Gromov Hyperbolic Graphs Arising From Iterations
aa r X i v : . [ m a t h . M G ] J un Gromov Hyperbolic Graphs Arising From Iterations
Shi-Lei Kong, Ka-Sing Lau and Xiang-Yang Wang
Abstract
For a contractive iterated function system (IFS), it is known that there is a natural hyperbolicgraph structure (augmented tree) on the symbolic space of the IFS that reflects the relationshipamong neighboring cells, and its hyperbolic boundary with the Gromov metric is H¨older equivalentto the attractor K [Ka, LW1, LW3]. This setup was taken up to study the probabilistic potentialtheory on K [KLW1, KL], and the bi-Lipschitz equivalence on K [LL]. In this paper, we formulatea broad class of hyperbolic graphs, called expansive hyperbolic graphs , to capture the most essentialproperties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics,bounded degree property, metric doubling property, and H¨older equivalence). We also study a newsetup of “weighted” IFS and investigate its connection with the self-similar energy form in theanalysis of fractals. Contents · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
12 Expansive graphs and hyperbolicity · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
53 Hyperbolic boundaries · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
104 Index maps and augmented graphs · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
145 Separation conditions and doubling metrics · · · · · · · · · · · · · · · · · · · · · · · · · ·
186 Examples and more on IFSs · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
237 Appendix: IFSs and augmented trees · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Let { S i } Ni =1 be a contractive iterated function system (IFS) on R d , and let K be the attractor.It is well-known that the IFS is associated with a finite word space Σ ∗ (symbolic space or codingspace), and the limit set Σ ∞ is used to represent elements in K . With the intention to carry overthe probabilistic potential theory to K , Denker and Sato [DS1, DS2] first constructed a specialtype of Markov chain { Z n } ∞ n =0 on Σ ∗ of the Sierpinski gasket (SG), and showed that the Martinboundary of { Z n } ∞ n =0 is homeomorphic to the SG. Motivated by this, Kaimanovich [Ka] introducedan “augmented tree” (Σ ∗ , E ) by adding “horizontal” edges to the coding tree (Σ ∗ , E v ) according to . Primary 28A78; Secondary 28A80. Keywords : hyperbolic graph, hyperbolic boundary, compact metric space, doubling, self-similar set.The research is supported in part by the HKRGC grant, SFB 1283 of the German Research Counciland the NSFC (nos. 11971500, 11831007). he neighboring cells in each level of the SG. He showed that the graph is hyperbolic in the senseof Gromov [Gr, Wo], and the SG equipped with the Euclidean metric is H¨older equivalent to thehyperbolic boundary of (Σ ∗ , E ). He also suggested that this approach would also work for moregeneral IFSs, and could provide another tool to study the geometry and analysis on fractals.These initiatives were carried out by the authors in a series of papers [LW1,LW2,LW3,JLW,Wa,KLW1,KLW2,KL]. In [LW1,LW3], we investigated the systems of neighboring cells to general IFSson which the hyperbolicity of the augmented trees is valid; it was shown that the hyperbolic bound-aries and the attractors K are H¨older equivalent (or homeomorphic) under various circumstances.The H¨older equivalence was applied to the bi-Lipschitz classification of the totally disconnectedself-similar sets [LL, DLL]. More importantly, this setup was taken up to study the probabilisticpotential theory on the self-similar sets [KLW1]: for an augmented tree ( X, E ) defined by an IFS on R d satisfying the open set condition, we introduced a class of reversible transient random walks on( X, E ). By identifying the Matin boundary, the hyperbolic boundary and K , we obtain an inducedDirichlet form of Gagliardo-type on K . The relevant energy forms and function spaces were furtherstudied in [KL].The augmented tree has very rich structure inherited from the iterations and the attractors. Inthis paper, our first goal is to formulate a broad class of hyperbolic graphs to capture the mostessential concepts, such as the special geodesics, the bounded degree property, and the doublingproperty of the Gromov metric on the hyperbolic boundaries, which were used extensively inthe study of augmented trees. This new setup allows us to carry out the idea of augmentationmuch further, and beyond IFS. Besides extending the previous study on the identification of thehyperbolic boundaries with the attractors K , we are able to use the hyperbolic techniques developedto construct new metrics on K that are useful in the analysis of fractals.Let ( X, E ) be a locally finite connected graph with a root ϑ ∈ X . We have the decomposition E = E v ∪ E h , the set of vertical edges (which does not necessarily form a tree) and the set ofhorizontal edges. Let J m ( x ) denote the set of descendants in the m -th generation of x ∈ X .Denote by d and d h the graph distances on ( X, E ) and (
X, E h ) respectively, and let | x | := d ( ϑ, x ).We call a rooted graph expansive if for x, y ∈ X with | x | = | y | , d h ( x, y ) > ⇒ d h ( u, v ) > , ∀ u ∈ J ( x ) , v ∈ J ( y ) , and call it ( m, k ) -departing if d h ( x, y ) > k ⇒ d h ( u, v ) > k, ∀ u ∈ J m ( x ) , v ∈ J m ( y ) . Intuitively, the two definitions describe the distances of the descendants that are drifted apartcompared to the non-neighboring predecessors. The expansive property provides a rather simpleform of geodesics ( convex geodesics , see Proposition 2.3, Figure 2), and the ( m, k )-departing prop-erty gives more details of how the graph evolves. These two lead to the following criteria for thehyperbolicity (Theorem 2.11), which are crucial in the paper.
Theorem 1.1.
Let ( X, E ) be an expansive graph. The following assertions are equivalent.(i) ( X, E ) is hyperbolic;(ii) ∃ L < ∞ such that the lengths of all horizontal geodesics are bounded by L ;(iii) ( X, E ) is ( m, k ) -departing for some positive integers m and k . By a horizontal geodesic , we mean a geodesic in (
X, E ) consisting of edges in E h only.) We willcall such ( X, E ) an expansive hyperbolic graph : together with the expansiveness, (ii) gives a cleargeometry of the geodesics in X ; the ( m, k )-departing property in (iii) will serve as the workhorsein many of the proofs in this study.We use ∂X to denote the hyperbolic boundary of ( X, E ), which is a compact space with anassociated metric θ a , a > Gromov metric , see Definition 2.4). By using the ( m, k )-departing property, we obtain a sharp description of the equivalent rays in X that converge to thesame boundary elements (Proposition 3.1) which will be used in a number of estimates. One of themain results is (Theorem 3.6) Theorem 1.2.
Suppose ( X, E ) is an expansive hyperbolic graph, and has bounded degree (i.e., sup x ∈ X deg( x ) < ∞ ). Then ( ∂X, θ a ) is a doubling metric space. Recall that an augmented tree of an IFS with attractor K is based on the tree of the symbolicspace, together with the added horizontal edges that connect neighboring cells in the same level(see Appendix). This can easily be reformulated on any vertical rooted graph ( X, E v ): let ( M, ρ )be a complete metric space, and let C M denote the family of nonempty compact subsets of M . Wedefine an index map Φ : X → C M that satisfies Φ( y ) ⊂ Φ( x ) whenever y ∈ J ( x ), and T ∞ i =0 Φ( x i )is a singleton for any geodesic ray [ x i ] i from the root ϑ (see Definition 4.1); likewise, we also havean attractor K . This setup is very general, which includes all IFSs (where Φ( x ) is defined as thecell K x ), as well as cases that are not from IFS, e.g., refinement systems of sets.With the vertical graph ( X, E v ) and the index map Φ, we define E ( ∞ ) h := (cid:8) ( x, y ) ∈ X × X : | x | = | y | , x = y, and Φ( x ) ∩ Φ( y ) = ∅ (cid:9) . Let E ( ∞ ) = E v ∪ E ( ∞ ) h , and call ( X, E ( ∞ ) ) an AI ∞ -graph ( augmented index graph of type- ( ∞ ),or intersection type ). In the case that { Φ( x ) } x ∈ X is of exponential type-( b ) (i.e., the diameter | Φ( x ) | ρ = O ( e − b | x | ) for some b >
0, as | x | → ∞ ), for some fixed γ >
0, we define a horizontal edgeset by E ( b ) h (= E ( b ) h ( γ )) := (cid:8) ( x, y ) ∈ X × X : | x | = | y | , x = y, and dist ρ (Φ( x ) , Φ( y )) ≤ γe − b | x | (cid:9) , Let E ( b ) = E v ∪ E ( b ) h , and call ( X, E ( b ) ) an AI b -graph ( augmented index graph of type- ( b )).Both AI ∞ - and AI b -graphs are expansive. The condition that defines an AI ∞ -graph is moreintuitive, and E ( ∞ ) h consists of fewer edges. However, concerning the hyperbolicity, the AI b -graphhas the advantage that we do not need to know the fine structure of K a priori. More precisely, byusing Theorem 1.1, we show that (Theorem 4.5) Theorem 1.3.
The AI b -graph is ( m, -departing for some m ≥ , and hence hyperbolic. More-over, the index map Φ induces a bijection κ : ∂X → K that is a H¨older equivalence, i.e., ρ ( κ ( ξ ) , κ ( η )) a/b ≍ θ a ( ξ, η ) for all ξ, η ∈ ∂X . (Here by f ≍ g we mean that there exists C ≥ C − f ( x ) ≤ g ( x ) ≤ Cf ( x ) for allvariables x in a given domain.) The AI ∞ -graph is not always hyperbolic (Example 6.1); in orderto have that, we need an additional separation condition ( S b ) on { Φ( x ) } x ∈ X (Definition 5.1, whichis satisfied by IFS of similitudes with the OSC). heorem 1.4. Suppose { Φ( x ) } x ∈ X is of exponential type-(b), and satisfies condition ( S b ) forsome b > . Then the AI ∞ -graph is hyperbolic, and the induced bijection κ : ∂X → K is H¨oldercontinuous, i.e., ρ ( κ ( ξ ) , κ ( η )) a/b ≤ Cθ a ( ξ, η ) for all ξ, η ∈ ∂X . The proof of the theorem is in Theorem 5.4 and Corollary 4.7. We also provide an example(Example 6.2) to show that unlike the AI b -graph, the H¨older continuity of κ for the AI ∞ -graphcannot be improved to H¨older equivalence.For the augmented tree of an IFS, the bounded degree property is important because it allowsus to consider certain random walks on it [KLW1, KL]. This property has been characterized interms of the separation properties such as the open set condition and weak separation conditionfor IFSs [LW3, Wa]. In Section 5, we prove (Theorem 5.5) Theorem 1.5.
The AI b -graph has bounded degree if and only if condition ( S b ) is satisfied. Also,the AI ∞ -graph has bounded degree provided that ( S b ) is satisfied. In our previous consideration of IFS, the structure of augmented trees (or AI ∞ - and AI b -graphshere) arose from the geometry of K x under a given metric (usually Euclidean metric). In theanalysis of fractals, there are situations that involve weighted IFS, which give rise to new metricson K (e.g., the resistance metric in the study of Dirichlet form [Ki1, HW] or the metrics involve inthe time change of Brownian motions [Ki2,Ki3,Ki4,GLQR]). This requires new graph structure toaccommodate the new parameter of weights.In this regard, we let { S j } Nj =1 be a contractive IFS on a complete metric space ( M, ρ ), and let K be the attractor. Let s = ( s , . . . , s N ), s j ∈ (0 , S j ’s. Weregroup the finite words in the symbolic space to form a new coding tree ( X ( s ) , E v ) (see (6.2)) suchthat in each level, the K x ’s have comparable weights. In this case the AI ∞ -graph is more naturalfor use (see Section 6). While we cannot check the hyperbolicity directly (as { K x } x ∈ X ( s ) does notsatisfy the separation condition ( S b ) as in Theorem 1.4), we still obtain some rather satisfactoryconclusions (Theorem 6.3) for the class of p.c.f. sets [Ki1]. Theorem 1.6.
