ( χ λ ) ] = 0. Thus[ D, π ψ ( χ λ )] = [ D , π ψ ( χ λ ) ] ⊕ [ D , π ψ ( χ λ ) ] ⊕ [ D , π ψ ( χ λ ) ]= [ D , π ψ ( χ λ ) ] ⊕ ⊕ , which is finite rank. This implies that Q s given in (9) is valid for all f, g ∈ Dom( Q s ). Now thesimilar arguments given in the proof of Theorem 4 of [19, Section 8] shows that Q s is a closableDirichlet form for all s ∈ R with dense domain Dom( Q s ). Then Theorem 7 of [19] implies thatthere exists a non-positive definite self-adjoint operator ∆ s such that h− ∆ s f, g i = Q s ( f, g ) , and that ∆ s generates a Markovian semigroup. (See details in Section 8.3 of [19]).According to [14, Section 4.1], one can compute the formula for ∆ s ( χ γ ) as follows, and henceone can obtain the eigenvalues and the corresponding eigenvectors explicitly. Fix a finite path γ ∈ F B Λ with | γ | = n . For 0 ≤ k ≤ n = | γ | , we write γ k = γ (0 , k ) for the initial sub-path of γ with length k . Then we have∆ s ( χ γ ) = − n − X k =0 G s ( γ k ) (cid:0) ( µ [ γ k ] − µ [ γ k +1 ]) χ γ − µ [ γ ]( χ γ k − χ γ k +1 ) (cid:1) , where G s ( η ) = 12 w δ ( η ) − s X ( e,e ′ ) ∈ ext ( η ) µ [ ηe ] µ [ ηe ′ ]as in (4.2a), (4.2b) of [14]. Moreover, Theorem 4.3 of [14] implies that we can show that ∆ s hasa pure point spectrum, and one can compute the eigenvalues and eigenspaces for ∆ s explicitlyas follows. In particular, we note that the values of λ s,γ are non-positive and compute theeigenvalues λ s,γ such that ∆ s χ γ = λ s,γ χ γ . For γ ∈ F B Λ , λ s,γ = n − X k =0 µ [ γ k +1 ] − µ [ γ k ] G s ( γ k ) − µ [ γ ] G s ( γ ) (11)is an eigenvalue with the eigenspace E s,γ = span n χ γe µ [ γe ] − χ γe ′ µ [ γe ′ ] : ( e, e ′ ) ∈ ext ( γ ) o . (12)We investigate for which values of s the eigenvalue λ s,γ goes to −∞ as | γ | → ∞ . Note thatthis behavior has not been investigated before. Proposition 3.2.
For any δ ∈ (0 , , let w δ be the weight given in (6) and d w δ be the associatedultrametric given in Proposition 2.9 on ∂ B Λ . Let µ be the probability measure on ∂ B Λ given in (3) . Let { ρ i : 1 ≤ i ≤ k } be the set of the spectral radii of all vertex matrices of Λ , and supposethat ρ i > for all ≤ i ≤ k . Let ∆ s , s ∈ R be the Laplace-Beltrami operator associated to theDirichlet form Q s given in (9) and let λ s,γ be its eigenvalues given in (11) , where γ ∈ F B Λ . If s < δ , then for γ ∈ F B Λ , the eigenvalue λ s,γ goes to −∞ as | γ | → ∞ . roof. Fix a finite path γ in B Λ . For simplicity, we drop the subscript s from λ s,γ for the proof.In order to simplify the computation, we write the formula of λ γ in (11) as λ γ = A − B, where A = | γ |− X n =0 µ [ γ n +1 ] − µ [ γ n ] G s ( γ n ) , B = µ [ γ ] G s ( γ ) . Using the formulas for the measure µ and the weight w δ on ∂ B , we compute A = | γ |− X n =0 (cid:0) w δ ( γ n ) (cid:1) s − ( µ [ γ n +1 ] − µ [ γ n ]) P ( e,e ′ ) ∈ ext ( γ n ) µ [ γ n e ] · µ [ γ n e ′ ] . If n = qk + t with 0 ≤ t ≤ k −
1, then we have that µ [ γ n ] = ( ρ . . . ρ t ) − ( q +1) ( ρ t +1 . . . ρ k ) − q κ Λ s ( γ n ) ,µ [ γ n +1 ] = ( ρ . . . ρ t +1 ) − ( q +1) ( ρ t +2 . . . ρ k ) − q κ Λ s ( γ n +1 ) ,µ [ γ n e ] = ( ρ . . . ρ t +1 ) − ( q +1) ( ρ t +2 . . . ρ k ) − q κ Λ s ( e ) ,µ [ γ n e ′ ] = ( ρ . . . ρ t +1 ) − ( q +1) ( ρ t +2 . . . ρ k ) − q κ Λ s ( e ′ ) , (cid:0) w δ ( γ n ) (cid:1) s − = (cid:16) ( ρ . . . ρ t ) − ( q +1) /δ ( ρ t +1 . . . ρ k ) − q/δ κ Λ s ( γ n ) (cid:17) s − . Thus, we obtain that A = 2 | γ |− X n =0 ( ρ . . . ρ t ) − ( q +1) ( ρ t +1 . . . ρ k ) − q (cid:16) κ Λ s ( γ n +1 ) ρ t +1 − κ Λ s ( γ n ) (cid:17) × (cid:16) ( ρ . . . ρ t ) − ( q +1) /δ ( ρ t +1 . . . ρ k ) − q/δ κ Λ s ( γ n ) (cid:17) s − X ( e,e ′ ) ∈ ext ( γ n ) ( ρ . . . ρ t +1 ) − q +1) ( ρ t +2 . . . ρ k ) − q κ Λ s ( e ) κ Λ s ( e ′ ) , so that A = 2 | γ |− X n =0 (cid:0) κ Λ s ( γ n +1 ) ρ t +1 − κ Λ s ( γ n ) (cid:17) ( ρ . . . ρ t ) − ( q +1)( s − δ +( q +1) ( ρ t +1 . . . ρ k ) − q ( s − δ + q ( κ Λ s ( γ n ) ) s − ρ − t +1 X ( e,e ′ ) ∈ ext ( γ n ) κ Λ s ( e ) κ Λ s ( e ′ ) . To compute B , let | γ | = q ′ k + t ′ with 0 ≤ t ′ ≤ k −
