RENORMALIZATION OF GENERALIZED “abc” GASKETS

ANTONIO-MIHAIL NUIC ˘A

We obtain existence of eigenforms (eigenoperators) for the renormalization map of the so-called generalized “abc” gasket. The method we used is based on Hilbert projective metric approach (developed by Volker Metz). We also com- pute the “highly-symmetric” irreducible eigenoperator of the renormalization map for a particular example, the so-called “affine Sierpinski gasket”.

AMS 2010 Subject Classification: 31XX, 60J45, 28A80.

Key words: post critical finite self-similar structure, Laplacian, Dirichlet form, electrical networks, Hilbert projective metric, renormalization map, eigenform, diffusion process, shortcut procedure.

1. INTRODUCTION

The post critical finite self-similar structures (introduced by Jun Kigami in [4]) are finitely ramified structures. Theaffine nested fractals are connected post critical finite self-similar structures [3].

The renormalization map of a post critical finite self-similar structure is defined on the cone of all possible energy forms (or associated operators) of the structure initial boundary. The existence of a Dirichlet form on a finitely ramified fractal is equivalent to the existence of aneigenform of the renormal- ization map of the structure (Kusuoka – [7]). This eigenvalue problem was studied by T. Lindstrom in [8] for nested fractals and for affine nested fractals in [3]. The existence of the eigenforms of the renormalization map for post critical finite self-similar structures is an open problem; significant results were obtained by V. Metz in [9–11]. If we start with anirreducible eigenformfor the renormalization map of a connected post critical finite self-similar structureF, then we can construct a local regular Dirichlet form onL²(F, µ), where µis a well chosen probability measure on F ([4, 6]).

In the next section, we overview the basic facts on self-similar structures, energy forms and Laplacians on finite sets, Hilbert projective metric on cones;

we remember the most important properties of the renormalization map of a post critical finite self-similar structure (Cf. [9, 11]). In the last section, we present the main result on the existence of eigenforms for a particular class of post critical finite self-similar structures, named generalized “abc” gaskets.

Finally, we compute the irreducible eigenoperator for a particular affine nested fractal,the affine Sierpinski gasket.

MATH. REPORTS15(65),4(2013), 477–487

2. THE RENORMALIZATION PROBLEM

2.1. SELF-SIMILAR STRUCTURES n

F,{χ_i}_i=1,No

is called self-similar structure if and only if (F, d) is a compact metric space, χ_i : F −→ F, i = 1, N are injective contractions and F =

N

[

i=1

χ_i(F) =: Ψ(F). Associated to such a structure is its natu- ral map π : Σ := {1,2, . . . , N}^N −→ F. We consider the ramifying set

B := [

1≤i<j≤N

(χi(F)∩χj(F)), the critical set Γ := π⁻¹(B), the postcritical set P := [

n≥1

σⁿ(Γ) and the initial boundary of the structure V₀ := π(P); for w=w1. . . wn. . .∈Σ we consideredσw:=w2. . . wn. . .. The self-similar struc- ture is calledfinitely ramified ifB is finite and post critical finiteifP is finite.

The boundaryV0 generates then-cells: Vn:= [

w∈Wn

Ψⁿ(V0),n≥1.

By restricting the similitudes of an iterated function system on R^d to its attractor (see [2]) we get a self-similar structure, which is called affine nested fractal if it satisfies three axioms: connectivity, symmetry and a so- called nesting axiom (Cf. [3]). An affine nested fractal is a connected post critical finite self-similar structure (Cf. Theorem 5.24 from [1]).

2.2. DIRICHLET FORMS, LAPLACIANS AND CONDUCTANCES Consider a finite setV, the subspace of linear symmetric operatorsL_s(V) inR^V =n

f

f :V →R o

and the subspace of bilinear symmetric forms Fs(V) in R^V. Then Ξ : L_s(V) → Fs(V), Ξ(H) := H_H is an isomorphism, where H_H(u, v) :=−(u, Hv), u,v∈R^V, (u, v) := X

p∈V

u(p)v(p) for u,v∈R^V.