Let { S j } Nj =1 be a contractive IFS that has the p.c.f. property. Then for any weight s ∈ (0 , N , the AI ∞ -graph ( X ( s ) , E ( ∞ ) ) is an ( m, -departing expansive graph of bounded degree.Consequently the AI ∞ -graph is a hyperbolic augmented tree. The theorem is applied to study self-similar energy forms and resistance metrics in Section 6.By using the fact that the natural identification κ : ( ∂X, θ a ) → ( K, ρ ) is a homeomorphism, wecan impose a new metric ˜ θ a on K , as a consequence of Theorem 1.6. We show that if K admits aregular harmonic structure, then ˜ θ a is H¨older equivalent to the associated resistance metric on K (Theorem 6.7).We remark that in [Ki5], Kigami proposed another construction of metrics through the weightedtrees associated with successive partitions on compact metrizable spaces, while he stated that suchconstruction is possible if and only if the “resolution” graph (which is similar to our AI ∞ -graphhere) is hyperbolic. He also studied different types of properties among metrics and measures (e.g.,Lipschitz equivalence, Ahlfors-regularity, volume doubling, etc.) via the weight functions on trees.In a forthcoming paper [KLWa], by showing that a hyperbolic graph is near-isometric to anexpansive hyperbolic graph, we present in greater generality the framework of index maps andaugmented index graphs. We can also extend the scope of underlying spaces to quasi-metric spaces particularly the spaces of homogeneous type [Ch,CW]), and study random walks on such hyperbolicgraphs.For the organization of the paper, we introduce the basics of expansive graphs and ( m, k )-departing property in Section 2, and prove Theorem 1.1. We study the boundaries of hyperbolicexpansive graphs in Section 3, and prove Theorem 1.2. In Section 4, we define the index maps,as well as the associated AI b -graphs, AI ∞ -graphs, and prove the hyperbolicity of AI b -graphs inTheorem 1.3. For the AI ∞ -graphs in Theorems 1.4 and 1.5, we need some separation propertiesof the index family, which are detailed in Section 5. In Section 6, we give the two examples asasserted above, and also apply the techniques developed to consider the weighted IFS. An appendixon the augmented tree defined by IFS of similitudes and some related results are included for theconvenience of the reader. A graph ( X, E ) is a countable set X of vertices together with a set E of edges which is a symmetricsubset of X × X \ ∆ (∆ := { ( x, x ) : x ∈ X } ). It is called locally finite if for any vertex x ∈ X ,deg( x ) := { y : ( x, y ) ∈ E } < ∞ . For x, y ∈ X , we use π ( x, y ) to denote the geodesic (path withsmallest path length) from x to y , and define the graph distance d ( x, y ) by the length of π ( x, y ) ifsuch path exists ( d ( x, y ) = ∞ otherwise). If d ( x, y ) is finite for all x, y ∈ X , we say that ( X, E ) is connected ; in this case d is an integer-valued metric on X .In this paper, we assume that ( X, E ) is a rooted graph , i.e., a locally finite connected graph inwhich a vertex ϑ ∈ X is fixed as a root . We write | x | := d ( ϑ, x ) for x ∈ X , and let X n := { x ∈ X : | x | = n } . Then X = S ∞ n =0 X n . We define a partial order ≺ on X with y ≺ x if and only if x lieson some π ( ϑ, y ). For an integer m ≥ x ∈ X , let J m ( x ) := { y ∈ X : y ≺ x, | y | = | x | + m } , J − m ( x ) := { z ∈ X : x ∈ J m ( z ) } be the m -th descendant set and the m -th precedessor set of x respectively; in general, J m ( x ) isallowed to be empty. We also write J ∗ ( x ) := { y ∈ X : y ≺ x } and J −∗ ( x ) := { z ∈ X : x ≺ z } forfurther use.Let E v = { ( x, y ) ∈ E : | x | − | y | = ± } and E h = { ( x, y ) ∈ E : | x | = | y |} denote the vertical edge set and the horizontal edge set respectively. Clearly E = E v ∪ E h , and E v = { ( x, y ) ∈ X × X : x ∈J ( y ) or y ∈ J ( x ) } . We say that a rooted graph ( X, E ) is vertical if E = E v . A (rooted) tree is avertical rooted graph satisfying J − ( x ) = 1 for all x ∈ X \ { ϑ } .We refer to the horizontal distance d h ( · , · ) as the graph distance on ( X, E h ). We write x ∼ h y for each pair ( x, y ) ∈ E h ∪ ∆. It is clear that d h ( x, y ) = ∞ for | x | 6 = | y | , and d ( x, y ) ≤ d h ( x, y ). Inthe case that d ( x, y ) = d h ( x, y ), there is a geodesic π ( x, y ) that lies in ( X, E h ), called a horizontalgeodesic of ( X, E ). Definition 2.1.
We call ( X, E ) an expansive graph if it is a rooted graph that satisfies for x, y ∈ X , d h ( x, y ) > ⇒ d h ( u, v ) > , ∀ u ∈ J ( x ) , v ∈ J ( y ) , (2.1) or equivalently if each u ∼ h v implies x ∼ h y whenever x ∈ J − ( u ) and y ∈ J − ( v ) . t follows that in such a graph ( X, E ), if x, y are predecessors of u (i.e., x, y ∈ J −∗ ( u )) with | x | = | y | , then x ∼ h y . It is easy to see that the above condition is also equivalent tomax { d h ( u, v ) , } ≥ d h ( x, y ) , u ∈ J ( x ) , v ∈ J ( y ) . (2.2)Intuitively, in an expansive rooted graph the children are drifted farther apart than their non-neighboring parents (see Figure 1). (a) (b) (c) Figure 1: (a) is an expansive graph, while (b) and (c) are not.
Clearly every rooted tree is expansive. Note that an expansive rooted graph (
X, E ) is calleda pre-augmented (rooted) tree in [LW3, KLW1] if the vertical part (
X, E v ) is a tree. There areimportant examples in which the vertical parts of expansive graphs are not trees; in the followingwe display one of those, which arises from the well-known Bernoulli convolutions. Example 2.2.
Let Y = { ϑ } , Y n = { x = ε · · · ε n : ε i = 0 or 1 } for n ≥
1, and Y = S ∞ n =0 Y n .Clearly there is a natural tree structure E ∗ v on Y : x = ε · · · ε n ∈ J ( y ) if and only if y = ε · · · ε n − .For 0 < ρ <
1, let I n = { ξ ( x ) = P ni =1 ρ n ε i : x = ε · · · ε n ∈ Y n } . If 0 < ρ ≤ /
2, then for each n ≥ ξ : Y n → I n is bijective. However for 1 / < ρ <
1, they may not be bijective, for example, if ρ equals the golden ratio √ − (the positive solution of x + x − ξ (011) = ξ (100).We define an equivalence relation ≃ n on Y n by x ≃ n y if and only if ξ ( x ) = ξ ( y ). Let X n be thequotient of Y n with respect to ≃ n , and use E v to denote the edge set on X = S ∞ n =0 X n induced by( Y, E ∗ v ). Then ( X, E v ) is a vertical graph, but not a tree unless all relations ≃ n are trivial. We canaugment ( X, E v ) by adding a set of horizontal edges:( x, y ) ∈ E h ⇔ x, y ∈ X n , and | ξ ( x ) − ξ ( y ) | ≤ aρ n for some n ≥ , where a = P ∞ n =1 ρ n = ρ − ρ . Let E = E v ∪ E h . Then ( X, E ) is an expansive graph, because for x, y ∈ X n with | ξ ( x ) − ξ ( y ) | > aρ n , | ξ ( u ) − ξ ( v ) | > aρ n − ρ n +1 = aρ n +1 , u ∈ J ( x ) , v ∈ J ( y ) . (cid:3) For more discussion of this example in connection with the iterations and augmented trees, thereader can refer to Appendix and [Wa].A geodesic path [ x i ] i in a rooted graph ( X, E ) is said to be convex if | x i | ≤ ( | x i − | + | x i +1 | )for all i . It is easy to see that for each convex geodesic [ x i ] i , there exist u = x k , v = x ℓ ( k ≤ ℓ andthey can be equal) such that x i − ∈ J ( x i ) , i ≤ k ;( x i − , x i ) ∈ E h , k < i ≤ ℓ ; x i − ∈ J − ( x i ) , i > ℓ. e denote a convex geodesic segment from x to y by π ( x, u, v, y ); such geodesic may not be unique(see Figure 2). Figure 2: Two convex geodesics between x and y . The following simple result is an important property of the expansive graphs.
Proposition 2.3.
Let ( X, E ) be an expansive graph. Then any two vertices x, y ∈ X can be joinedby a convex geodesic.Proof. Let π be a geodesic path from x to y . Then it is clear that π does not contain any segment[ u, v, w ] such that | u | = | w | = | v | −
1; indeed in this case, we have d h ( u, w ) ≤ π contains a segment [ u, v, w ] such that | v | = | w | = | u | + 1 (or | u | = | v | = | w | + 1), then by the expansive property, we can replace this segmentby [ u, w − , w ] (or [ u, u − , w ] respectively), where w − ∈ J − ( w ) (see [Ka, Proposition 3.4]). Such areplacement does not change the end points or increase the length of the segment. Repeating thisprocess, we get a convex geodesic from x to y eventually. Definition 2.4. [Gr]
On a rooted graph ( X, E ) , we define the Gromov product of x, y ∈ X to be ( x | y ) := 12 (cid:0) | x | + | y | − d ( x, y ) (cid:1) . ( X, E ) is said to be (Gromov) hyperbolic if there is δ ≥ such that ( x | y ) ≥ min { ( x | z ) , ( z | y ) } − δ, ∀ x, y, z ∈ X. In this case for a > with e δa < √ , we can define a metric θ a ( · , · ) (Gromov metric) on X by θ a ( x, y ) = inf { X ni =1 e − a ( x i − | x i ) : n ≥ , x = x , x , · · · , x n = y ∈ X } (2.3) for distinct x, y ∈ X , and θ a ( x, x ) = 0 for x ∈ X . Let b X denote the θ a -completion of X , and call ∂X := b X \ X the hyperbolic boundary of ( X, E ) . The reader can refer to [CDP, Gr, GH, Wo] for more details of the hyperbolic graphs and thehyperbolic boundaries. We observe that if (
X, E ) is expansive, then for x, y ∈ X and a convexgeodesic π ( x, u, v, y ), we have( x | y ) = 12 (cid:0) | x | + | y | − d ( x, y ) (cid:1) = 12 (cid:0) | u | + | v | − d h ( u, v ) (cid:1) = ( u | v ) . (2.4)Next we introduce another important notion to describe the departing behavior of the descen-dant vertices in conjunction with the expansive property. Together they provide a useful criterionfor the hyperbolicity of the expansive graphs. efinition 2.5. Let m, k be two positive integers. A rooted graph ( X, E ) is said to be ( m, k )-departing if for x, y ∈ X , d h ( x, y ) > k ⇒ d h ( u, v ) > k, ∀ u ∈ J m ( x ) , v ∈ J m ( y ) . (2.5)It follows from the definitions that every (1 , m, k )-departing for any m, k , and every finite graph is ( m, k )-departing for sufficiently large m, k .However, an infinite expansive graph may not be ( m, k )-departing for any m, k (see Figure 3 for asimple example; in that graph, if both x, y are on the left or right side, then d h ( x, y ) = d h ( u, v ) for u ∈ J m ( x ) and v ∈ J m ( y ), m ≥ Figure 3: An expansive graph that is not ( m, k )-departing for any m, k . Lemma 2.6.
Suppose ( X, E ) is ( m, k ) -departing. Then it is ( m, ℓk ) -departing for any positiveinteger ℓ .Proof. Let x, y ∈ X , u ∈ J m ( x ) and v ∈ J m ( y ) that satisfy d h ( u, v ) ≤ ℓk . Then there exists aset of vertices { z , z , · · · , z s } ⊂ X | x | + m such that z = u , z s = v , s ≤ ℓ and d h ( z i − , z i ) ≤ k for i = 1 , , · · · , s . By the ( m, k )-departing property, we have d h ( x, y ) ≤ s X i =1 d h ( z ( − m ) i − , z ( − m ) i ) ≤ sk ≤ ℓk, where z ( − m ) i ∈ J − m ( z i ) and z ( − m )0 = x , z ( − m ) s = y . This completes the proof.It follows that ( m, m, k )-departing for any k . Also, it is straightforwardto check inductively that (1 , k )-departing implies ( m, k )-departing for any m . Hence we have Corollary 2.7.
Suppose ( X, E ) is (1 , -departing. Then it is ( m, k ) -departing for any positiveintegers m, k . We give two examples to illustrate the ( m, k )-departing property.
Example 2.8. (SG graph)
Let X = S ∞ n =0 Σ n with Σ = { , , · · · , d } ( d ≥
1) be the symbolicspace representing the d -dimensional Sierpinski gasket K , and let K x be the cell associated to theword x ∈ X . Let E v be the natural tree structure on X , and define E h = { ( x, y ) ∈ X × X : | x | = | y | , x = y, and K x ∩ K y = ∅} [Ka]. Consider the rooted graph ( X, E ) with E = E v ∪ E h . It iseasy to show that ( X, E ) is expansive and (1 , ndeed, it is easy to check that u ∼ h v if and only if either (i) u = wi, v = wj for some w ∈ X, i, j ∈ Σ, or (ii) u = wij k , v = wji k for some w ∈ X, i, j ∈ Σ , i = j, k ≥ u − = v − , and (ii) implies d h ( u − , v − ) = 1. Therefore u ∼ h v implies u − ∼ h v − ,i.e., ( X, E ) is expansive. Moreover, for any horizontal segment [ x, z, y ], we can conclude that oneof x − , y − must equal z − , hence d h ( x − , y − ) ≤
1. This shows that (
X, E ) is (1 , (cid:3) Example 2.9. (Hata tree)
The Hata tree K is defined on C by the iterated function system S ( z ) = c ¯ z and S ( z ) = (1 − | c | )¯ z + | c | where | c | , | − c | ∈ (0 ,
1) (see the left figure of Figure 4,the detailed description of K can be found in [Ki1, p.16]). The graph ( X, E ) is the symbolic space X with E = E v ∪ E h , where E h = { ( x, y ) ∈ X × X : | x | = | y | , x = y, and K x ∩ K y = ∅} (the rightpicture). It is clear that ( X, E ) is expansive.From the graph, we see that d h (11 ,
22) = d h (112 , X, E ) is not (1 , X, E )is ( m, m ≥ (cid:3)
122 121111 112211 212221 222
Figure 4: Hata tree ( c = 0 . . i ) and the graph ( X, E ). Proposition 2.10.