1. Then we have B = µ [ γ ] 2 (cid:0) w δ ( γ ) (cid:1) s − P ( e,e ′ ) ∈ ext ( γ ) µ [ γe ] · µ [ γe ′ ]= 2( ρ . . . ρ t ′ ) − ( q ′ +1) ( ρ t ′ +1 . . . ρ k ) − q ′ κ Λ s ( γ ) (cid:16) ( ρ . . . ρ t ′ ) − ( q ′ +1) δ ( ρ t ′ +1 . . . ρ k ) − q ′ δ κ Λ s ( γ ) (cid:17) s − P ( e,e ′ ) ∈ ext ( γ ) ( ρ . . . ρ t ′ +1 ) − q ′ +1) ( ρ t ′ +2 . . . ρ k ) − q ′ κ Λ s ( e ) κ Λ s ( e ′ ) = 2( ρ . . . ρ t ′ ) − ( q ′ +1)( s − δ +( q ′ +1) ( ρ t ′ +1 . . . ρ k ) − q ′ ( s − δ + q ′ ( κ Λ s ( γ ) ) s − ρ − t ′ +1 P ( e,e ′ ) ∈ ext ( γ ) κ Λ s ( e ) κ Λ s ( e ′ ) . λ γ = A − B = 2 | γ |− X n =0 (cid:16) κ Λ s ( γ n +1 ) ρ t +1 − κ Λ s ( γ n ) (cid:17) ( ρ . . . ρ t ) − ( q +1)( s − δ +( q +1) ( ρ t +1 . . . ρ k ) − q ( s − δ + q ( κ Λ s ( γ n ) ) s − ρ − t +1 P ( e,e ′ ) ∈ ext ( γ n ) κ Λ s ( e ) κ Λ s ( e ′ ) − ρ . . . ρ t ′ ) − ( q ′ +1)( s − δ +( q ′ +1) ( ρ t ′ +1 . . . ρ k ) − q ′ ( s − δ + q ′ ( κ Λ s ( γ ) ) s − ρ − t ′ +1 P ( e,e ′ ) ∈ ext ( γ ) κ Λ s ( e ) κ Λ s ( e ′ ) . We first note that λ γ are eigenvalues of a non-positive operator ∆ s , we must have κ Λ s ( γ n +1 ) ρ t +1 − κ Λ s ( γ n ) < κ Λ s ( γ n +1 ) ρ t +1 − κ Λ s ( γ n ) = 0for all n .Now suppose that κ Λ s ( γn +1) ρ t +1 − κ Λ s ( γ n ) <
0. Then there are three cases to consider.
Case 1 : If 2 < s < δ , then s − > ( s − δ <
1. We have − ( q + 1) (cid:0) s − δ (cid:1) + ( q + 1) = ( q + 1) (cid:0) − ( s − δ (cid:1) > , so that ( ρ · · · ρ t ) − ( q +1)( s − δ )+( q +1) → ∞ as q → ∞ since ρ i > i ∈ { , . . . , k } . Similarly, we have − q ( s − δ ) + q > , which implies that ( ρ t +1 · · · ρ k ) − q ( s − δ )+ q → ∞ as q → ∞ . Therefore, we have that A → −∞ as | γ | → ∞ . A similar argument also shows that B → ∞ as | γ | → ∞ , and hence λ γ = A − B → −∞ as | γ | → ∞ . Case 2 : If s <
2, then s − <
0. So, we have − ( q + 1)( s − δ ) + ( q + 1) > , and hence one can show that λ γ → −∞ as | γ | → ∞ . Case 3 : If s = 2, then we get − ( q + 1)( s − δ ) + ( q + 1) = q + 1 > . Thus, a similar argument shows that λ γ → −∞ as | γ | → ∞ . Therefore if s < δ , then theeigenvalue λ γ → −∞ as | γ | → ∞ .Because of Proposition 3.2, the results in [16] imply that we can find an ultrametric d ( s ) ,called the intrinsic metric in [16], on ∂ B Λ associated to the eigenvalues of ∆ s as follows.14 roposition 3.3. Let ∆ s and λ s,γ be given in Proposition 3.2 and { ρ i : 1 ≤ i ≤ k } be the setof the spectral radii of all vertex matrices of Λ . Suppose that ρ i > for all ≤ i ≤ k . (a) If s < , then λ s,α < for any α ∈ F B Λ with | α | ≥ , and { λ s,x (0 ,n ) : n ∈ N } is strictlydecreasing for any x ∈ ∂ B Λ , i.e. λ s,x (0 ,n ) > λ s,x (0 ,n +1) . (b) Fix s ∈ R , and we define d ( s ) ( x, y ) = for x = y ∈ ∂ B Λ , − λ s,x ∧ y for x = y ∈ ∂ B Λ . Then d ( s ) is an ultrametric on ∂ B Λ . (c) Let the open ball of center x ∈ ∂ B Λ and radius a > for the metric d ( s ) be defined by B s ( x, a ) = { y | d ( s ) ( x, y ) < a } . Then for each x ∈ ∂ B Λ , we have B s ( x, a ) = [ x (0 , n )] if and only if − λ s,x (0 ,n ) < a ≤ − λ s,x (0 ,n − . Proof.