Definition 2.1. a) Let H ∈ Fs(V). H can satisfy one of the axioms:

(F.1): H(u, u) ≥ 0, for all u ∈ R^V; (F.2): H(u, u) = 0 ⇔ u constant; (F.3):

H(u, u)≤ H(u, u), for allu∈R^V (u:= 0∨(1∧u), foru∈R^V). Instead of axiom (F.2) we consider (F.2)⁰: H(1,1) = 0 or (F.2)⁰⁰: H(u, u) = 0⇒u constant.

b) We consider the cone F12⁰(V) of forms with (F.1), (F.2)⁰, the cone F12⁰3(V) of forms which satisfy (F.1), (F.2)⁰ and (F.3) and the cone F123(V) of forms satisfying (F.1), (F.2) and (F.3). The forms in F12⁰3(V) are called Dirichlet forms on V and the forms in F123(V) are calledirreducible Dirichlet

forms. It can be verified that F12⁰3(V) is polyhedral andF12⁰(V) is not always polyhedral.

c) LetH ∈ L_s(V). H can satisfy one of the properties: (O.1): (Hu, u)≤ 0, for all u ∈R^V; (O.2): Hu= 0 ⇔u constant; (O.3): H_pq := (Hχ_q)(p) ≥0, for all p 6= q ∈ V. Instead of (O.2) we consider: (O.2)⁰: H1 = 0, that is, X

q∈V

H_pq = 0, for all p ∈ V; (O.2)⁰⁰: Hu = 0 ⇒ u constant. We denote by L₁₂⁰(V), L₁₂⁰₃(V), L₁₂(V), L₁₂₃(V) the cones of operators H which satisfies (O.1), (O.2)⁰, or (O.1), (O.2)⁰ and (O.3), and so on. If H ∈ L₁₂(V), then N := (V, H) is called electrical network on V.

It’s enough for H to satisfy property (O.2)⁰, in order to have (2.1) H_H(h, g) =−(h, Hg) = 1

2 X

p,q∈V

Hpq(h(p)−h(q))(g(p)−g(q)), h, g∈R^V. The following Proposition is an extension of Prop. 2.1.3 from [5]:

Proposition 2.2. Ξ(L₁₂⁰(V)) = F12⁰(V); Ξ(L₁₂⁰₃(V)) = F12⁰3(V);

Ξ(L₁₂(V)) =F12(V);Ξ(L₁₂₃(V)) =F12⁰3(V).

Definition 2.3. 1) Aconductanceis a matrixc:= (c(p, q))_p,q∈V withcpq :=

c(p, q), having properties (C.1): cpq =cqp, for all p, q∈ V; (C.2): cpp = 0, for allp∈V; (C.3): c_pq ≥0, for allp6=q ∈V. We associate the graph Γ := (V, G), G := {{p, q} ⊂ V |c_pq >0}. There is an one-to-one correspondence between L₁₂⁰₃(V) and conductance matrices.

2) For U $ V and H ∈ L_s(V), consider X : R^U −→ R^U, Y : R^U −→

R^V\U, Z : R^V\U −→ R^V\U, so that H ∼=

X Y^t

Y Z

. If H ∈ L₁₂(V), by Lemma 2.1.5 and Theorem 2.1.6 from [5] Z is negative definite, [H]U :=

X −Y^tZ⁻¹Y is well defined on V and [H]_U ∈ L₁₂(U). [H]_U is called the trace, orrestriction of H toU. (V, H) and (U,[H]_U) are also called equivalent networks. The well known Ohm’s law and ∆−Y transform are particular cases of equivalence.

2.3. HILBERT’S PROJECTIVE METRIC AND THE RENORMALIZATION PROBLEM In this subsection, we fix r:= (r₁, . . . , r_n), r_i >0,S :=n

F,{χ_i}_i=1,No a connected post critical finite self-similar structure, V0 the initial boundary of S, the conesF12⁰(V₀) =:F12⁰,F12⁰3(V₀) =:F12⁰3,F123(V₀) =:F123onV₀defined as above and their associated cones of operators.

If B(V₀) :=F12⁰3(V₀)−F12⁰3(V₀) is the real normed space with respect to the norm ||E||² := sup

E(u, u)

u∈R^V0,||u||_V0 = 1 , then it’s easy to verify that

◦

F12\⁰3(V₀) =

E ∈F12⁰3

E(χ_p, χ_q)<0, p6=q∈V₀ ,

◦

F\12⁰(V₀) =L₁₂⁰(V₀).