Suppose ( X, E ) is ( m, k ) -departing, then the lengths of all horizontal geodesicsin ( X, E ) are bounded by L = L ( m, k ) := ⌈ m +1 k ⌉ k + 2 m . In particular, for ( m, -departing rootedgraphs, L = 4 m + 1 .Proof. Suppose otherwise, then there exists a horizontal geodesic π ( x, y ) with length L + 1. Notethat π ( x, ϑ ) ∪ π ( ϑ, y ) is a path joining x and y . Comparing the lengths of two paths, we have2 | x | ≥ L + 1 > m , i.e., | x | > m . Let x ( − m ) ∈ J − m ( x ) and y ( − m ) ∈ J − m ( y ). Then thereexists a horizontal path joining x ( − m ) and y ( − m ) (otherwise d h ( x ( − m ) , y ( − m ) ) = + ∞ , and Lemma2.6 implies that d h ( x, y ) = + ∞ , a contradiction). Now consider a new path: x → x ( − m ) (alonghorizontal edges) → y ( − m ) → y , and by comparing the length of this new path with the one ofgeodesic π ( x, y ) we have 2 m + d h ( x ( − m ) , y ( − m ) ) ≥ L + 1 . It follows that d h ( x ( − m ) , y ( − m ) ) ≥ ⌈ m +1 k ⌉ k + 1 > ⌈ m +1 k ⌉ k . Making use of Lemma 2.6, we have d h ( x, y ) > ⌈ m +1 k ⌉ k ≥ L + 1. This is a contradiction, and completes the proof.The following theorem provides two useful criteria for the hyperbolicity of expansive graphs. heorem 2.11. Let ( X, E ) be an expansive graph. Then the following assertions are equivalent.(i) ( X, E ) is hyperbolic;(ii) ∃ L < ∞ such that the lengths of all horizontal geodesics are bounded by L ;(iii) ( X, E ) is ( m, k ) -departing for some positive integers m, k .Proof. (i) ⇔ (ii) follows from a similar proof as in [LW1, Theorem 2.3].(iii) ⇒ (ii) follows from Proposition 2.10.(ii) ⇒ (iii): We claim that ( X, E ) is ( L + 1 , L + 2)-departing. Indeed, let x, y ∈ X , x ′ ∈ J L +1 ( x )and y ′ ∈ J L +1 ( y ) satisfying L + 2 < d h ( x ′ , y ′ ) ≤ L + 2) (see Figure 5). By Proposition 2.3, thereexists a convex geodesic segment π ( x ′ , u, v, y ′ ) between x ′ and y ′ , and u = x ′ (by the first inequalityand (ii)). Let u, v ∈ X j . Then2( L + 2) ≥ d h ( x ′ , y ′ ) > d ( x ′ , y ′ ) = 2( | x ′ | − | j | ) + d h ( u, v ) ≥ | x ′ | − j ) . As ℓ := | x ′ | − j ≤ L + 1 = | x ′ | − | x | , we have j ≥ | x | . Let u ′ ∈ J ∗ ( x ) ∩ J −∗ ( x ′ ) ∩ X j and v ′ ∈ J ∗ ( y ) ∩ J −∗ ( y ′ ) ∩ X j . Since x ′ ∈ J ℓ ( u ) ∩ J ℓ ( u ′ ), u and u ′ are predecessors of x ′ , and we have u ∼ h u ′ by the expansive property. Similarly v ∼ h v ′ . Hence by (2.2) and (ii), d h ( x, y ) ≤ max { d h ( u ′ , v ′ ) , } ≤ d h ( u, v ) + 2 ≤ L + 2 . This proves the claim. b bbbb bb b x ′ y ′ yxu ′ v ′ u v Figure 5: Illustration for the proof of Theorem 2.11.
We will call the graphs in the above theorem expansive hyperbolic graphs . In an infinite graph (
X, E ) with root ϑ , we let R v := { x = [ x i ] ∞ i =0 : x = ϑ, and x i +1 ∈ J ( x i ) , ∀ i ≥ } (3.1)denote the class of (geodesic) rays starting from the root ϑ . For brevity, we shall use the boldsymbols such as x , y , z to denote the rays [ x i ] i , [ y i ] i , [ z i ] i in R v respectively.In this section, we assume that ( X, E ) is hyperbolic. It follows from (2.3) that θ a ( x, y ) ≍ e − a ( x | y ) , ∀ x, y ∈ X (3.2)(see [Wo, p.245]), and hence the topology on the θ a -completion b X is independent of the value of a (as θ a ( · , · ) ≍ θ b ( · , · ) a/b for a, b > b X and ∂X are compact. very ray in R v is θ a -Cauchy. Note that for x , y ∈ R v , the Gromov product ( x i | y i ) is increasingin i by the triangle inequality. By using this we define ( x | y ) := lim i →∞ ( x i | y i ). We say that x , y ∈ R v are equivalent if they converge to the same point in ∂X ; this holds if and only if( x | y ) = ∞ . With such equivalence, the quotient of R v is identified with ∂X .For an integer k ≥ x , y ∈ R v , we define | x ∨ y | k := sup { i ≥ d h ( x i , y i ) ≤ k } . (3.3)Clearly if E = E v , | x ∨ y | k = sup { i ≥ x i = y i } for all k ≥
0, and equals ( x | y ) when ( X, E ) is atree. Note that if | x ∨ y | k = ∞ , i.e., d h ( x i , y i ) ≤ k for all i ≥
0, then by( x i | y i ) = | x i | − d ( x i , y i ) ≥ i − k , (3.4)we have lim i →∞ ( x i | y i ) = ∞ , i.e., x and y are equivalent.Furthermore, if the hyperbolic graph ( X, E ) is expansive, then by Theorem 2.11, it is ( m, k )-departing for some m, k >
0. This provides a more concrete characterization of the equivalenceclasses in R v as follows, which is used for some estimations in the sequel. Proposition 3.1.
Suppose the rooted graph ( X, E ) is expansive and ( m, k ) -departing. Then thereexists a constant D = D ( m, k ) > such that | ( x | y ) − | x ∨ y | k | ≤ D , ∀ x , y ∈ R v . (3.5) Consequently, two rays x , y in R v are equivalent if and only if d h ( x i , y i ) ≤ k for all i ≥ .Proof. Note that | x ∨ y | k = ∞ implies ( x | y ) = ∞ . Hence we need only prove (3.5) for the casethat | x ∨ y | k < ∞ . Set ℓ := | x ∨ y | k + 1. Then d h ( x ℓ − , y ℓ − ) ≤ k , and d h ( x ℓ , y ℓ ) > k . Using (3.4)for i = ℓ −
1, we have ( x | y ) ≥ ( x ℓ − | y ℓ − ) ≥ ℓ − − k/ | x ∨ y | k − k/ . On the other hand, by using the ( m, k )-departing in Lemma 2.6 repeatedly, we have d h ( x mj + ℓ , y mj + ℓ ) > j k, ∀ j ≥ . (3.6)For j ≥
0, we choose u, v ∈ X such that π ( x mj + ℓ , u, v, y mj + ℓ ) is a convex geodesic. Then x | u | ∼ h u and v ∼ h y | u | (by the remark after Definition 2.1). By Proposition 2.10, there exists L = L ( m, k ) > d h ( x | u | , y | v | ) ≤ d h ( x | u | , u ) + d h ( u, v ) + d h ( v, y | v | ) ≤ L + 1 = L + 2 . (3.7)Let j = j ( m, k ) be the smallest integer such that 2 j k ≥ L + 2. It follows from (3.6) and (3.7)that | u | = | v | ≤ mj + ℓ −
1. Hence by (2.4), we have( x mj + ℓ | y mj + ℓ ) = ( u | v ) = | u | − d h ( u, v ) ≤ mj + ℓ − , and this implies that ( x | y ) ≤ mj + ℓ − | x ∨ y | k + mj . Setting D = max { k/ , mj } , inequality(3.5) follows. y Proposition 3.1, we can extend the Gromov product to b X by letting( x | ξ ) = sup { lim i →∞ ( x | x i ) } , ( ξ | η ) = sup { lim i →∞ ( x i | y i ) } , (3.8)where x ∈ X , ξ, η ∈ ∂X , and the supremum is taking over all rays [ x i ] i and [ y i ] i in R v that convergeto ξ and η respectively; the differences of the limits for different rays are at most k if ( X, E ) is( m, k )-departing. Then the Gromov metric θ a ( · , · ) on b X satisfies the same estimate as in (3.2).Throughout the rest of the paper, we will assume that J ( x ) = ∅ for all x ∈ X . In this sectionwe will investigate the metric doubling property of ( ∂X, θ a ). For x ∈ X and ξ ∈ ∂X , we extendthe partial order ≺ in Section 2 by writing ξ ≺ x if x lies on some ray [ x i ] i ∈ R v that converges to ξ . Define J ∂ ( x ) := { ξ ∈ ∂X : ξ ≺ x } and call it the cell of x in ∂X . Then it is clear that each J ∂ ( x ) is a nonempty compact subset of ∂X . We remark that in the context of IFS with attractor K and an associated graph structure( X, E ) of the symbolic space, J ∂ ( x ) plays the role of the x -cell K x in K (see Appendix); this willbe discussed in detail in Section 4. Proposition 3.2.
Let ( X, E ) be an ( m, k ) -departing expansive graph. Then there exists a constant γ > (depending on a ) such that for x, y ∈ X n , n ≥ , d h ( x, y ) > k ⇒ dist θ a ( J ∂ ( x ) , J ∂ ( y )) > γe − an . Proof.
For ξ ∈ J ∂ ( x ) and η ∈ J ∂ ( y ), we choose two rays x , y ∈ R v that pass through x, y andconverge to ξ, η respectively. Then | x ∨ y | k < n . It follows from (3.5) that( ξ | η ) ≤ ( x | y ) + k ≤ | x ∨ y | k + k + D < n + k + D . Hence θ a ( ξ, η ) ≥ ce − a ( ξ | η ) > γe − a | x | with γ = ce − a ( k + D ) .For an integer k ≥
0, define the k -shadow of x in ∂X by J k∂ ( x ) := [ {J ∂ ( y ) : d h ( x, y ) ≤ k } . (3.9)(Hence J ∂ ( x ) = J ∂ ( x ).) Proposition 3.3.
Let ( X, E ) be an ( m, k ) -departing expansive graph. Then there exists a constant C ≥ (depending on a ) such that B θ a ( ξ, C − e − a | x | ) ⊂ J k∂ ( x ) ⊂ B θ a ( ξ, Ce − a | x | ) , ∀ x ∈ X, ξ ∈ J ∂ ( x ) . (3.10) Proof.
Suppose x ∈ X n . For η ∈ ∂X \J k∂ ( x ), we choose y ∈ X n with η ∈ J ∂ ( y ). Then d h ( x, y ) > k ,and by Proposition 3.2 we have θ a ( ξ, η ) ≥ dist θ a ( J ∂ ( x ) , J ∂ ( y )) > γe − an , ∀ ξ ∈ J ∂ ( x ) . This shows that B θ a ( ξ, γe − a | x | ) ⊂ J k∂ ( x ) for all ξ ∈ J ∂ ( x ).For ζ ∈ J k∂ ( x ), we choose z ∈ X n that satisfies d h ( x, z ) ≤ k and ζ ∈ J ∂ ( z ). Then it followsfrom (3.8) that ( ξ | ζ ) ≥ ( x | z ) = n − d ( x, z ) ≥ n − k , ∀ ξ ∈ J ∂ ( x ) . Hence θ a ( ξ, ζ ) ≤ C e − a ( ξ | ζ ) ≤ C e − a | x | with C = C e ak/ , and J k∂ ( x ) ⊂ B θ a ( ξ, C e − a | x | ) for all ξ ∈ J ∂ ( x ). e remark that in (3.10), J k∂ ( x ) cannot be simply replaced by J ∂ ( x ), as is shown in the followingsimple example that the left inclusion does not hold. Example 3.4.
Let (
X, E ) be the rooted graph as in Figure 6. It is an expansive graph, in whichany (nontrivial) horizontal geodesic has length 1. By Theorem 2.11, the graph is hyperbolic. Let x = x be as in the figure. Then it is seen that J ∂ ( x ) = { ξ } , where ξ is the limit of the ray x = [ x i ] ∞ i =0 in ∂X . There is another ray a = [ a i ] ∞ i =0 converging to ξ .For n ≥
1, let η n be the limit of the ray [ ϑ, a , a , · · · , a n +1 , b n , · · · ] in ∂X . Then ∂X = { ξ, η , η , · · · } . By (3.8), we have ( ξ | η n ) = n + 1 and θ a ( ξ, η n ) ≍ e − a ( n +1) → n → ∞ . We seethat J ∂ ( x ) cannot contain any ball B ( ξ, r ) , r > (cid:3) = a = x a x = xa b a a a b b b x x x x Figure 6: J ∂ ( x ) does not contain any ball. Definition 3.5. [He]
A metric space ( M, ρ ) is called (metric) doubling if there is an integer ℓ > such that for any ξ ∈ M and r > , the ball B ρ ( ξ, r ) can be covered by a union of at most ℓ ballsof radii r/ . Theorem 3.6.