A straightforward computation gives (a) so we leave it to the readers. Also, (b) and (c)follow by similar arguments given in the proofs of Proposition 6.4(1)(3) of [16].
According to [16], we can prove various interesting results for the metric measure space thathas a volume doubling property with respect to an ultrametric on the space. The measure µ on ∂ B Λ has a volume doubling property with respect to the both ultrametric d ( s ) and d w δ on ∂ B Λ . The former will be proved in Proposition 4.1 and the latter will become clear when weshow Theorem 4.7. Moreover, we show that there exists a heat kernel p t of a process associatedto the Dirichlet form Q s and discuss the asymptotic behavior of p t after. For a metric measure space (
X, µ, d ), we define an open ball with radius r by B ( x, r ) = { y ∈ X | d ( x, y ) < r } for x ∈ X and r >
0. We say that the measure µ has the volume doubling property with respectto a metric d if there exists a constant c > µ ( B ( x, r )) ≤ c · µ ( B ( x, r ))15or any x ∈ X and any r > d ( s ) and d w δ defined onthe measure space ( ∂ B Λ , µ ) if the spectral radii ρ i of vertex matrices of Λ satisfy ρ i > ≤ i ≤ k . The volume doubling property of µ with respect to d w δ is essentially included inthe proof of Theorem 4.7. We first show that the measure µ has the volume doubling propertywith respect to d ( s ) as below. Proposition 4.1.
Suppose that the spectral radius ρ i of vertex matrices of Λ satisfies ρ i > for all ≤ i ≤ k . Let µ be the probability measure on ∂ B Λ given in (3) and d ( s ) be the intrinsicultrametric on ∂ B Λ given in Proposition 3.3. Then µ has the volume doubling property withrespect to d ( s ) for s < .Proof. Since s <
2, first note that sequence of eigenvalues { λ s,x (0 ,n ) } is strictly decreasing foreach x ∈ ∂ B Λ and the associated intrinsic metric d ( s ) is an ultrametric on ∂ B Λ by Proposition 3.3.We apply Theorem 6.5 of [16] to the metric measure space ( ∂ B Λ , µ, d ( s ) ) in order to prove theproposition. So it suffices to prove two things. First, we need to find c ∈ (0 ,
1) such that c ≤ µ [ x (0 , n )] µ [ x (0 , n − x ∈ ∂ B Λ and n ∈ N , (13)and second, we need to show that there exist m ≥ c ∈ (0 ,
1) such that λ s,x (0 ,n ) λ s,x (0 ,n + m ) < c for all x ∈ ∂ B Λ and n ∈ N . (14)To see the first claim, fix x ∈ ∂ B Λ and n ∈ N . Then let | x (0 , n ) | = n = qk + t for some q ∈ N and 0 ≤ t ≤ k −
1. Then we have that µ [ x (0 , n )] = (cid:16) ρ q ρ . . . ρ t (cid:17) κ Λ s ( x (0 ,n )) ,µ [ x (0 , n − (cid:16) ρ q ρ . . . ρ t − (cid:17) κ Λ s ( x (0 ,n )) , so that we get µ [ x (0 , n )] µ [ x (0 , n − ρ t · κ Λ s ( x (0 ,n )) κ Λ s ( x (0 ,n − > . Since the right-hand side of above equation is positive, there exists c ∈ (0 ,
1) that satisfies(13).For the second claim, recall that the sequence of eigenvalues { λ s,x (0 ,n ) } is strictly decreasingfor each x ∈ ∂ B Λ if s <
2. Thus, we have that0 < λ s,x (0 ,n ) λ s,x (0 ,n + m ) < m ≥ . Hence there exists c ∈ (0 ,
1) that satisfies (14). Therefore µ has the volume doubling propertywith respect to d ( s ) if s <
2. 16 .2 Kernels and their asymptotic behaviors
We show in this section that the Dirichlet form Q s coincides with the Dirichlet form Q J s ,µ associated to a jump kernel J s in Proposition 4.3, and show that there exists a heat kernelassociated to the Dirichlet form Q s in Proposition 4.4. Then we discuss the asymptotic behaviorof the heat kernel in Proposition 4.5, Theorem 4.6 and Theorem 4.8. Definition 4.2 (Definition 10.6 [16]) . Let µ be the probability measure on ∂ B Λ given in (3).Suppose that the spectral radius ρ i of vertex matrices of Λ satisfies ρ i > ≤ i ≤ k and that J is a non-negative function on F B Λ . We define W J : ( ∂ B Λ × ∂ B Λ ) → [0 , ∞ ) by W J ( x, y ) = J ( x ∧ y ) for x = y ∈ ∂ B Λ . Then we let D J,µ = { f ∈ L (Λ ∞ , µ ) : Z ∂ B Λ × ∂ B Λ W J ( x, y )( f ( x ) − f ( y )) dµ ( x ) dµ ( y ) < ∞} and for f, g ∈ D J,µ , we define the bilinear form by Q J,µ ( f, g ) = Z ∂ B Λ × ∂ B Λ W J ( x, y )( f ( x ) − f ( y ))( g ( x ) − g ( y )) dµ ( x ) dµ ( y ) . The Dirichlet form of the above form is called a jump type Dirichlet form and the correspondingkernel J is called a generalized jump kernel . As in [16], we now identify a generalized jump kernel J s for the Dirichlet form Q s of (9) asfollows. Note that the result below is more general than the one in Section 13 of [16] that onlyconcerns a tree case. Furthermore, our proof is involved with the spectral triple while there isno spectral triple involved in [16]. Proposition 4.3.