Definition 2.4.1. The renormalization mapis defined by:

Λ := Λr:F12⁰(V0)−→F12⁰(V0),Λ := Φ◦Ψ,

where Ψ is the replication “operation” and Φ is the trace “operation”: for A₀ ∈ F12⁰(V0), put Ψ(A₀) =: A₁, with A₁(u, u) :=

N

P

i=1

r_i⁻¹A₀(u◦χi, u◦χi) for all u ∈ R^V1; then Ψ(A₀) = A₁ ∈ F12⁰(V₁). For A₁ ∈ F12⁰(V₁), define Φ(A₁) =: A₀, where A₀(u, u) := inf

A₁(v, v)

v ∈R^V1, v_|V0 =u , u ∈ R^V0; thenA₀ ∈F12⁰(V₀). Proposition 2.2 allows Λ to be defined on operators instead of forms.

2. For K∈ {F12⁰,F12⁰3} and A,B ∈K\{0}, we can define A ∼_K B if and only if there existsα,β >0 withαA ≤_K B ≤_K βA, whereA ≤_K Bif and only ifB − A ∈K; then “∼_K” is an equivalence relation onK\{0}, the equivalence classes are called parts (subcones of K), K^◦ is the most important one and obviously every F12⁰3-part is contained in aF12⁰-part.

IfA ∈F12⁰3, then we define the graph Γ(A) := (V₀, G(A)), with G(A) :=

{p, q} ⊂V0

cA(p, q)>0 . If A, B ∈ F12⁰3, from (2.1) written for conduc- tances we get A ∼_F

1203 B if and only if Γ(A) = Γ(B). So, we can define Γ(F⁰₁₂⁰₃) := Γ(A), forA ∈F⁰₁₂⁰₃ andF⁰₁₂⁰₃ aF12⁰3 part. Γ(Ψ(F⁰₁₂⁰₃)) denotes the graph with vertices V1 associated with Ψ(A), for every A ∈ F⁰₁₂⁰₃. Similarly, if A, B ∈ F12⁰\{0}, then A ∼_F

120 B if and only if KerA = KerB (KerA is the kernel of the operator associated with A). The following lemma is a very useful tool:

Lemma 2.5 ([12], Prop. 1.15). For A ∈ F12⁰3 and p 6= q ∈ V₀ we have cΛ(A)(p, q) > 0 if and only if p and q are Ψ(A)-connected avoiding V0\{p, q}

(c_Λ(A) is the conductance matrix associated to Λ(A)).

Definition 2.6.Let K∈ {F12⁰,F12⁰3} andA,B ∈K\{0}.

– for A ∼_K B, define m(B|A) := sup α >0

αA ≤ B , M(B|A) := inf

β >0

B ≤βA , h(A,B) := lnM(B|A) m(B|A); – for A_K B, consider h(A,B) :=∞,h(0,0) := 0.

h is called the Hilbert projective “metric”.

In fact, his just a semi-metric onK\{0}. h(A,B) = 0 if and only if there exists α > 0, such that A = αB; h(A,B) = ∞ if and only if A _K B; also h(A,B) =h(αA, βB), for allA,B ∈K\{0},α,β >0 (Cf. [9]).

The next proposition summarizes the properties of Λ (Cf. Sections 2 and 4 from [9], Cor. 4, Lemma 7 from [11]). “≤_F

120” is denoted simply by “≤”.

Proposition2.7. (a) Λ F^◦₁₂⁰

⊂F^◦₁₂⁰,Λ F^◦₁₂⁰₃

⊂F^◦₁₂⁰₃,Λ(F123)⊂F^◦₁₂⁰₃; Λ is positive homogeneous, ≤-monotone and superadditive;

(b) Λ is non-expansive with respect to h on Λ-invariant F12⁰-parts;

(c) Λ “acts” on F12⁰3-parts and also on F12⁰-parts;

(d)There exists an eigenform in every closed convex Λ-invariant subcone (for example F12⁰3-part) of F12⁰3; if A, B ∈ F12⁰, A ∼_F

120 B, Λ(A) = αA, Λ(B) =βB, then α=β.

The renormalization map is studied with respect to subcones which are invariant under specific symmetry groups associated to the structure S:

Definition 2.8.1) Θ is called symmetry group for S if and only if Θ is formed with continuous bijectionsg:F −→F such thatg(V₀)⊂V₀, for every g∈Θ and if for all i∈ {1, . . . , N} and g∈Θ, there existsj ∈ {1, . . . , N} and g⁰∈Θ withg◦χ_i =χ_j ◦g⁰.