Suppose ( X, E ) is an expansive hyperbolic graph, and has bounded degree (i.e., sup x ∈ X deg( x ) < ∞ ). Then the hyperbolic boundary ( ∂X, θ a ) is doubling.Proof. Fix a > e δa < √
2, and suppose (
X, E ) is ( m, k )-departing (by Theorem 2.11). Let C ≥ r ∈ (0 , m ∗ ( r ) ( m ∗ ( r )) be the smallest (largestrespectively) nonnegative integers such that C − e − a ( m ∗ ( r )+1) < r < Ce − a ( m ∗ ( r ) − . It follows that m ∗ ( r ) − m ∗ ( r ) < ⌈ log(2 C ) a ⌉ + 2 =: ℓ . For a ball B θ a ( ξ, r ) ⊂ ∂X , let x be the vertexsuch that | x | = m ∗ ( r ) and ξ ∈ J ∂ ( x ). We claim that B θ a ( ξ, r ) ⊂ J k∂ ( x ). Indeed, if m ∗ ( r ) = 0,then x = ϑ , and trivially B θ a ( ξ, r ) ⊂ ∂X = J k∂ ( ϑ ); if m ∗ ( r ) ≥
1, then by Proposition 3.3 we have B θ a ( ξ, r ) ⊂ B θ a ( ξ, C − e − am ∗ ( r ) ) ⊂ J k∂ ( x ) as well. By the claim and Proposition 3.3, we have B θ a ( ξ, r ) ⊂ J k∂ ( x ) = [ y : d h ( x,y ) ≤ k [ z ∈J ℓ ( y ) J ∂ ( z ) ⊂ [ y : d h ( x,y ) ≤ k [ z ∈J ℓ ( y ) B θ a ( η z , Ce − am ∗ ( r ) ) (by (3.10)) ⊂ [ y : d h ( x,y ) ≤ k [ z ∈J ℓ ( y ) B θ a ( η z , r/ , here each η z is chosen arbitrarily from J ∂ ( z ). Let t = sup x ∈ X deg( x ), which is finite by assump-tion. Then { y : d h ( x, y ) ≤ k } ≤ t k and J ℓ ( y ) ≤ t ℓ . Hence the ball B ̺ a ( ξ, r ) is covered by a union of at most t k + ℓ many balls of radii r/
2, and ( ∂X, θ a )is doubling. Remark 1.
A similar result was proved in another setting by Bonk and Schramm [BS, Theorem9.2]; here our proof on expansive hyperbolic graphs is more straightforward.
Remark 2.
Note that the doubling property of ( ∂X, θ a ) does not imply the bounded degreeproperty of ( X, E ). For example, if we take E h = { ( x, y ) ∈ X × X \ ∆ : | x | = | y |} , then it istrivially an expansive hyperbolic graph, and ∂X is a singleton (so it is doubling trivially); butdeg( x ) ≥ X n − x ∈ X n , which has no bound if X n tends to ∞ .However, this converse can be verified with some further separation properties among the cells(Theorem 5.5). In this section, we fix a complete metric space (
M, ρ ), and let C M denote the family of all nonemptycompact subsets of M . By our convention in last section, any graph ( X, E ) mentioned below isassumed to satisfy J ∂ ( x ) = ∅ for all x ∈ X . Definition 4.1.
Let ( X, E v ) be a vertical graph. A map Φ : X → C M is called an index map (on ( X, E v ) over ( M, ρ ) ) if it satisfies(i) Φ( y ) ⊂ Φ( x ) for all x ∈ X and y ∈ J ( x ) ;(ii) T ∞ i =0 Φ( x i ) is a singleton for all x = [ x i ] i ∈ R v .In particular, such Φ is called saturated if (i) is strengthened to Φ( x ) = S y ∈J ( x ) Φ( y ) .We call K := T ∞ n =0 (cid:16)S x ∈ X n Φ( x ) (cid:17) the attractor of Φ , and K x := Φ( x ) ∩ K a cell of K . Remark 1.
An index map Φ induces another index map Φ ′ : X → C K with Φ ′ ( x ) = K x ; the image { K x } x ∈ X is the family of cells of K indexed by X . Since Φ and Φ ′ behave the same at infinity (i.e., T ∞ i =0 Φ( x i ) = T ∞ i =0 Φ ′ ( x i ) for all x = [ x i ] i ∈ R v ), we use these two interchangeably. Remark 2.
For a saturated index map Φ, the attractor K = Φ( ϑ ), and the cell K x = Φ( x ) for all x ∈ X . In fact, every index map Φ induces a saturated index map e Φ : X → C K with e Φ( x ) := \ ∞ n =0 (cid:16)[ y ∈J n ( x ) Φ( y ) (cid:17) , x ∈ X, (4.1)which also behaves the same as Φ at infinity. It is clear that e Φ( x ) ⊂ K x for all x ∈ X , but thereverse inclusion does not hold in general.The index map Φ defines a mapping κ : R v → K by { κ ( x ) } = \ ∞ i =0 Φ( x i ) , ∀ x = [ x i ] i ∈ R v . (4.2)Using the local finiteness of ( X, E v ) and a diagonal argument (cf. [Wo,LW3]), we can show that theimage of κ is equal to K . From Section 3, we see that for an expansive hyperbolic graph ( X, E ),the hyperbolic boundary ∂X can be identified with a quotient set of R v . Hence the induced map : ∂X → K from the quotient is well-defined if κ satisfies: x , y are equivalent ⇒ κ ( x ) = κ ( y );furthermore κ : ∂X → K is one-to-one if the converse is also satisfied. With these, we see that if κ satisfies x and y are equivalent ⇔ κ ( x ) = κ ( y ) , (4.3)then it induces a bijection κ : ∂X → K via the quotient. Definition 4.2.
We call ( X, E, Φ) an admissible index triple if it satisfies(i) ( X, E ) is an expansive hyperbolic graph;(ii) Φ : X → C M is an index map on ( X, E v ) over ( M, ρ ) ;(iii) κ : ∂X → K is a well-defined bijection, i.e., (4.3) holds.In such case, ( X, E ) is said to be an admissible (augmented) graph (associated to Φ ); if ( X, E v ) isa tree, then we call ( X, E ) an admissible augmented tree . By Remark 2, κ ( J ∂ ( x )) = e Φ( x ) ⊂ Φ( x ) , ∀ x ∈ X ; (4.4)and the inclusion is an “=” if and only if the index map Φ is saturated. Via the bijection κ , theGromov metric θ a on ∂X defines naturally a metric ˜ θ a on the attractor K by˜ θ a ( ξ, η ) = θ a ( κ − ( ξ ) , κ − ( η )) , ξ, η ∈ K. (4.5) Proposition 4.3.
For an admissible index triple, the bijection κ is a homeomorphism. We will prove this in [KLWa], as the proof requires more preparatory work and we will not needthe proposition here.For a subset E in ( M, ρ ), we denote the diameter of E by | E | ρ (or simply by | E | ). In Definition4.1, we see that the family { Φ( x ) } x ∈ X satisfies lim n →∞ sup x ∈J n | Φ( x ) | ρ = 0. For b ∈ (0 , ∞ ), we saythat { Φ( x ) } x ∈ X (or Φ) is of exponential type- ( b ) (under ρ ) if the diameter | Φ( x ) | ρ is decreasing in arate of e − b | x | , i.e., | Φ( x ) | ρ = O ( e − b | x | ) as | x | → ∞ , and call Φ an exponential type if it is of type-( b )for some b ∈ (0 , ∞ ).We mainly consider the following two classes of expansive graphs associated to index maps,which are motivated by the augmented trees of the IFS’s [Ka, LW1, LW3, Wa] (see Appendix). Definition 4.4.
Let Φ be an index map on the vertical graph ( X, E v ) . We define a horizontal edgeset by E ( ∞ ) h := [ ∞ n =1 (cid:8) ( x, y ) ∈ X n × X n \ ∆ : Φ( x ) ∩ Φ( y ) = ∅ (cid:9) , (4.6) and let E ∞ = E v ∪ E ( ∞ ) h . We call ( X, E ( ∞ ) ) an AI ∞ -graph , augmented index graph of type-( ∞ ) (or intersection type ).In addition, assume that Φ is of exponential type- ( b ) . For a fixed γ > , we define E ( b ) h := [ ∞ n =1 (cid:8) ( x, y ) ∈ X n × X n \ ∆ : dist ρ (Φ( x ) , Φ( y )) ≤ γe − bn (cid:9) . (4.7) Let E ( b ) = E v ∪ E ( b ) h . We call ( X, E ( b ) ) an AI b -graph , augmented index graph of type-( b ) . It is clear that both (
X, E ( ∞ ) ) and ( X, E ( b ) ) are expansive. Comparing the two definitions,the AI ∞ -graph is more intuitive but needs more information on the neighborhood of Φ( x ) , x ∈ X under the given metric ρ ; the AI b -graph is more flexible on the neighboring cells, which actuallymakes it easier to handle. First we prove heorem 4.5. For an index map Φ on ( X, E v ) over ( M, ρ ) of exponential type- ( b ) , the associated AI b -graph is ( m, -departing, and is an admissible graph. Moreover, κ : ( ∂X, θ a ) → ( K, ρ ) is aH¨older equivalence with exponent b/a , i.e., ρ ( κ ( ξ ) , κ ( η )) a/b ≍ θ a ( ξ, η ) , ∀ ξ, η ∈ ∂X. (4.8) Proof.
It is easy to check from the definition of E ( b ) h that ( X, E ( b ) ) is expansive. To show that it is( m, m ≥
1, let δ := sup z ∈ X e b | z | | Φ( z ) | . Let u ∈ J m ( x ) and v ∈ J m ( y ) with d h ( u, v ) = 2. Using the triangle inequality twice, we havedist(Φ( x ) , Φ( y )) ≤ dist(Φ( u ) , Φ( v )) ≤ (2 γ + δ ) e − b ( | x | + m ) ≤ γe − b | x | , where the positive integer m is chosen to give the last inequality, i.e., (2 γ + δ ) e − bm ≤ γ . Therefore x ∼ h y , and this shows that ( X, E ) is ( m, AI b -graph followsfrom Theorem 2.11.By Proposition 3.1 (with k = 1) and (4.7), we see that two rays x , y are equivalent if and onlyif dist(Φ( x i ) , Φ( y i )) ≤ γe − bi for all i . This verifies (4.3) so that κ : ∂X → K is a well-definedbijection, and hence ( X, E ( b ) ) is an admissible graph.We now prove that κ is a H¨older equivalence. For distinct ξ, η ∈ ∂X , we take two rays x , y ∈ R v that converge to ξ, η respectively with ( ξ | η ) = ( x | y ) (by (3.8) and the following remark). Let n = | x ∨ y | as in (3.3) with k = 1, i.e., d h ( x n , y n ) ≤ d h ( x n +1 , y n +1 ) ≥
2. By Proposition 3.1,we have | ( ξ | η ) − n | = | ( x | y ) − n | ≤ D for some D := D ( m, >
0. As κ ( ξ ) ∈ Φ( x n +1 ) ⊂ Φ( x n )and κ ( η ) ∈ Φ( y n +1 ) ⊂ Φ( y n ), we get the lower bound of (4.8) by ρ ( κ ( ξ ) , κ ( η )) ≥ dist(Φ( x n +1 ) , Φ( y n +1 )) ≥ γe − b ( n +1) ≥ γe − b ( D +1) e − b ( ξ | η ) ≥ C θ a ( ξ, η ) b/a , and the upper bound by ρ ( κ ( ξ ) , κ ( η )) ≤ | Φ( x n ) | + dist(Φ( x n ) , Φ( y n )) + | Φ( y n ) |≤ (2 δ + γ ) e − bn ≤ (2 δ + γ ) e bD e − b ( ξ | η ) ≤ C θ a ( ξ, η ) b/a . This completes the proof.Now we turn to study the AI ∞ -graphs. Unlike the AI b -graph, the AI ∞ -graph is not alwayshyperbolic (see Example 6.1), and sufficient conditions for its hyperbolicity will be provided inSection 5. The following result shows that if it is hyperbolic, then the admissibility follows. Proposition 4.6.