Let µ be the probability measure on ∂ B Λ given in (3) . Suppose that thespectral radius ρ i of vertex matrices of Λ satisfies ρ i > for all ≤ i ≤ k . For a finite path γ ∈ Λ , we let J s ( γ ) = ( w δ ( γ )) s − P ( e,e ′ ) ∈ ext µ [ γe ] µ [ γe ′ ] . (15) The Dirichlet form ( Q s , Dom( Q s )) on L ( ∂ B , µ ) given in (9) coincides with ( Q J s ,µ , D J s ,µ ) givenin Definition 4.2.Proof. We prove the proposition by computing the Dirichlet form Q s given in (9) explicitlyas follows. Note that | D | = √ D ∗ D and the Dirac operator D acts on the Hilbert space H = ℓ ( F B Λ ) ⊗ C . For ξ ∈ H and λ ∈ F B Λ , we have | D | ξ ( λ ) = 1 w δ ( λ ) (cid:18) (cid:19) (cid:18) (cid:19) ξ ( λ ) = 1 w δ ( λ ) (cid:18) (cid:19) ξ ( λ ) . For any s ∈ R , we get | D | − s = w δ ( λ ) s (cid:18) (cid:19) . Note that the definition works for an arbitrary tree as long as there exists a measure on the associatedinfinite path space. See also Definition 10.6 of [16].
D, π φ ( g )] ξ ( λ ) = g ( φ ( λ )) − g ( φ ( λ ))( w δ ( λ )) (cid:18) −
11 0 (cid:19) ξ ( λ ) , [ D, π φ ( f )] ∗ ξ ( λ ) = f ( φ ( λ )) − f ( φ ( λ ))( w δ ( λ )) (cid:18) − (cid:19) ξ ( λ ) . So, Tr (cid:0) | D | − s [ D, π φ ( f )] ∗ [ D, π φ ( g )] (cid:1) is given by2 X λ ∈ F B Λ w δ ( λ ) s − ( f ( φ ( λ )) − f ( φ ( λ )))( g ( φ ( λ )) − g ( φ ( λ ))) . Thus, we have that Q s ( f, g ) = 12 Z Ξ Tr( | D | − s [ D, π φ ( f )] ∗ [ D, π φ ( g )]) dν ( φ )= Z Ξ X λ ∈ F B Λ w δ ( λ ) s − ( f ( φ ( λ )) − f ( φ ( λ )))( g ( φ ( λ )) − g ( φ ( λ ))) dν ( φ ) . (16)Now we apply the formula of ν on the set Ξ of the choice functions given in (10). Then weobtain that Q s ( f, g ) = Z ∂ B Λ × ∂ B Λ ( w δ ( x ∧ y )) s − P ( e,e ′ ) ∈ ext ( x ∧ y ) µ [( x ∧ y ) e ] µ [( x ∧ y ) e ′ ] (cid:0) f ( x ) − f ( y ) (cid:1)(cid:0) g ( x ) − g ( y ) (cid:1) dµ dµ, where x, y ∈ ∂ B Λ and x ∧ y is the longest common path of x and y . (Note that a choice function φ only picks up a finite path λ = x ∧ y for ( x, y ) ∈ ∂ B Λ × ∂ B Λ , so that the summation in λ of(16) goes away as above). Therefore, by letting J s ( γ ) = ( w δ ( γ )) s − P ( e,e ′ ) ∈ ext ( γ ) µ [ γe ] µ [ γe ′ ] , we have shown that Q s ( f, g ) = Q J s ,µ ( f, g ) . Also, it is straightforward to check that their domains coincide.Recall that each eigenspace E s,γ corresponding to the eigenvalue λ s,γ of the non-positivedefinite self-adjoint Laplace-Beltrami operator ∆ s associated to the above Dirichlet form Q s isgiven in (12): E s,γ = span n χ γe µ [ γe ] − χ γe ′ µ [ γe ′ ] : ( e, e ′ ) ∈ ext ( γ ) o ⊂ L ( ∂ B Λ , µ ) , which can be realized as n ψ ∈ L ( ∂ B Λ , µ ) : ψ = X α = γe,e ∈ s ( γ ) F B Λ , | e | =1 a α χ [ α ] , X α = γe,e ∈ s ( γ ) F B Λ , | e | =1 a α = 0 o . { ψ γ, , . . . , ψ γ,m γ } be an L ( ∂ B Λ , µ )-orthonormal basis of the above E s,γ . Then Lemma 10.2of [16] implies that there exists a complete orthonormal system { ψ , ψ γ,n : γ ∈ F B Λ , ≤ n ≤ m γ } of L ( ∂ B Λ , µ ), where ψ = χ [ ∂ B ] , m γ = |{ e ∈ s ( γ ) B }| − B is the set of edges in B .Therefore, Lemma 7.1 of [16] gives the formula m γ − X j =1 ψ γ,j ( x ) ψ γ,j ( y ) = X e ∈ s ( γ ) B (cid:16) χ [ γe ] ( x ) χ [ γe ] ( y ) µ [ γe ] − χ [ γ ] ( x ) χ [ γ ] ( y ) µ [ γ ] (cid:17) , which is the same form as the formula given in the Section 7 of [16].Now we are ready to give the formula of the heat kernel which we are interested in. As inthe equation (7.1) of [16], the heat kernel associated to the Dirichlet form ( Q s , Dom( Q s )) isgiven by p ( t, x, y ) = 1 + X γ ∈ F B Λ e λ s,γ t m γ − X j =1 ψ γ,j ( x ) ψ γ,j ( y ) , (17)where { ψ , ψ γ,n : γ ∈ F B Λ , ≤ n ≤ m γ } is a complete orthonormal system of L ( ∂ B Λ , µ ) givenas above.In particular, we obtain that p ( t, x, y ) = ∞ X n =0 (cid:16) µ [ x (0 , n + 1)] − µ [ x (0 , n )] (cid:17) e λ s,x (0 ,n ) t if x = y, | x ∧ y | X n =0 µ [ x ∧ y (0 , n )] (cid:16) e λ s,x ∧ y (0 ,n − t − e λ s,x ∧ y (0 ,n ) t (cid:17) if x = y, (18)where | x ∧ y | is the length of the path x ∧ y . Note that we have e λ s,γ t instead of e − λ s,γ t since λ s,γ < γ ∈ F B Λ in our case.Since the measure µ on ∂ B Λ has the volume doubling property with respect to the intrinsicmetric d ( s ) and the ultrametric d w δ induced by the weight w δ on ∂ B given in (6), we expect toobtain similar results to those in [16, Section 7], such as asymptotic behaviors of heat kernel andjump kernel of the process associated to the Dirichlet form Q s given in (9). But, we first notethat if the eigenvalues of the associated Laplace-Beltrami operator ∆ s blow up at infinity, thenone can find a Hunt process ( { Y t } t> , { P x } x ∈ ∂ B ) on ∂ B Λ whose transition density is p ( t, x, y ) asfollows. Proposition 4.4.