2) Consider A ∈Fs(V0), H ∈ L_s(V0), c a conductance matrix onV0 and r= (r1, . . . , r_N) withri >0,i= 1, N. Then: Ais called Θ-invariantif and only if for allθ∈Θ andu,v∈R^V0,A(u◦θ, v◦θ) =A(u, v);His called Θ-invariant if and only if for allθ∈Θ andp,q ∈V0,Hpq =H_θ(p)θ(q);cis called Θ-invariant if and only if for all θ ∈ Θ and p, q ∈ V0, c(p, q) = c(θ(p), θ(q)); r is called Θ-invariantif and only if θ∈Θ such thatθ(χ_i(V₀)) =χ_j(V₀) =⇒r_i =r_j

. 3) Denote by F^Θ₁₂⁰,F^Θ₁₂⁰₃ and F^Θ123 theR-subcones of Θ-invariant forms in F12⁰,F12⁰3 and F123,respectively.

There are one-to-one correspondences between Θ-invariant forms, opera- tors and conductance matrices.

Remark 2.9.1) For affine nested fractals Θ is usually the symmetry group generated by the reflections in the hyperplanes bisecting the line segments which connects points in V0 and shall be denoted by G_s. Denote, also F^s₁₂⁰ :=

F^G₁₂^s0,F^s₁₂⁰₃ :=F^G₁₂^s0₃,F^s₁₂₃:=F^G₁₂₃^s .

2) For an affine nested fractal S :=

n

F,{χ_i}_i=1,No

and r G_s-invariant there exists α > 0 and A₀ ∈ F^s123 (unique up to multiplication by a positive constant) such that Λ(A₀) =αA₀ (Cf. [3, 8, 12]).

3. RENORMALIZATION OF GENERALIZED “abc” GASKETS

In this section, we prove the existence of eigenforms for the renormal- ization map of a generalized “abc” gasket. Then we derive a necessary and sufficient condition in terms of r_i, i = 1, N for the existence and uniqueness of the “highly symmetric” irreducible eigenoperator for the affine Sierpinski gasket and effectively determine it.

First, we recall a very useful tool called the “shortcut procedure”.

3.1. THE “SHORTCUT” PROCEDURE We consider S =

n

F,{χ_i}_i=1,No

a connected post critical finite self- similar structure and r := (r₁, . . . , r_n), with r_i > 0; consider also the initial boundary V0 of S and V1.

ConsiderP^k:={A^k₁, . . . , A^k_p}a partition ofV_k,k= 0,1 andv:V_k−→R, v_|Ak

i =ct, i = 1, p. Define v : P^k −→ R, v(A^k_i) := v(x), x ∈ A^k_i, i = 1, p.

Consider H₀ ∈ F12⁰(V0), so H₁ := Ψ(H) ∈ F12⁰(V1); let cH_k : Vk×Vk −→ R be the conductance matrix associated to H_k. Define cH_k : P^k× P^k −→ R, cH_k(A^k_i, A^k_j) := P

p∈A^k_i,q∈A^k_j

cH_k(p, q) for i6=j and cH_k := 0 for i=j. Then (see Lemma 17 from [11])

(3.1) H_k(v, v) = 1 2

l

X

i,j=1

(v(A^k_i)−v(A^k_j))²cH_k(A^k_i, A^k_j) =:H_k(v, v) and H_k∈F12⁰(P^k) for k= 0,1.