Suppose the AI ∞ -graph ( X, E ( ∞ ) ) is hyperbolic. Then for two rays x , y ∈ R v ,the following assertions are equivalent.(i) x and y are equivalent; (ii) κ ( x ) = κ ( y ) ; (iii) d h ( x i , y i ) ≤ for all i ≥ .It follows from (4.3) that ( X, E ( ∞ ) ) is an admissible graph (Definition 4.2).Proof. (iii) ⇒ (i) is clear. For (ii) ⇒ (iii), since κ ( x ) ∈ Φ( x i ) and κ ( y ) ∈ Φ( y i ), we haveΦ( x i ) ∩ Φ( y i ) = ∅ for all i ≥
0. This yields (iii) by the definition (4.6) of E ( ∞ ) h .For (i) ⇒ (ii), as ( X, E ( ∞ ) ) is expansive and hyperbolic, by Theorem 2.11(iii) and Proposition3.1, there exists an integer k > d h ( x i , y i ) ≤ k for all i ≥
0. We show inductively that anysuch k will imply (ii). When k = 1, it follows that Φ( x i ) ∩ Φ( y i ) = ∅ for all i ≥
0. By the compactness f Φ( x i ) ∩ Φ( y i ), the intersection (cid:0)T ∞ i =0 Φ( x i ) (cid:1) ∩ (cid:0)T ∞ i =0 Φ( y i ) (cid:1) = T ∞ i =0 (Φ( x i ) ∩ Φ( y i )) = ∅ , hence κ ( x ) = κ ( y ).Inductively, suppose (ii) is verified when k = n for some n >
0. Let x , y ∈ R v satisfy d h ( x i , y i ) ≤ n + 1. Then for each i ≥
0, we choose z i ∈ X such that d h ( x i , z i ) ≤ d h ( z i , y i ) ≤ n . Thesequence { z i } ∞ i =0 may not be a ray; however, using the local finiteness of ( X, E v ) and a diagonalargument, we can choose a ray w ∈ R v such that each w i contains an infinite subsequence of { z i } ∞ i =0 , and the expansive property (2.2) implies that d h ( x i , w i ) ≤ d h ( w i , y i ) ≤ n for all i ≥
0. Using the induction hypothesis, we have κ ( x ) = κ ( w ) = κ ( y ), and the proof is completedby induction.When Φ is of exponential type-( b ), comparing the AI ∞ -graph with the AI b -graph, it is clearthat E ( ∞ ) ⊂ E ( b ) . The following is a consequence of Theorem 4.5. Corollary 4.7.
Suppose the index map Φ is of exponential type- ( b ) , and the associated AI ∞ -graph ( X, E ( ∞ ) ) is hyperbolic. Then κ : ( ∂X, θ a ) → ( K, ρ ) is H¨older continuous with exponent b/a , i.e., ρ ( κ ( ξ ) , κ ( η )) a/b ≤ Cθ a ( ξ, η ) , ∀ ξ, η ∈ ∂X. (4.9) Proof.
We consider the associated AI b -graph ( X, E ( b ) ), and denote its graph distance and Gromovproduct by d ′ ( · , · ) and ( ·|· ) ′ respectively. By Theorem 4.5, ( X, E ( b ) ) is hyperbolic, and the bijection κ ′ : ∂X ′ → K satisfies ρ ( κ ′ ( ξ ) , κ ′ ( η )) ≍ e − b ( ξ | η ) ′ for all ξ, η ∈ ∂X ′ . As κ = κ ′ on R v , it followsthat ∂X = ∂X ′ and κ = κ ′ .Note that E ( ∞ ) ⊂ E ( b ) . This implies d ( x, y ) ≥ d ′ ( x, y ), and( x | y ) = 12 ( | x | + | y | − d ( x, y )) ≤
12 ( | x | + | y | − d ′ ( x, y )) = ( x | y ) ′ for all x, y ∈ X . Taking the limits in (3.8), we have ( ξ | η ) ≤ ( ξ | η ) ′ , and θ a ( ξ, η ) ≥ c e − a ( ξ | η ) ≥ c e − a ( ξ | η ) ′ ≥ c ρ ( κ ′ ( ξ ) , κ ′ ( η )) a/b = c ρ ( κ ( ξ ) , κ ( η )) a/b . This verifies (4.9), the H¨older continuity of κ .In general, we cannot expect this κ to be a H¨older equivalence (see Example 6.2). In thefollowing theorem, we give a characterization for the hyperbolicity, or equivalently the ( m, k )-departing property, of the AI ∞ -graph associated to a saturated index map (Remark 2 of Definition4.1), together with the H¨older equivalence to hold. Theorem 4.8.
Suppose Φ is a saturated index map on ( X, E v ) over ( M, ρ ) . Then for b ∈ (0 , ∞ ) and an integer k > , the following assertions are equivalent.(i) The AI ∞ -graph ( X, E ( ∞ ) ) is ( m, k ) -departing for some m > , and κ : ( ∂X, θ a ) → ( K, ρ ) is a H¨older equivalence with exponent b/a , i.e., ρ ( κ ( ξ ) , κ ( η )) a/b ≍ θ a ( ξ, η ) , ∀ ξ, η ∈ ∂X. (4.10) (ii) Φ is of exponential type- ( b ) under ρ , and there exists γ > such that ( X, E ( ∞ ) ) satisfiesfor x, y ∈ X , | x | = | y | and d h ( x, y ) > k ⇒ dist ρ (Φ( x ) , Φ( y )) > γe − b | x | . (4.11) roof. For (i) ⇒ (ii), since ( X, E ( ∞ ) ) is expansive and ( m, k )-departing, it is hyperbolic (Theorem2.11). By Proposition 4.6, it is an admissible graph, and the bijection κ : ∂X → K is well-defined.Using (4.4) and (4.10), we have | Φ( x ) | ρ = | κ ( J ∂ ( x )) | ρ ≤ C |J ∂ ( x ) | b/aθ a ≤ C e − b | x | for all x ∈ X . Therefore Φ is of exponential type-( b ) under ρ . Moreover, for x, y ∈ X n with d h ( x, y ) > k , it follows from (4.4), (4.10) and Proposition 3.2 thatdist ρ (Φ( x ) , Φ( y )) = dist ρ ( κ ( J ∂ ( x )) , κ ( J ∂ ( y ))) ≥ C dist θ a ( J ∂ ( x ) , J ∂ ( y )) b/a > C γ e − bn . This proves (4.11) with γ = C γ .For (ii) ⇒ (i), the proof is similar to the one of Theorem 4.5 on the AI b -graph. Let δ =sup z ∈ X e b | z | | Φ( z ) | ρ , which is finite as Φ is of exponential type-( b ). Let u ∈ J m ( x ) and v ∈ J m ( y )with d h ( u, v ) = ℓ < k . Then Φ( u ) and Φ( v ) are joined by a chain { Φ( u j ) } ℓj =0 with u = u and u ℓ = v , in which | u j | = | u | = | x | + m and Φ( u j − ) ∩ Φ( u j ) = ∅ for all j ∈ { , , · · · , ℓ } . Therefore,dist ρ (Φ( x ) , Φ( y )) ≤ dist ρ (Φ( u ) , Φ( v )) ≤ X ℓ − j =1 | Φ( u j ) | ρ ≤ (2 k − δ e − b ( | x | + m ) ≤ γe − b | x | , where the integer m > k − δ e − bm ≤ γ . By (4.11),we have d h ( x, y ) ≤ k , and this proves that ( X, E ( ∞ ) ) is ( m, k )-departing.To prove (4.10), for ξ = η ∈ ∂X , we take x , y ∈ R v that converge to ξ, η respectively with( ξ | η ) = ( x | y ), and let n = | x ∨ y | k as in (3.3). It follows from Proposition 3.1 that | ( ξ | η ) − n | = | ( x | y ) − n | ≤ D for some D = D ( m, k ) >
0. Using κ ( ξ ) ∈ Φ( x n +1 ), κ ( η ) ∈ Φ( y n +1 ), d h ( x n +1 , y n +1 ) > k and (4.11), we get the lower bound of (4.10) by ρ ( κ ( ξ ) , κ ( η )) ≥ dist ρ (Φ( x n +1 ) , Φ( y n +1 )) > γe − b ( n +1) ≥ γe − b ( D +1) e − b ( ξ | η ) ≥ cθ a ( ξ, η ) b/a ;the upper bound is proved by Corollary 4.7. This completes the proof. Remark.
Letting k = 1 in (4.11), the condition becomes | x | = | y | and Φ( x ) ∩ Φ( y ) = ∅ ⇒ dist(Φ( x ) , Φ( y )) > γe − b | x | . (4.12)This is the condition (H) in [LW1] in the setup where K is a self-similar set (see also Appendix,Theorem 7.1); the authors proved that this condition is sufficient for the H¨older equivalence between ∂X and K . Here a necessity part is also provided, together with an extra relation with the ( m, k )-departing property (or hyperbolicity) of these graphs. In this section, we aim to give some sufficient conditions for AI ∞ -graphs to be hyperbolic, andcharacterize the bounded degree property of AI b - and AI ∞ -graphs; both involve some separationconditions on index maps and the doubling property of attractors.Let Φ be an index map on a vertical graph ( X, E v ) over a complete metric space ( M, ρ ) withattractor K . We call a map ι : X → K a projection (with respect to Φ) if it satisfies ι ( x ) ∈ K x (:= ( x ) ∩ K ) for all x ∈ X . When there is no confusion, we shall denote the ball B ρ ( ξ, r ) by B ( ξ, r )for simplicity. Definition 5.1.
Let Φ be an index map with attractor K . For b ∈ (0 , ∞ ) , we say that Φ (or { K x } x ∈ K ) satisfies(i) condition ( S b ) if for any c > , there is a constant ¯ ℓ = ¯ ℓ ( c ) such that { x ∈ X n : K x ∩ F = ∅} ≤ ¯ ℓ, ∀ n ≥ and F ⊂ M with | F | ρ < ce − bn ; (5.1) (ii) condition ( B b ) if there exist a projection ι : X → K and c ∈ (0 , ∞ ) such that B ( ι ( x ) , c e − b | x | ) ∩ K ⊂ K x , ∀ x ∈ X. (5.2)We remark that condition ( S b ) is motivated from the open set condition on self-similar sets (see(7.3) in Appendix, and condition ( S ′ b ) in Proposition 5.3). To study the above conditions, we needa preliminary result on doubling metric spaces (Lemma 5.2). For a subset F ⊂ M and r >
0, definethe r -covering number of F by N cr ( F ) := inf { ⊂ F and F ⊂ [ ξ ∈ Ξ B ( ξ, r ) } ; (5.3) F is called totally bounded if the r -covering number is finite for all r >
0. We also define the r -packing number of F to be N pr ( F ) := sup { ⊂ F and B ( ξ, r ) ∩ B ( η, r ) = ∅ , ∀ ξ, η ∈ Ξ , ξ = η } , and the r -separating number of F to be N sr ( F ) := sup { ⊂ F and ρ ( ξ, η ) ≥ r, ∀ ξ, η ∈ Ξ , ξ = η } . Recall that the metric space (
M, ρ ) is doubling (Definition 3.5) if and only ifsup { N cr ( B ( ξ, r )) : ξ ∈ M, r > } < ∞ . Lemma 5.2.
Let ( M, ρ ) be a metric space. Then the inequalities N s r ( F ) ≤ N pr ( F ) ≤ N cr ( F ) ≤ N sr ( F ) (5.4) hold for all totally bounded subset F ⊂ M and r > . As a consequence, the following assertionsare equivalent.(i) ( M, ρ ) is doubling (Definition 3.5).(ii) For some ( ⇔ any) t > , b N c ( t ) := sup { N cr ( B ( ξ, tr )) : ξ ∈ M, r > } < ∞ .(iii) For some ( ⇔ any) t > , b N p ( t ) := sup { N pr ( B ( ξ, tr )) : ξ ∈ M, r > } < ∞ .(iv) For some ( ⇔ any) t > , b N s ( t ) := sup { N sr ( B ( ξ, tr )) : ξ ∈ M, r > } < ∞ . The inequality (5.4) is straightforward by using the definitions, and the equivalence of (i)–(iv)follows directly from (5.4). We omit the detail.
Proposition 5.3.