Let µ be the probability measure on ∂ B Λ given in (3) . For any x, y ∈ ∂ B Λ and t > , define p t,x ( y ) = p ( t, x, y ) , where p ( t, x, y ) is the heat kernel given in (17) that isassociated to the Dirichlet form Q s in (9) . (a) For any bounded Borel measurable function f : ∂ B Λ → R , we define ( p t f )( x ) = Z ∂ B Λ p t,x f dµ ( t > . Then { p t : t > } is a Markovian transition function in the sense of [12, Section 1.4]. There exists a Hunt process ( { Y t } t> , { P x } x ∈ ∂ B Λ ) on ∂ B Λ whose transition density is p ( t, x, y ) , i.e., E x ( f ( Y t )) = Z ∂ B Λ p ( t, x, y ) f ( y ) µ ( dy ) , for x ∈ ∂ B Λ and for a bounded Borel measurable function f : ∂ B Λ → R , where E x ( · ) isthe expectation with respect to P x .Proof. Since | λ s,γ | → ∞ as | γ | → ∞ , we see that the result follows from Proposition 7.2 andTheorem 7.3 of [16].Since { λ s,x (0 ,n ) : n ∈ N } is strictly decreasing for any x ∈ ∂ B Λ and the measure µ has thevolume doubling property with respect to d ( s ) , the heat kernel satisfies the estimates in termsof the intrinsic metric d ( s ) as follows. Note that the proofs of Proposition 4.5 and Theorem 4.6are very similar to the ones in [16]. Proposition 4.5. (c.f. Proposition 7.5 of [16]) Let µ be the probability measure on ∂ B Λ givenin (3) and p ( t, x, y ) be the heat kernel given in (17) . Suppose that s < and the spectral radius ρ i of vertex matrices of Λ satisfies ρ i > for every ≤ i ≤ k . Let d ( s ) be the intrinsic metricon ∂ B Λ given in Proposition 3.3 and we denote B s ( x, t ) be the open ball with radius t centeredat x ∈ ∂ B Λ with respect to d ( s ) . Then the following statements are true. (a) For x ∈ ∂ B Λ and t > , we have p ( t, x, x ) ≥ e · µ ( B s ( x, t )) . (b) For < t ≤ d ( s ) ( x, y ) , we have p ( t, x, y ) ≤ d ( s ) ( x, y ) µ [ x ∧ y ] . Proof.
The results follow from Proposition 7.5 of [16].Now we give the heat kernel estimations in terms of the intrinsic metric d ( s ) as follows. Wefirst recall that f ( x ) ≍ g ( x ) means that there exist two positive numbers c , c such that c g ( x ) ≤ f ( x ) ≤ c g ( x ) . (19) Theorem 4.6. (c.f. Theorem 7.6 of [16]) Suppose that the spectral radius ρ i of vertex matricesof Λ satisfies ρ i > for all ≤ i ≤ k . Let µ be the probability measure on ∂ B Λ as given in (3) and d ( s ) be the intrinsic ultrametric on ∂ B Λ given in Proposition 3.3. Let p ( t, x, y ) be the heatkernel given in (17) . Then the heat kernel p ( t, x, y ) is continuous on (0 , ∞ ) × ∂ B Λ × ∂ B Λ andsatisfies p ( t, x, y ) ≍ td ( s ) ( x, y ) µ [ x ∧ y ] if < t ≤ d ( s ) ( x, y ) , µ ( B s ( x, t )) if t > d ( s ) ( x, y ) . Proof.