For E ∈F12⁰3(V0)\{0},F ∈B(V0) withE+F ∈F12⁰3(V0) and u:V_k −→

R, obviously (F_k+nE_k)(u, u) % ∞ for E_k(u, u) > 0 and (F_k+nE_k)(u, u) = F_k(u, u) forE_k(u, u) = 0, that isu =ct. on E_k-connected components. So, we can define (F_k+∞E_k)(u, u) :=F_k(u, u), foru∈KerE_k, (F_k+∞E_k)(u, u) :=∞, for u /∈ KerE_k. We say that we shortcut the edges of E_k. In (3.1) we consider H_k = F_k +nE_k and v ∈ KerE_k and we obtain H_k(v, v) = F_k(v, v) = (F_k+

∞E_k)(v, v) =H_k(v, v), so “shortcutting the edges of E_k” “the energy does not modify”.

3.2. RENORMALIZATION OF “abc” SIERPINSKI GASKETS I. Let ∆z1z2z3 ⊂R² an equilateral triangle and H := co{z₁, z2, z3}. Let {χ_i}_i=1,M M similitudes so that H_i := χ_i(H) touches Hi−1 and H_i+1 in just one point; consider M = a+b+c, a ≥ 1, b ≥ 1, c ≥ 1; the choice of the

similitudes is so that we should havec+ 1 trianglesH_i on z₁z₂,a+ 1 triangles H_i on the edge z₂z₃ and b+ 1 onz₃z₁ (see Figure 1, witha= 4,b= 5,c= 3).

H2 touches HM if and only if a=b= c= 1 (the classical Sierpinski gasket).

We denotei= 1, j:=c+ 1, k:=c+a+ 1, byF the attractor of the iterated function system

n

R²,{χ_i}_i=1,Mo

and byχi also, the restrictions ofχi toF.

Fig. 1. The construction of a “453” gasket.

ThenS_abc:=n

F,{χ_i}_i=1,Mo

is a self-similar structure. B= S

1≤s<t≤M

(H_s∩H_t) hasM points and each ramifying point is coded by two addresses; for example, H₁∩H₂={π(1˙i) =π(2 ˙j)}, whereπ is the natural map and ˙l=ll . . .. So Γ has 2M codes, P ={(˙i),( ˙j),( ˙k)}andV0 ={z₁, z2, z3}. S_abcis a post critical finite self-similar structure and is calledgeneralized “abc” gasket. For certainki and ratios we get an affine nested fractal with respect to the “maximal” groupG_s. We take Θ ={Id_V0}(IdV0 the identity map onV0) andr:= (r1, . . . , rM), with r_i >0. For simplicity we consider equal weights for the similitudes with χi(H) on the same edge of the initial triangle (for example, if a = 4, b = 5, c= 3, then r2 =r3 =r4 =: 1/η2,r5 =r6 =r7 =r8 =: 1/η3,r9 =r10=r11= r₁₂=r₁ =: 1/η₁, where η₁,η₂,η₃ >0).

Because Θ ={Id_V0}, the renormalization map will be considered on the largest cone. The coneF12⁰ :=F^Θ₁₂⁰ can be determined by looking at the eigen- values of an operatorH=H(α₁, α₂, α₃) =





−α₂−α3 α3 α2

α₃ −α₁−α₃ α₁ α2 α1 −α₁−α2



,

α₁,α₂,α₃≥0. Cf. [9].

F12⁰ '

(α₁, α₂, α₃)∈R³

α₁+α₂+α₃ ≥0, α₁α₂+α₂α₃+α₃α₁≥0 , F12⁰3 'R³+ and the F12⁰3-parts are F^(1,1,1)₁₂⁰₃ '(0,∞)³,F^(0,1,1)₁₂⁰₃ ' {0} ×(0,∞)², F^(1,0,1)₁₂⁰₃ ' (0,∞)× {0} ×(0,∞), F^(1,1,0)₁₂⁰₃ ' (0,∞)² × {0}, F^(1,0,0)₁₂⁰₃ ' (0,∞)× {0} × {0},F^(0,1,0)₁₂⁰₃ ' {0} ×(0,∞)× {0},F^(0,0,1)₁₂⁰₃ ' {0} × {0} ×(0,∞).