Let Φ be an index map with attractor K , and is of exponential type- ( b ) . Suppose ( K, ρ ) is doubling, then the following conditions are equivalent:(i) condition ( S b ) ; ii) condition ( S ′ b ) : there exist c > and ℓ > such that { x ∈ X n : K x ∩ B ( ξ, c e − bn ) = ∅} ≤ ℓ , ∀ n ≥ , ξ ∈ K ; (5.5) (iii) condition ( S ′′ b ) : there exist a projection ι : X → K , c > and ℓ > such that { x ∈ X n : ρ ( ι ( x ) , ι ( y )) < c e − bn } ≤ ℓ , ∀ n ≥ , y ∈ X n . (5.6) Proof. (i) ⇒ (ii) ⇒ (iii) is obvious. We need only show (ii) ⇒ (i) and (iii) ⇒ (ii).(ii) ⇒ (i): For any c > F ⊂ M with | F | ρ < ce − bn , we will show that (5.1) holds for some¯ ℓ . Without loss of generality, we assume that c > c and K ∩ F = ∅ . Fix a point ξ ∈ K ∩ F ,and denote the open ball B ( ξ, ce − bn ) ∩ K in ( K, ρ ) by B K . Since ( K, ρ ) has doubling property,by Lemma 5.2(ii), B K can be covered by a union of N := b N c ( cc ) open balls B ( ξ i , c e − bn ) ∩ K in( K, ρ ), where ξ i ∈ K , i = 1 , · · · , N , i.e., K ∩ F ⊂ B K ⊂ [ N i =1 (cid:16) K ∩ B ( ξ i , c e − bn ) (cid:17) . It follows from (5.5) that { x ∈ X n : K x ∩ F = ∅} ≤ X N i =1 { x ∈ X n : K x ∩ B ( ξ i , c e − bn ) = ∅} ≤ ℓ N . We see that (5.1) holds for the constant ¯ ℓ = ℓ N .(iii) ⇒ (ii): For n ≥ ξ ∈ K , denote X n ( ξ ) := { x ∈ X n : K x ∩ B ( ξ, c e − bn ) = ∅} , on which we define an edge set e E = { ( x, y ) ∈ X n ( ξ ) × X n ( ξ ) \ ∆ : ρ ( ι ( x ) , ι ( y )) < c e − bn } . By(5.6), we see that the maximal degree in the graph ( X n ( ξ ) , e E ) does not exceed ( ℓ − K : X n ( ξ ) → Σ with ≤ ℓ such that K ( x ) = K ( y ) if ( x, y ) ∈ e E .Since Φ is of exponential type-( b ), there exists a constant δ > | Φ( x ) | ρ ≤ δ e − b | x | for all x ∈ X . Consider the discrete sets M ( i ) := { ι ( x ) : x ∈ X n ( ξ ) , K ( x ) = i } , i ∈ Σ . Then ρ ( ι ( x ) , ι ( y )) ≥ c e − bn for x = y ∈ M ( i ) . By the triangle inequality, it is easy to see that M ( i ) ⊂ B ( ξ, ( c + δ ) e − bn ) holds for all i ∈ Σ. Using Lemma 5.2(iii), we have X n ( ξ ) = X i ∈ Σ M ( i ) ≤ ℓ N sc e − bn ( B ( ξ, ( c + δ ) e − bn ) ∩ K ) ≤ ℓ b N s (( c + δ ) /c ) < ∞ . This completes the proof by letting ℓ = ℓ b N s (( c + δ ) /c ).The following theorem provides sufficient conditions for AI ∞ -graphs to be hyperbolic, andcompletes the study of AI ∞ -graphs in the last section. Theorem 5.4.
Let Φ be an index map with attractor K , and is of exponential type- ( b ) . If either(i) condition ( S b ) is satisfied; or(ii) the attractor ( K, ρ ) is doubling, and condition ( B b ) is satisfied, hen the AI ∞ -graph is hyperbolic, and hence an admissible graph.Proof. We prove the hyperbolicity of (
X, E ∞ ) by using a similar argument as in [LW3, Theorem1.2]. Suppose it is not hyperbolic. Then by Theorem 2.11, for any integer m >
0, there exists ahorizontal geodesic [ x , x , · · · , x m ] with length 3 m . Clearly | x | =: n > m , and by the expansiveproperty (2.1), there is a horizontal path [ y , y , · · · , y ℓ ] in X n − m (see Figure 7) such that(a) y ∈ J − m ( x ), y ℓ ∈ J − m ( x m ), and y i ∈ S mj =0 J − m ( x j ) for all 0 < i < ℓ ;(b) ( y i , y j ) ∈ E h if and only if | i − j | = 1. x x x x j x m y i y y y ` x x y Figure 7: Two horizontal paths [ x j ] j and [ y i ] i . As 3 m = d ( x , x m ) ≤ d ( x , y ) + d ( y , y ℓ ) + d ( y ℓ , x m ) ≤ m + ℓ + m , we have ℓ ≥ m . Let F := S ℓi =0 Φ( y i ). Note that by (a), for any η ∈ F , there exists k ∈ { , , . . . , m } such that bothΦ( x k ) and η are contained in some Φ( y i ). Thus for any projection ι : X → K , ρ ( ι ( x ) , η ) ≤ X k − j =0 ρ ( ι ( x j ) , ι ( x j +1 )) + | Φ( y i ) |≤ kδ e − bn + δ e b ( m − n ) ≤ δ (6 me − bm + 1) e b ( m − n ) , where δ := sup z ∈ X e b | z | | Φ( z ) | < ∞ . Take m large enough such that 6 me − bm <
1. Then F = [ ℓi =0 Φ( y i ) ⊂ B ( ι ( x ) , δ e b ( m − n ) ) =: B. (i) If condition (S b ) is satisfied, by noting that | F | ≤ δ e b ( m − n ) =: ce b ( m − n ) , we have ℓ + 1 ≤ (cid:8) y ∈ X n − m : Φ( y ) ⊂ F (cid:9) ≤ (cid:8) y ∈ X n − m : K y ∩ F = ∅ (cid:9) ≤ ¯ ℓ ( c ) . This is impossible since ℓ ≥ m can be arbitrarily large.(ii) If ( K, ρ ) is doubling and condition ( B b ) holds, then we choose the above ι to satisfy (5.2).By (b), the ball B contains at least ⌊ ℓ/ ⌋ + 1 > m/ B ( ι ( y i ) , c e b ( m − n ) ),0 ≤ i ≤ ⌊ ℓ/ ⌋ . On the other hand by considering the packing number and the constant of doubling(see Lemma 5.2(iii)), we have N pe b ( m − n ) ( B ∩ K ) ≤ b N p (2 δ /c ) < ∞ . As m can be arbitrarily large, this is impossible. Hence ( X, E ∞ ) is hyperbolic in either case. ByProposition 4.6, the AI ∞ -graph is an admissible graph.Recall that for an expansive hyperbolic graph with bounded degree, the hyperbolic boundarypossesses the doubling property (Theorem 3.6). With the separation property, we have a strongerresult for AI b -graphs as well as a sufficient condition for AI ∞ -graphs. heorem 5.5. Let Φ be an index map of exponential type- ( b ) . Then the AI b -graph has boundeddegree if and only if condition ( S b ) is satisfied. As a consequence, ( S b ) condition is sufficient forthe AI ∞ -graph to have bounded degree.Proof. It is clear that the last statement follows from the first one, since E ∞ ⊂ E ( b ) .To show the necessity of the first part, by Theorem 4.5, the AI b -graph is hyperbolic, and thehyperbolic boundary ( ∂X, θ a ) is H¨older equivalent to the attractor ( K, ρ ). From Theorem 3.6 weknow that ( ∂X, θ a ) has the doubling property, which is preserved by the H¨older equivalence κ , andthus ( K, ρ ) is also doubling.Next we will show that the index map Φ satisfies condition ( S ′′ b ). We fix an arbitrary projection ι from X to K . For x, y ∈ X n with ρ ( ι ( x ) , ι ( y )) < γe − bn (where γ > E ( b ) h ), we have x ∼ h y since dist ρ (Φ( x ) , Φ( y )) ≤ ρ ( ι ( x ) , ι ( y )) < γe − bn . Hence for all y ∈ X n , { x ∈ X n : ρ ( ι ( x ) , ι ( y )) < γe − bn } ≤ { x ∈ X n : x ∼ h y } ≤ deg( y ) . As ℓ := sup y ∈ X deg( y ) < ∞ , (5.6) holds for c = γ . Making use of Proposition 5.3, we see thatcondition ( S b ) is satisfied.For the sufficiency, we calculate the degree of a fixed vertex y ∈ X n . Set δ := sup z ∈ X e b | z | | Φ( z ) | .For x ∈ J ( y ), we see that K x ⊂ Φ( x ) ⊂ Φ( y ). As | Φ( y ) | ≤ δ e − bn =: ce − b ( n +1) , condition ( S b )implies that J ( y ) ≤ { x ∈ X n +1 : K x ∩ Φ( y ) = ∅} ≤ ¯ ℓ ( c ) . For x ∈ J − ( y ), it follows thatΦ( y ) ⊂ Φ( x ) ⊂ F := { ξ ∈ M : ρ ( ξ, Φ( y )) ≤ δ e b (1 − n ) } . Using the triangle inequality, we get | F | < δ e b (1 − n ) =: c ′ e b (1 − n ) , and thus by condition ( S b ), J − ( y ) ≤ { x ∈ X n − : K x ∩ F = ∅} ≤ ¯ ℓ ( c ′ ) . For x ∈ X n with ( x, y ) ∈ E h , using the triangle inequality we have K x ⊂ Φ( x ) ⊂ G := { ξ ∈ M : ρ ( ξ, Φ( y )) ≤ ( γ + δ ) e − bn } . It follows in a similar way that | G | < γ + δ ) e − bn =: c ′′ e − bn , and hence { x ∈ X : ( x, y ) ∈ E h } ≤ { x ∈ X n : K x ∩ G = ∅} ≤ ¯ ℓ ( c ′′ ) . As deg( y ) = J − ( y ) + J ( y ) + { x ∈ X : ( x, y ) ∈ E h } , we conclude from the above estimatesthat deg( y ) is uniformly bounded by ¯ ℓ ( c ) + ¯ ℓ ( c ′ ) + ¯ ℓ ( c ′′ ) for all y ∈ X , so that the AI b -graph is ofbounded degree. We complete the proof.As a consequence of Theorems 3.6, 5.5 and Proposition 5.3, we have Corollary 5.6.
Let Φ be an index map with attractor K , and is of exponential type- ( b ) . Then the AI b -graph has bounded degree if and only if the attractor ( K, ρ ) is doubling, and condition ( S ′′ b ) (or ( S ′ b ) ) in Proposition 5.3 is satisfied. Examples and more on IFSs
We first give an example that the AI ∞ -graph is not hyperbolic. Example 6.1. (An anisotropic binary partition of unit square)
We define a binary subdi-vision scheme to partition [0 , iteratively into subrectangles. By a Type-I subdivision, we meandividing a rectangle J horizontally into two equal subrectangles; Type-II means dividing J verti-cally instead. Let L = { ℓ ( ℓ + 1) : integer ℓ > } = { , , · · · } . Now let J ϑ = [0 , . Supposewe have defined J x with x = x · · · x n , x i = 0 ,
1. If n + 1 ∈ L , we use Type-I subdivision on J x toobtain J x and J x ; otherwise, we use Type-II for the subdivision (see Figure 8). Φ (1) Φ (0) Φ (00) Φ (01) Φ (10) Φ (11) Φ (101) Φ (100) Φ (001) Φ (000) Φ (010) Φ (011) Φ (110) Φ (111) Figure 8: The binary partition { Φ( x ) } x ∈ X n , n = 1 , , , ,
5; 1 , ∈ L . Let (
X, E v ) be the corresponding binary tree , on which we define the index map Φ( x ) = J x and consider the AI ∞ -graph ( X, E ). Fix an integer ℓ >
0. Let n := ℓ ( ℓ +1)2 , x = 0 n and y =1010 · · · ℓ − . Then Φ( x ) = [0 , − n + ℓ ] × [0 , − ℓ ] and Φ( y ) = [0 , − n + ℓ ] × [1 − − ℓ , , respectively. It is clear that d h ( x, y ) = 2 ℓ −
1. Taking u = 0 n + ℓ ∈ J ℓ ( x ) and v = 1010 · · · ℓ − ℓ ∈ J ℓ ( y ) , we can also checkthat d h ( u, v ) = d h ( x, y ) = 2 ℓ −
1. This shows that (
X, E ) is not ( m, k )-departing whenever m ≤ ℓ and k ≤ ℓ −
2. As ℓ can be arbitrary, ( X, E ) is not hyperbolic by Theorem 2.11. (cid:3)
Our next example gives an IFS that is homogeneous (the contraction ratios are equal) andsatisfies the OSC. The associated AI ∞ -graph is hyperbolic (by Theorem 5.4(i)), but the hyperbolicboundary is not H¨older equivalent to the attractor. This shows that (unlike the AI b -graphs) theone-sided H¨older inequality in Corollary 4.7 cannot be improved. Example 6.2. In M = R , let p = (0 , p = (1 , p = (3 , p = ( η, p = (0 ,