We leave the proof to the reader since it is very similar to the proof in [16, Theorem 7.6].We show in the main theorem of this paper that the ultrametric d w δ associated to the weights w δ on B Λ is equivalent to the intrinsic metric d ( s ) associated to the eigenvalues λ s,γ of ∆ s .20 heorem 4.7. Suppose that the spectral radius ρ i of vertex matrices Λ satisfies ρ i > forevery ≤ i ≤ k . Let d w δ be the ultrametric on ∂ B Λ associated to the weight w δ given in (6) for δ ∈ (0 , , and let d ( s ) be the intrinsic metric given in Proposition 3.3. If < s < δ , thenwe have that d ( s ) ( x, y ) ≍ ( d w ( x, y )) δ − s (20) for x, y ∈ ∂ B Λ .Proof. We have to find two positive constants a, b such that a · d ( s ) ( x, y ) ≤ ( d w ( x, y )) δ − s ≤ b · d ( s ) ( x, y ) . To see this, fix a finite path γ ∈ F B Λ with | γ | = n = qk + t where q ∈ N and 1 ≤ t ≤ k − s ∈ R , (11) gives λ s,γ = n − X k =0 µ [ γ k +1 ] − µ [ γ k ] G s ( γ k ) − µ [ γ ] G s ( γ ) , (21)where G s ( η ) = (cid:0) w ( η ) (cid:1) − s P ( e,e ′ ) ∈ ext ( η ) µ [ ηe ] µ [ ηe ′ ]. In particular, for any finite path η withlength | η | = m = pk + ℓ , one can compute G s ( η ) = (cid:0) ρ p +11 . . . ρ p +1 ℓ ρ pℓ +1 . . . ρ pk (cid:1) s − δ − ( κ Λ s ( η ) ) − s (2( ρ ℓ +1 ) − ) X ( e,e ′ ) ∈ ext ( η ) κ Λ s ( e ) κ Λ s ( e ′ ) . To simplify the computations, the first sum and the second sum of λ s,γ are denoted by A and B , respectively. Then we compute B = µ [ γ ] G s ( γ ) = ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) − κ Λ s ( γ ) ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) s − δ − · ( κ Λ s ( γ ) ) − s · P ( e,e ′ ) ∈ ext1( γ ) κ Λ s ( e ) κ Λ s ( e ′ ) ρ t +1 ) = ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) − s − δ · ( κ Λ s ( γ ) ) s − ( α γ ) − , where α γ = P ( e,e ′ ) ∈ ext1( γ ) κ Λ s ( e ) κ Λ s ( e ′ ) ρ t +1 ) .To compute A , now let γ i be a sub-path of γ with | γ i | = i = q ′ k + t ′ . Then A = n − X i =0 µ [ γ i +1 ] − µ [ γ i ] G s ( γ i )= n − X i =1 ( ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ +1 t ′ +1 ρ q ′ t ′ +2 . . . ρ q ′ k ) − · κ Λ s ( γ i +1 ) − ( ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . . ρ q ′ k ) − · κ Λ s ( γ i ) ( ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . . ρ q ′ k ) s − δ − · ( κ Λ s ( γ i ) ) − s · ( α γ i ) , where α γ i = P ( e,e ′ ) ∈ ext1( γi ) κ Λ s ( e ) κ Λ s ( e ′ ) ρ t ′ +1 ) . Then we can simplify A and obtain A = n − X i =1 ( ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . . ρ q ′ k ) δ − sδ · ( κ Λ s ( γ i ) ) s − ( α γ i ) − · (cid:0) ρ − t ′ +1 · κ Λ s ( γ i +1 ) − κ Λ s ( γ i ) (cid:1) .
21e see that λ s,γ = A − B is given by( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) δ − sδ · (cid:16) n − X i =1 ( ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . . ρ q ′ k ) δ − sδ · ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) − δ − sδ · ( κ Λ s ( γ i ) ) s − ( α γ i ) − ( ρ − t ′ +1 κ Λ s ( γ i +1 ) − κ Λ s ( γ i ) ) − ( κ Λ s ( γ ) ) s − · ( α γ ) − (cid:17) . Therefore, we have d ( s ) ( x, y ) = − ( λ s,x ∧ y ) − = − ( A − B ) − = ( B − A ) − if γ = x ∧ y , and hencewe get d ( s ) ( x, y ) = ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) δ − sδ · F γ , (22)where F γ is a finite sum given by F γ = (cid:18) ( κ Λ s ( γ ) ) s − α γ − n − X i =1 (cid:0) ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . .ρ q ′ k (cid:1) δ − sδ (cid:0) ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk (cid:1) − δ − sδ · ( κ Λ s ( γ i ) ) s − α γ i · (cid:0) κ Λ s ( γ i +1 ) ρ t ′ +1 − κ Λ s ( γ i ) (cid:1)(cid:19) − . (23)Since d ( s ) ( x, y ) > F γ is positive. We notice that F γ only involves with a finite number ofterms. Thus, there exists an upper bound of F γ . In particular, there exists the least upper boundof F γ , denoted by sup γ F γ , which is not zero. Since d w ( x, y ) = ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) − δ · κ Λ s ( γ ) ,the equation (22) implies d ( s ) ( x, y ) = ( d w ( x, y )) δ − sδ · F γ ( κ Λ s ( γ ) ) δ − s (24)where γ = x ∧ y with | γ | = n = qk + t . This gives the equality( κ Λ s ( γ ) ) δ − s F γ d ( s ) ( x, y ) = ( d w ( x, y )) δ − s . We claim that there exists the minimum value of ( κ Λ s ( γ ) ) δ − s F γ that does not depend on s .Indeed, we observe that ( κ Λ s ( γ ) ) δ − s F γ ≥ ( κ Λ s ( γ ) ) δ − s sup γ F γ . We also note that 2 + δ − s > < κ Λ s ( γ ) <