Using Lemma 2.5 we get: Λ(F^(1,0,0)₁₂⁰₃ ) ⊂F^(1,0,0)₁₂⁰₃ (in Figure 2, just z₂ and z3 can be connected by a path in Γ(Ψ(F^(1,0,0)₁₂03 )) with points from V1\V₀) and similarly Λ(F^(0,1,0)₁₂⁰₃ ) ⊂ F^(0,1,0)₁₂⁰₃ , Λ(F^(0,0,1)₁₂⁰₃ ) ⊂ F^(0,0,1)₁₂⁰₃ and Λ(F^(1,1,1)₁₂⁰₃ ) ⊂ F^(1,1,1)₁₂⁰₃ . It can be shown that the only Λ_r-invariant F12⁰3-parts are F^◦₁₂⁰₃ ' (0,∞)³, F¹ := F^(1,0,0)₁₂⁰₃ ' (0,∞) × {0} × {0}, F² := F^(0,1,0)₁₂⁰₃ ' {0} × (0,∞) × {0}, F³ :=F^(0,0,1)₁₂⁰₃ ' {0} × {0} ×(0,∞).

Fig. 2. Λ(D(1,0,0))⊂D(1,0,0).

Now, we can state the following

Proposition 3.1. a)The eigenvalues of{Fⁱ}_i=1,3 are λ₁ =

1 η₂ + a

η₃ −1

, λ₂ = b

η₂ + 1 η₁

−1

, λ₃ = 1

η₃ + c η₁

−1

. b) The eigenvalues of

Λr F^◦₁₂⁰₃+∞Fⁱ _i=1,3, are λ⁰₁ =





 1 η1

+

"

η₂ b +

c−1 η1

+ 1 η3

−1#−1





−1

,

λ⁰₂ =





 1 η₃ +

"

η₁ c +

a−1 η₃ + 1

η₂

−1#−1





−1

,

λ⁰₃ =





 1 η2

+

"

η₃ a +

b−1 η2

+ 1 η1

−1#−1





−1

.

If λi < λ⁰_i, i= 1,3, then there exists irreducible eigenforms for Λr associated to S_abc.

Proof. a) Applying successively Ohm’s law (resistors in series add their resistances and resistors in parallel add their conductances) we getλ_i.

b) By “shortcutting” (3.1) and Ohm’s law we get the above λ⁰_i.

BecauseF^◦₁₂⁰₃is Λ-invariant and for each Λ-invariantF12⁰3-partFⁱ ⊂∂F12⁰

the eigenvalueλ⁰_iis strictly greater thanλ_i, then Theorem 25 and Corollary 28 from [11] allow us to say that there exists an eigenform in F123.

II. Consider the particular case of “222” gasket; that is, we take M = 6, a = b = c = 2, and the scaling factors of the similitudes s1 = s2 = s3 = 2/5, s4 = s5 = s6 = 1/5. We take the “maximal” symmetry group Θ = G_s. If we denote by F the attractor of the iterated function system n

(R²,k · k);{χ_i, si}_i=1,6o

and χi for their restrictions to F also, it is easy to verify thatS₂₂₂ :=

n

F,{χ_i}_i=1,6o

is an affine nested fractal with respect toG_s and is calledthe affine Sierpinski gasket. The attractor is depicted in Figure 3.

Fig. 3. The attractor ofS222.

Consider r= (u, u, u, uv, uv, uv) (u,v >0) and D=





−2 1 1

1 −2 1

1 1 −2



.

D andr areG_s-invariant and Γ(A_D) = Γ((F^s₁₂⁰₃)^◦) is shown in Figure 4–(7).

Fig. 4. The renormalization of the affine Sierpinski gasket.

The graph Γ(Ψ((F^s₁₂⁰₃)^◦)) is shown in (1). Using ∆ −Y and Y − ∆ transforms and Ohm’s law (Figure 4) we get

Proposition3.2. ForS₂₂₂,Θ =G_sand the weightsr= (u, u, u, uv, uv, uv), the following statements are equivalent:

(1) there exists G_s invariant irreducible eigenoperators for the renormal- ization map Λr associated to S₂₂₂.

(2) u(2v+ 5) = 3.

In this case,Dis the unique (up to a multiplication by a positive constant) eigenoperator of Λr for eigenvalue λ= _u(2v+5)³ = 1.

So, in order to have existence and uniqueness even on this particular

“highly symmetric case”, the weights have to be considered with extreme care.

REFERENCES

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Received 10 October 2013 University of Pite¸sti,

Department of Mathematics and Informatics, Romania,

antonio 74nm@yahoo.com