3) and p = (3 , η := X ∞ ℓ =0 (4 − n ℓ + 4 − n ℓ − ) with n ℓ = 1 + ℓ ( ℓ + 7)2 , ℓ = 0 , , · · · Let S j ( x ) = ( x + p j ), j ∈ Σ = { , , · · · , } , and let K be the self-similar set of the IFS { S j } j =1 .We set X = Σ ∗ (the symbolic space) and Φ( x ) = S x ([0 , ), x ∈ X . Clearly S j =1 Φ( j ) ⊂ [0 , , andevery Φ( x ) is a square with side length 4 −| x | (see Figure 9). Hence Φ is an index map of exponentialtype-( b ) with b = ln 4, and it is easy to see that the associated saturated map e Φ( x ) = S x ( K ) forany x ∈ X (see (4.1)) .Fix an integer ℓ >
0. Let x = 5221 · · · ℓ +1 and y = 45 n ℓ − . Clearly | x | = | y | = n ℓ , andΦ( y ) has the same upper-left corner ζ = ( η/ , /
4) as Φ(4). Moreover, we can calculate that thelower-left corner of Φ( x ) is ( ξ/ , / ξ := X ℓ − k =0 (4 − n k + 4 − n k − ) = η − X ∞ k = ℓ (4 − n k + 4 − n k − ) . Let u = x
23 and v = y
56. Then | u | = | v | = n ℓ + 2, the lower-right corner of Φ( u ) is ( ξ/ · − n ℓ − , / v ) is ( η/ − n ℓ − − − n ℓ − , / (1) Φ (2) Φ (3) Φ (6) Φ (5) Φ (4) ζ ζ Φ ( x ) Φ ( y ) Φ ( u ) Φ ( v ) Figure 9: The squares in Example 6.2. S u ( K ) ∩ S v ( K ) = Φ( u ) ∩ Φ( v ) = ∅ . By checking the two nearest corners of Φ( u ) and Φ( v ), we seethat dist( e Φ( u ) , e Φ( v )) = dist(Φ( u ) , Φ( v )), hencedist( S u ( K ) , S v ( K )) = dist(Φ( u ) , Φ( v ))= ( η/ − n ℓ − − − n ℓ − ) − ( ξ/ · − n ℓ − )= X ∞ k = ℓ +1 (4 − n k − + 4 − n k − ) < − n ℓ +1 = 4 − ℓ − e − b | u | . (6.1)Consider the AI ∞ -graph ( X, E ) associated to e Φ, i.e., E h = { ( x, y ) ∈ X × X \ ∆ : | x | = | y | , S x ( K ) ∩ S y ( K ) = ∅} . As the IFS { S j } j =1 satisfies the OSC, so that condition ( S b ) in Definition 5.1 is satisfied, by Theorem5.4, ( X, E ) is hyperbolic. Since neither Φ( u ) nor Φ( v ) intersects other cells in { Φ( z ) } z ∈ X nℓ +2 , wehave d h ( u, v ) = ∞ . As ℓ can be arbitrarily large, (6.1) implies that the condition (4.11) fails forany k ≥
1. By Theorem 4.8, the hyperbolic boundary ( ∂X, θ a ) of ( X, E ) is not H¨older equivalentto K . (cid:3) We now return to the setup in which { S j } Nj =1 is a contractive IFS on a complete metric space( M, ρ ) (see Appendix). In the previous studies, the augmented tree was established according tothe geometric sizes of K x [LW1, LW3]. Within the framework of augmented index graphs (Section4), we are able to extend the consideration to some weighted IFSs. In the rest of this section, weare given a vector of weights s ∈ (0 , N on { S j } Nj =1 (instead of contraction ratio r = ( r j ) Nj =1 ), andexpect some new metric induced on K .Let Σ ∗ be the symbolic space of a contractive IFS { S j } Nj =1 . Let s = ( s , s , · · · , s N ) be a vectorof weights on Σ with s j ∈ (0 ,
1) (not necessarily probability weight). Write s ∗ := min j ∈ Σ { s j } , and s x := s i s i · · · s i m for x = i i · · · i m ∈ Σ ∗ ( s ϑ = 1 by convention). We consider a regrouping onΣ ∗ by setting X := { ϑ } , and for n ≥ X n = X n ( s ) := { x = i i · · · i m ∈ Σ ∗ : s x ≤ s n ∗ < s i s i · · · s i m − } . (6.2)Let X = X ( s ) := S ∞ n =0 X n denote the modified coding space with respect to s . This X hasa natural tree structure E v that consists of edges between each x = i i · · · i m ∈ X n and y = i · · · i m i m +1 · · · i k ∈ X n +1 . Let Φ( x ) = S x ( K ), x ∈ X . Then Φ is a saturated index map on( X, E v ) over ( K, ρ ).Let r ∗ = max j ∈ Σ { r j } be the maximal contraction ratio of { S j } j ∈ Σ . Then it is clear that { Φ( x ) } x ∈ X ( s ) is always of exponential type-( b ) with b = | log r ∗ | . From (4.5) and Theorem 4.5, theassociated AI b -graph ( X ( s ) , E ( b ) ) brings a metric ˜ θ a on K that is H¨older equivalent to the original ρ , which is similar to the previous investigation; also, note that the index map Φ on ( X ( s ) , E ( b ) )rarely satisfies the separation conditions in Section 5, and hence we cannot expect that ( X ( s ) , E ( b ) )has bounded degree. The more interesting question is to investigate the hyperbolicity of AI ∞ -graph ( X ( s ) , E ( ∞ ) ) without assuming any separation property. The problem is difficult in general.However, for post critically finite (p.c.f.) sets, we have some rather complete conclusions, as wellas a new connection to the harmonic structure and resistance networks in analysis on fractals.We recall the notion of p.c.f. sets (without assuming self-similarity) [Ki1]. Let Σ ∞ := { i i · · · : i k ∈ Σ , k ≥ } be the set of infinite words, and let ̟ : Σ ∞ → K be the natural surjection definedby { ̟ ( i i · · · ) } = \ ∞ k =1 S i i ··· i k ( K ) . The shift map σ : Σ ∞ → Σ ∞ is given by σ ( i i i · · · ) = i i i · · · . Define the critical set and the post critical set by C := ̟ − (cid:16)[ i,j ∈ Σ ,i = j (cid:0) S i ( K ) ∩ S j ( K ) (cid:1)(cid:17) and P := [ ∞ n =1 σ n ( C ) (6.3)respectively. We call the IFS { S j } Nj =1 (or K ) post critically finite (p.c.f.) if P is a finite set.Also we define V = ̟ ( P ) as the boundary of K . A p.c.f. set K has the property that every twocells S i ( K ) , S j ( K ), i, j ∈ Σ , i = j are disjoint or intersect at finitely many points. An importantconsequence is that [Ki1, Proposition 1.3.5] for any distinct x, y in the same X n ( s ), S x ( K ) ∩ S y ( K ) = S x ( V ) ∩ S y ( V ) . Theorem 6.3.
Let { S j } Nj =1 be a contractive IFS that satisfies the p.c.f. property. Then there existsan integer m > such that for any s ∈ (0 , N , the AI ∞ -graph ( X ( s ) , E ( ∞ ) ) is an ( m, -departingexpansive graph, and hence ( X ( s ) , E ( ∞ ) ) is admissible. Moreover, it has bounded degree.Proof. It is clear that the AI ∞ -graph ( X ( s ) , E ( ∞ ) ) is expansive. We show that ( X ( s ) , E ( ∞ ) ) is( m, m , i.e., d h ( x, y ) = 2 with | x | = | y | ≥ m ⇒ d h ( x ( − m ) , y ( − m ) ) ≤ , where J − m ( x ) = { x ( − m ) } .Set ℓ ∗ := min ξ = η ∈ V { ρ ( ξ, η ) } , ℓ ∗ := max ξ,η ∈ V { ρ ( ξ, η ) } and m := ⌊ log( ℓ ∗ /ℓ ∗ )log r ∗ ⌋ + 1, where r ∗ = max i { r i } . Let [ x, z, y ] be a horizontal geodesic. Suppose the statement is not true. Then d h ( x ( − m ) , y ( − m ) ) = 2, so that [ x ( − m ) , z ( − m ) , y ( − m ) ] is also a horizontal geodesic. As d h ( x, y ) = 2,we can choose distinct ξ , ξ such that ξ ∈ S x ( K ) ∩ S z ( K ) and ξ ∈ S y ( K ) ∩ S z ( K ). Observe that S x ( V ) ∩ S z ( V ) = S x ( K ) ∩ S z ( K ) ⊂ S x ( − m ) ( K ) ∩ S z ( − m ) ( K ) = S x ( − m ) ( V ) ∩ S z ( − m ) ( V ) . Thus ξ ∈ S z ( V ) ∩ S z ( − m ) ( V ), and so does ξ . Let w ∈ Σ ∗ satisfy z = z ( − m ) w . It follows that ξ i ∈ S z ( V ) ∩ S z ( − m ) ( V ) = S z ( − m ) ( V ∩ S w ( V )) , i = 1 , . herefore there exists η i ∈ V such that S w ( η i ) ∈ V (and S z ( η i ) = ξ i ) for i = 1 ,
2. The fact that w ∈ S k ≥ m Σ k implies ℓ ∗ ≤ ρ ( S w ( η ) , S w ( η )) ≤ ( r ∗ ) m ρ ( η , η ) ≤ ( r ∗ ) m ℓ ∗ . Thus m ≤ log( ℓ ∗ /ℓ ∗ )log r ∗ , contradicting the choice of m above. Hence ( X ( s ) , E ( ∞ ) ) is ( m, s ∗ := max j ∈ Σ { s j } . For x ∈ X n ( s ) and xv ∈ X n +1 ( s ), using (6.2) we have s v = s xv · s − x ≥ s n +2 ∗ · s − n ∗ = s ∗ . Therefore v ∈ S m ′ k =1 Σ k where m ′ := ⌊ s ∗ log s ∗ ⌋ , and hence J ( x ) ≤ S m ′ k =1 Σ k ) < N m ′ +1 . Using(6.3), it follows that sup ξ ∈ K { ̟ − ( ξ )) } ≤ C < ∞ . For x, y ∈ X ( s ), note that S x ( K ) ∩ S y ( K ) ⊂ S x ( V ). Therefore { y ∈ X : ( x, y ) ∈ E h } ≤ ̟ − ( S x ( V ))) ≤ V · sup ξ ∈ K { ̟ − ( ξ )) } ≤ V · C . As deg( x ) ≤ J ( x ) + { y ∈ X : ( x, y ) ∈ E h } for all x ∈ X , the graph ( X ( s ) , E ( ∞ ) ) hasbounded degree.Consequently, the map κ : ∂X ( s ) → K is a bijection as in Definition 4.2 (actually κ is ahomeomorphism by Proposition 4.3). Hence the Gromov metric θ a on ∂X induces a new metric˜ θ a on K by (4.5). Next we show that the index map Φ over ( K, ˜ θ a ) satisfies the condition ( B a ) inDefinition 5.1. Proposition 6.4.
Let { S j } Nj =1 be a contractive p.c.f. IFS. For s ∈ (0 , N , let ˜ θ a be the metric on K induced by ( X ( s ) , E ( ∞ ) ) . Then there exists c > such that for any x ∈ X ( s ) , S x ( K ) containsa ball of radius c e − a | x | in ( K, ˜ θ a ) (i.e., condition ( B a ) ).Proof. Note that ( X ( s ) , E ( ∞ ) ) is ( m, γ > x, y ∈ X ( s ), | x | = | y | and S x ( K ) ∩ S y ( K ) = ∅ ⇒ dist ˜ θ a ( S x ( K ) , S y ( K )) > γe − a | x | . (6.4)Let ℓ := ⌊ log( C )log N ⌋ + 1, where C is the critical set. Then for x ∈ X , there is y ∈ J ℓ ( x ) such that S y ( K ) ⊂ S x ( K \ V ). Choose ι ( x ) ∈ S y ( K ) arbitrarily. It follows from (6.4) thatinf η ∈ K \ S x ( K ) { ˜ θ a ( ι ( x ) , η ) } ≥ dist ˜ θ a (cid:0) S y ( K ) , ∪ z ∈ X | x | \{ x } S z ( K ) (cid:1) > γ · e − a ( | x | + ℓ ) . Hence B ˜ θ a ( ι ( x ) , c e − a | x | ) ⊂ S x ( K ) with c = γe − aℓ . This completes the proof.Let α = α ( s ) be the positive number such that P j ∈ Σ s αj = 1, and let µ s be the self-similarmeasure with respect to the vector of probability weights ( s α , s α , · · · , s αN ), i.e., the unique regularBorel probability measure on K that satisfies µ s ( · ) = X j ∈ Σ s αj · µ s ( S − i ( · )) . (6.5)In particular if the IFS is p.c.f., then µ s ( S x ( K )) = s αx for all x ∈ X . roposition 6.5. Let { S j } Nj =1 be a contractive IFS that satisfies the p.c.f. property. For s ∈ (0 , N , the self-similar measure µ s is Ahlfors-regular with exponent ( − α log s ∗ /a ) on ( K, ˜ θ a ) , i.e., µ s ( B ˜ θ a ( ξ, r )) ≍ r − α log s ∗ /a , ∀ ξ ∈ K, r ∈ (0 , . (6.6) Proof.