1. Since there exists the minimum value of afinite set of positive numbers and Λ is a finite k -graph, the set { ( κ Λ s ( γ ) ) δ − s : s ( γ ) ∈ Λ } hasthe minimum value, denoted by min { ( κ Λ s ( γ ) ) δ − s } . Thus we have( κ Λ s ( γ ) ) δ − s F γ ≥ min { ( κ Λ s ( γ ) ) δ − s } sup γ F γ . By letting a := min { ( κ Λ s ( γ ) ) δ − s } sup γ F γ , we obtain a d ( s ) ( x, y ) ≤ ( d w ( x, y )) δ − s .22o obtain the other inequality, note that (24) gives d w ( x, y ) δ − s = d ( s ) ( x, y ) ( κ Λ s ( γ ) ) δ − s F γ . Since 0 < κ Λ v < δ − s >
0, we have d w ( x, y ) δ − s = d ( s ) ( x, y ) ( κ Λ s ( γ ) ) δ − s F γ < d ( s ) ( x, y ) 1 F γ . Thus, we only need to show that there exists an upper bound of the set { F γ : γ ∈ F B} thatdoes not depend on s . From (23), we see that1 F γ = ( κ Λ s ( γ ) ) s − α γ − n − X i =1 ( ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . . ρ q ′ k ) δ − sδ ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) − δ − sδ · ( κ Λ s ( γ i ) ) s − α γ i · (cid:18) κ Λ s ( γ i +1 ) ρ t ′ +1 − κ Λ s ( γ i ) (cid:19) . We note that each γ i with | γ i | = q ′ k + t ′ is a sub-path of γ with | γ | = qk + t . Since δ − sδ > ρ i > ≤ i ≤ k , we have that (cid:18) ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . . ρ q ′ k ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk (cid:19) δ − sδ < . Thus, we have that1 F γ = ( κ Λ s ( γ ) ) s − α γ + n − X i =1 ( ρ q ′ +11 . . . ρ q ′ +1 t ′ ρ q ′ t ′ +1 . . . ρ q ′ k ) δ − sδ ( ρ q +11 . . . ρ q +1 t ρ qt +1 . . . ρ qk ) δ − sδ · ( κ Λ s ( γ i ) ) s − α γ i · (cid:16) κ Λ s ( γ i ) − κ Λ s ( γ i +1 ) ρ t ′ +1 (cid:17) ≤ ( κ Λ s ( γ ) ) s − α γ + n − X i =1 ( κ Λ s ( γ i ) ) s − α γ i · (cid:16) κ Λ s ( γ i ) − κ Λ s ( γ i +1 ) ρ t ′ +1 (cid:17) = 2( ρ t +1 ) ( κ Λ s ( γ ) ) s − P ( e,e ′ ) ∈ ext ( γ ) κ Λ s ( e ) κ Λ s ( e ′ ) + n − X i =1 ρ t ′ +1 ) ( κ Λ s ( γ i ) ) s − P ( e,e ′ ) ∈ ext ( γ i ) κ Λ s ( e ) κ Λ s ( e ′ ) · (cid:16) κ Λ s ( γ i ) − κ Λ s ( γ i +1 ) ρ t ′ +1 (cid:17) . Since Λ has only finitely many vertices, there exists the positive minimum value of the set { P ( e,e ′ ) ∈ ext ( η ) κ Λ s ( e ) κ Λ s ( e ′ ) : η ∈ F B Λ } denoted by MIN. Also, there exists the maximum value of { ρ j : 1 ≤ j ≤ k } , which we denote by ρ max . Thus, we have that1 F γ < ρ max ) MIN (cid:18) ( κ Λ s ( γ ) ) s − + n − X i =1 ( κ Λ s ( γ i ) ) s − (cid:16) κ Λ s ( γ i ) − κ Λ s ( γ i +1 ) ρ t ′ +1 (cid:17)(cid:19) = 2( ρ max ) MIN (cid:18) ( κ Λ s ( γ ) ) s − + n − X i =1 ( κ Λ s ( γ i ) ) s − (cid:16) − κ Λ s ( γ i +1 ) ρ t ′ +1 · κ s ( γ i ) (cid:17)(cid:19) . Since 0 < κ Λ s ( γ ) < s − >
0, we have ( κ Λ s ( γ ) ) s − < F γ < ρ max ) MIN (cid:18) n − X i =1 ( κ Λ s ( γ i ) ) s − (1 − κ Λ s ( γ i +1 ) ρ t ′ +1 · κ Λ s ( γ i ) ) (cid:19) . (cid:16) − κ Λ s ( γi +1) ρ t ′ +1 · κ Λ s ( γi ) (cid:17) <
0, then we have the inequality F γ < ρ max ) MIN , which completes the proof.If 0 < (cid:16) − κ Λ s ( γi +1) ρ t ′ +1 · κ Λ s ( γi ) (cid:17) < i = 0 , , . . . , n −
1, then we have that1 F γ < ρ max ) MIN (cid:18) n − X i =1 ( κ Λ s ( γ i ) ) s − (cid:19) . Since κ Λ s ( γ i ) ’s are entries of the unimodular Perron-Frobenius eigenvector and s − > P n − i =1 ( κ Λ s ( γ i ) ) s − is uniformly bounded by some constant M . This gives the inequality1 F γ < ρ max ) MIN (1 + M ) , which completes the proof.We showed that there exists a heat kernel p ( t, x, y ) associated to the Dirichlet form Q s inProposition 4.4 and described its asymptotic behavior in Proposition 4.5 and Theorem 4.6 withrespect to the intrinsic metric d ( s ) . Due to the previous theorem, we can now give the heatkernel estimates in terms of the ultrametric d w δ associated to the weight w δ given in (6) on ∂ B Λ . In fact, (b) of the following theorem may be thought of as a heat kernel estimate for jumpprocesses associated to the Dirichlet form Q J s ,µ . Theorem 4.8.