Consider the AI ∞ -graph ( X ( s ) , E ∞ ). For x ∈ X ( s ), setΦ ( x ) := [ { Φ( y ) : d h ( x, y ) ≤ } = κ ( J ∂ ( x )) , where the index map Φ( x ) = S x ( K ). By Proposition 3.3 and Theorem 6.3, there is C ≥ B ˜ θ a ( ξ, C − e − a | x | ) ⊂ Φ ( x ) ⊂ B ˜ θ a ( ξ, C e − a | x | ) , ∀ x ∈ X ( s ) , ξ ∈ Φ( x ) . Using s α ( | x | +1) ∗ < s αx = µ s (Φ( x )) ≤ µ s (Φ ( x )) ≤ ts α | x |∗ where t := sup x ∈ X ( s ) deg( x )( < ∞ byTheorem 6.3), we have ( µ s ( B ˜ θ a ( ξ, C − e − an )) ≤ ts αn ∗ ,µ s ( B ˜ θ a ( ξ, C e − an )) ≥ s α ( n +1) ∗ , ∀ ξ ∈ K, n ≥ . This proves (6.6).In the following, we show that the metric measure space ( K, ˜ θ a , µ s ) plays a special role inconnection with the study of local regular Dirichlet forms (which give a Laplacian) on K with aregular harmonic structure. This will also extend a consideration by Hu and Wang [HW], in whichthey studied the relation between the resistance metric R and the Euclidean metric for IFS on R d .We first recall some notations. A (discrete) Laplacian H = [ H pq ] p,q ∈ V on V is a non-positivedefinite matrix on V that satisfies P q ∈ V H pq = 0 for all p ∈ V , and H pq ≥ p, q ∈ V . For a weight vector s ∈ (0 , N , let V n = V n ( s ) := S x ∈ X n ( s ) S x ( V ) for n ≥
1, and V ∗ = V ∗ ( s ) := S ∞ n =0 V n ( s ). Denote the collection of real-valued functions on V n (or V ∗ ) by ℓ ( V n )(or ℓ ( V ∗ ) respectively). For a Laplacian H on V , we have( u, Hv ) = X p ∈ V u ( p ) (cid:0)X q ∈ V H pq v ( q ) (cid:1) , u, v ∈ ℓ ( V ) . We define the energy form E n on V n by E [ u ] = − ( u, Hu ) , E n [ u ] = X x ∈ X n ( s ) s − x E [ u ◦ S x ] , for u ∈ ℓ ( V n ) , n ≥ E n [ u ] := P x ∈ X n ( s ) s − x P p,q ∈ V H pq | u ( S x ( p )) − u ( S x ( q )) | ). We say that the pair ( H, s ) is a regular harmonic structure [Ki1] of K ifmin {E n +1 [ v ] : v ∈ ℓ ( V n +1 ) , v = u on V n } = E n [ u ] , ∀ n ≥ , u ∈ ℓ ( V n ) . (6.7)This implies that for u ∈ ℓ ( V ∗ ), {E n [ u ] } ∞ n =0 is an increasing sequence (here in each E n [ u ], u isrestricted on V n ), and E [ u ] := lim n →∞ E n [ u ] = sup n ≥ E n [ u ] , u ∈ ℓ ( V ∗ ) . (6.8)The u ∈ ℓ ( V ∗ ) can be extended continuously to a function on K if E [ u ] < ∞ . This defines the localregular Dirichlet form ( E , D ) with D := { u ∈ C ( K ) : E [ u ] < ∞} , where C ( K ) denotes the space ofcontinuous functions on K . This energy form is self-similar in the sense that for any n ≥ E [ u ] = X x ∈ X n ( s ) s − x E [ u ◦ S x ] , ∀ u ∈ D . (6.9) efine the effective resistance between two nonempty compact subsets F, G ⊂ K by R ( F, G ) = (inf {E [ u ] : u = 1 on F, and u = 0 on G } ) − if F and G are disjoint, and = 0 otherwise. Lemma 6.6.
Let K be a connected p.c.f. set that possesses a regular harmonic structure ( H, s ) .Then there exists γ ′ > such that for any x, y ∈ X ( s ) with | x | = | y | , the inequality R ( S x ( K ) , S y ( K )) ≥ γ ′ · s | x |∗ holds whenever S x ( K ) ∩ S y ( K ) = ∅ .Proof. Let x, y ∈ X n ( s ) with S x ( K ) ∩ S y ( K ) = ∅ . The p.c.f. property implies R ( S x ( K ) , S y ( K )) = R ( S x ( V ) , S y ( V )) ≥ R ( S x ( V ) , V n \ S x ( V )) . (6.10)By (6.7) and (6.8), there exists a function u on K such that u = 1 on S x ( V ), u = 0 on V n \ S x ( V ),and E n [ u ] = E [ u ] = R ( S x ( V ) , V n \ S x ( V )) − . (6.11)Let H ( x ) := { z ∈ X n ( s ) \ { x } : S z ( K ) ∩ S x ( K ) = ∅} . Using the bounded degree property of the AI ∞ -graph (Theorem 6.3), we have H ( x ) = { z ∈ X ( s ) : ( z, x ) ∈ E h } < sup x ∈ X ( s ) { deg( x ) } =: t < ∞ . It follows that E n [ u ] = 12 X z ∈ H ( x ) s − z X p,q ∈ V H pq | u ( S z ( p )) − u ( S z ( q )) | ≤ γ ′− s − n ∗ , where γ ′− := ts − ∗ (cid:0) V (cid:1) max { H pq : p = q ∈ V } . This together with (6.10) and (6.11) proves thelemma.For ξ, η ∈ K , we write R ( ξ, η ) instead of R ( { ξ } , { η } ) for short. It is well-known that R ( · , · ) is ametric on K , called the resistance metric . By definition, for compact subsets F, G ⊂ K we havedist R ( F, G ) := inf { R ( ξ, η ) : ξ ∈ F, η ∈ G } ≥ R ( F, G ) . (6.12)As a consequence of Lemma 6.6, we see that the metric space ( K, R ) with the index map Φ hasthe property in (6.4).
Theorem 6.7.
Let K be a connected p.c.f. set that possesses a regular harmonic structure ( H, s ) .Then the metric ˜ θ a induced by ( X ( s ) , E ( ∞ ) ) satisfies ˜ θ a ( ξ, η ) ≍ R ( ξ, η ) − a/ log s ∗ , ∀ ξ, η ∈ K. (6.13) Proof.
Firstly we prove that the index map Φ is of exponential type-( b ) under R , where b := | log s ∗ | .For this, let x ∈ X n ( s ) and ξ, η ∈ S x ( K ). Then for u ∈ D , | u ( ξ ) − u ( η ) | = | u ( S x ( ξ ′ )) − u ( S x ( η ′ )) | (here S x ( ξ ′ ) = ξ and S x ( η ′ ) = η ) ≤ R ( ξ ′ , η ′ ) E [ u ◦ S x ] ≤ | K | R E [ u ◦ S x ] ≤ | K | R · s x E [ u ] ≤ | K | R · s n ∗ E [ u ] , here the diameter | K | R < ∞ (cf. [Ki1, Theorem 3.3.4]), and the third inequality follows from thethe energy self-similar identity (6.9). Therefore, by using an equivalent expression of the effectiveresistance [Ki1], R ( ξ, η ) = sup { | u ( ξ ) − u ( η ) | E [ u ] : u ∈ D , E [ u ] = 0 } ≤ | K | R · s n ∗ = | K | R · e − bn . This proves that | Φ( x ) | R ≤ | K | R · e − b | x | for all x ∈ X .By (6.12) and Lemma 6.6, the index map Φ satisfies (4.11) with k = 1 under R . It followsfrom Theorem 4.8 that the bijection κ : ( ∂X, θ a ) → ( K, R ) satisfies (4.10) with ρ = R . From thedefinition (4.5) of ˜ θ a , we see that ˜ θ a ( · , · ) ≍ R ( · , · ) a/b on K . In this Appendix, for the convenience of the reader, we summarize some notations and known factson iterated function systems, as well as some background of this paper.Let (
M, ρ ) be a complete metric space, and let { S j } Nj =1 ( N ≥
2) be a contractive iteratedfunction system (IFS) on (
M, ρ ) [Ki1], i.e., each S j : M → M satisfies r j := sup { ρ ( S j ( ξ ) , S j ( η )) ρ ( ξ, η ) : ξ, η ∈ M, ξ = η } < . (7.1)Then there exists a unique nonempty compact set K ⊂ M satisfying K = S Nj =1 S j ( K ) , called the attractor of { S j } Nj =1 ; K is called a self-similar set if M = R n and the S j ’s are similitudes, i.e., | S j ( ξ ) − S j ( η ) | = r j | ξ − η | .Let the alphabet set Σ := { , , · · · , N } . Write Σ := { ϑ } ( ϑ is the empty word ), and for n ≥ n := { x = i · · · i k · · · i n : i k ∈ Σ , ∀ k } . Let Σ ∗ := S ∞ n =0 Σ n denote the symbolic space of finitewords. This Σ ∗ has a natural N -ary tree structure with the root ϑ . For x = i · · · i n ∈ X , write r x := r i r i · · · r i n , S x := S i ◦ S i ◦ · · · ◦ S i n and K x := S x ( K ) for short.For a self-similar set K of a homogeneous IFS { S j } Nj =1 (i.e., r j = r for all j ), coding the iterationsby the tree of symbolic space Σ ∗ is natural, as each K x with x ∈ Σ n has a constant diameter r n | K | .But in a non-homogeneous case, the diameters of the cells on each level are not comparable. Acommon way is to regroup the indices as follows: let r ∗ = min j ∈ Σ { r j } , X = { ϑ } , X n = { x = i i · · · i k ∈ Σ ∗ : r x ≤ r n ∗ < r i r i · · · r i k − } , n ≥ , (7.2)and X = S ∞ n =0 X n . This X has a natural tree structure E v , and the diameters of the cells in { K x } x ∈ X n are comparable with r n ∗ .A contractive IFS of similitudes is said to satisfy the open set condition (OSC) if there is abounded nonempty open set U such that S j ( U ) ⊂ U for all j ∈ Σ, and S i ( U ) ∩ S j ( U ) = ∅ for all i = j . The OSC is one of the most fundamental conditions in fractal geometry. For a self-similarset K , the OSC yields an explicit expression of the Hausdorff dimension s of K by P Nj =1 r sj = 1,and K supports the s -Hausdorff measure. Furthermore, the OSC is equivalent to the followingproperty (cf. [Sc, Theorem 2.2]): for any c >
0, there is a constant ℓ = ℓ ( c ) such that ∀ η ∈ K and integer n > , B ( η, cr n ∗ ) ∩ K x = ∅ for at most ℓ of x ∈ X n . (7.3) e augment the tree ( X, E v ) by a set of horizonal edges E h = [ ∞ n =1 { ( x, y ) ∈ X n × X n \ ∆ : K x ∩ K y = ∅} , (7.4)and let E = E v ∪ E h (cf. [Ka, LW1]). It has been proved that Theorem 7.1. [LW1]
Let { S j } Nj =1 be an IFS of contractive similitudes that satisfies the OSC.Then ( X, E ) is a hyperbolic graph, and for the hyperbolic boundary ∂X , the canonical identification κ : ∂X → K is a homeomorphism. Furthermore, κ is a H¨older equivalence if the IFS satisfies thecondition (H), i.e., there exists a constant c > such that ∀ n > , x, y ∈ X n , K x ∩ K y = ∅ ⇒ dist( K x , K y ) ≥ cr n ∗ . (7.5)The identification of K and ∂X has been applied to study the Lipschitz equivalence of self-similar sets [LL, DLL]. If we enlarge the horizontal edge set to be E ′ h = [ ∞ n =1 { ( x, y ) ∈ X n × X n \ ∆ : dist( K x , K y ) ≤ γr n ∗ } (7.6)for some γ > E ′ = E v ∪ E ′ h (cf. [LW3]), then we can improve Theorem 7.1 as Theorem 7.2. [LW3]
For any IFS { S j } Nj =1 of similitudes, ( X, E ′ ) is a hyperbolic graph, and thecanonical identification ι : ∂X → K is a H¨older equivalence. Moreover, ( X, E ′ ) has bounded degreeif and only if { S j } Nj =1 satisfies the OSC. In [KLW1], thanks to the bounded degree property, we can introduce a class of transient re-versible random walks on (
X, E ′ ) such that the Martin boundary, ∂X and K are homeomorphic,by which we obtain a jump kernel (i.e., Na¨ım kernel) to study the induced energy form on K .For IFS { S j } Nj =1 with overlaps, it is possible that S x = S y for some different x, y ∈ X , where X is defined in (7.2). For example, let M = R , S ( x ) = rx and S ( x ) = rx + (1 − r ), where r = √ − is the golden ratio. Then S = S (see also Example 2.2). In this case, we can definean equivalence relation ≃ on X by x ≃ y if and only if S x = S y . Then there is a natural verticalgraph ( X ∼ , E ∼ v ) as the quotient of ( X, E v ) with respect to ≃ , which is not a tree unless the relation ≃ is trivial [LW3]. It was proved in [Wa] that the associated augmented tree of (7.4) in ( X ∼ , E ∼ )is hyperbolic if the self-similar set K has positive Lebesgue measure, or the IFS { S j } Nj =1 satisfiesthe weak separation condition (WSC) (cf. [LN], [LW, Theorem 2.1]), i.e., the condition (7.3) with X n replaced by the quotient X n / ≃ . Some more variants were discussed in [Wa]. Acknowledgements : The authors would like to thank Professor Alexander Grigor’yan for manyvaluable discussions. They also like to extend their thanks to Doctor Leung-Fu Cheung, ProfessorsQingsong Gu and Sze-Man Ngai for going through the manuscript and making some suggestions.
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Random Walks on Infinite Graphs and Groups . Cambridge University Press,Cambridge (2000).Shi-Lei Kong, Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld,[email protected] Lau, Department of Mathematics, The Chinese University of Hong Kong, Hong Kong.& Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, [email protected] Wang, School of Mathematics, Sun Yat-Sen University, Guangzhou, [email protected]. Cambridge University Press,Cambridge (2000).Shi-Lei Kong, Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld,[email protected] Lau, Department of Mathematics, The Chinese University of Hong Kong, Hong Kong.& Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, [email protected] Wang, School of Mathematics, Sun Yat-Sen University, Guangzhou, [email protected]