Suppose that the spectral radius ρ i of vertex matrices of Λ satisfies ρ i > forevery ≤ i ≤ k . Let d w δ be the ultrametric on ∂ B Λ associated to the weight w δ given in (6) for δ ∈ (0 , . Suppose that ≤ s < δ . Then we have the following facts. (a) There exists a jointly continuous transition density p ( t, x, y ) on (0 , ∞ ) × ∂ B Λ × ∂ B Λ forthe Hunt process ( { Y t } t> , { P x } x ∈ ∂ B Λ ) associated to the Dirichlet form ( Q J s ,µ , D J s ,µ ) on L ( ∂ B Λ , µ ) given in (4.3) . (b) The transition density p ( t, x, y ) in (a) satisfies p ( t, x, y ) ≍ td w δ ( x, y ) δ − sδ if d w δ ( x, y ) δ − sδ > t µ ( B d wδ ( x, t δ δ − s )) if d w δ ( x, y ) δ − sδ ≤ t for any ( t, x, y ) ∈ (0 , ∞ ) × ∂ B Λ × ∂ B Λ .(c) For any x ∈ ∂ B Λ and t ∈ (0 , E x ( d w δ ( x, Y t ) δ − sδ α ) ≍ t if α > ,t ( | log t | + 1) if α = 1 ,t γ if 0 < α < . Proof.
Note that (a) and (b) follow from Theorem 4.6 and Theorem 4.7. Also (c) follows byCorollary 7.9 of [16] since the measure µ has the volume doubling property with respect to theultrametric d w δ . 24 eferences [1] M. Abe and K. Kawamura, Branching laws for endomorphisms of fermions and the Cuntzalgebra O , J. Math. Phys. (2008), 043501, 10 pp.[2] S. Bezuglyi and P.E.T. Jorgensen, Representations of Cuntz-Krieger relations, dynamicson Bratteli diagrams, and path-space measures , Trends in harmonic analysis and its appli-cations, Contemp. Math., vol. 650, Amer. Math. Soc., Providence, RI, 2015, pp. 57–88.[3] T. Carlsen, S. Kang, J. Shotwell, and A. Sims,
The primitive ideals of the Cuntz-Kriegeralgebra of a row-finite higher-rank graph with no sources , J. Funct. Anal. (2014) 2570–2589.[4] Z.-Q. Chen and T. Kumagai,
Heat kernel estimates for jump processes of mixed types onmetric measure space , Prob. Theory Related Fields (2008), 277–317.[5] D.E. Dutkay and P.E.T. Jorgensen,
Analysis of orthogonality and of orbits in affine iteratedfunction systems , Math. Z. (4) (2007), 801–823.[6] D.G. Evans,
On the K -theory of higher rank graph C ∗ -algebras , New York J. Math. (2008) 1–31.[7] C. Farsi, E. Gillaspy, S. Kang and J. Packer, Separable representations, KMS states, andwavelets for higher-rank graphs , J. Math. Anal. Appl. (2015), 241–270.[8] C. Farsi, E. Gillaspy, A. Julien, S. Kang and J. Packer,
Wavelets and spectral triples forfractal representations of Cuntz algebras , Problems and recent methods in operator theory,103–133, Contemp. Math., 687, Amer. Math. Soc., Providence, RI, 2017.[9] C. Farsi, E. Gillaspy, A. Julien, S. Kang and J. Packer,
Spectral triples and wavelets forhigher-rank graphs , submitted, available at https://arxiv.org/abs/1803.09304 [10] C. Farsi, E. Gillaspy, P. Jorgensen, S. Kang and J. Packer,
Monic representations of finitehigher-rank graphs , to appear in Ergodic Theory and Dynamical System.[11] C. Farsi, E. Gillaspy, P. Jorgensen, S. Kang and J. Packer,
Representations of higher-rankgraph C ∗ -algebras associated to Λ -semibranching function systems , J. Math. Anal. Appl. (2018), 766–798.[12] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Pro-cesses, de Gruyter Stud. Math., vol. , de Gruyter, Berlin, 1994.[13] A. an Huef, M. Laca, I. Raeburn and A. Sims, KMS states on the C ∗ -algebra of a higher-rank graph and periodicity in the path space , J. Funct. Anal. (2015), 1840–1875.[14] A. Julien and J. Savinien, Transverse Laplacians for substitution tiling , Comm. Math.Phys. (2011), 285–318.[15] A. Julien and J. Savinien,
Embeddings of self-similar ultrametric Cantor sets , TopologyAppl. (2011), 2148–2157. 2516] J. Kigami,
Dirichlet forms and associated heat kernels on the Cantor set indued by randomwalks on trees , Adv. Math. (2010), 2674–2730.[17] A. Kumjian and D. Pask,
Higher rank graph C ∗ -algebras , New York J. Math. (2000),1–20.[18] M. Marcolli and A.M. Paolucci, Cuntz-Krieger algebras and wavelets on fractals , ComplexAnal. Oper. Theory (2011), 41–81.[19] J. Pearson and J. Bellissard, Noncommutative Riemannian geometry and diffusion on ul-trametric Cantor sets , J. Noncommut. Geom. (2009), 447–480.[20] E. Ruiz, A. Sims and A.P.W. Sørensen, UCT-Kirchberg algebras have nuclear dimensionone , Adv. Math. (2015), 1–28.[21] K.R. Davidson and D. Yang,
Periodicity in Rank graph algebras , Canad. J. Math. (2009), 1239–1261. Jaeseong Heo : Department of Mathematics, Research Institute for Natural Sci-ences, Hanyang University, Seoul 04763, Republic of Korea
E-mail address : [email protected] Sooran Kang : College of General Education, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, Republic of Korea
E-mail address : [email protected] Yongdo Lim : Department of Mathematics, Sungkyunkwan University, Suwon, Re-public of Korea
E-mail address : [email protected]@skku.edu