A P.C.F. Self-Similar Set with no Self-Similar Energy

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by  Roberto Peirone

A P.C.F. Self-Similar Set with no Self-Similar Energy
Roberto Peirone
Universit`a di Roma ”Tor Vergata”, Dipartimento di Matematica
via della Ricerca Scientifica, 00133, Roma, Italy
e.mail: [email protected]
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0 Abstract A general class of finitely ramified fractals is that of P.C.F. self-similar sets.
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An important open problem in analysis on fractals was whether there exists a self-similar

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energy on every P.C.F. self-similar set. In this paper, I solve the problem, showing an
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J example of a P.C.F. self-similar set where there exists no self-similar energy.

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1. Introduction.

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A
An important problem in analysis of fractals is the construction of a Laplace operator,
F
or equivalently, an energy, more precisely, a self-similar Dirichlet form. The construction
.
h of a self-similar Dirichlet form has been investigated specially on finitely ramified fractals.
t
a Roughly speaking, a fractal is finitely ramified if the intersection of each pair of copies
m
of the fractal is a finite set. The Sierpinski Gasket, the Vicsek Set and the Lindstrøm
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Snowflake are finitely ramified fractals, while the Sierpinski Carpet is not.

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More specially, weconsider theP.C.F.selfsimilar-sets, ageneral classoffinitelyramified
v
5 fractals introduced by Kigami in [3]. A general theory with many examples can be found
8 in [4]. On such a class of fractals, the basic tool used to construct a self-similar Dirichlet
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7 form is a discrete Dirichlet form defined on a special finite subset V(0) of the fractal. Such
0 discrete Dirichlet forms have to be eigenforms, i.e. the eigenvectors of a special nonlinear
.
1 operator Λ called renormalization operator, which depends on a set of positive weights
r
0
r placed on the cells of the fractal. In [5], [10] and [6] criteria for the existence of an
7 i
1 eigenform with prescribed weights are discussed. In particular, in [5], T. Lindstrøm proved
:
v that there exists an eigenform on the nested fractals with all weights equal to 1, C. Sabot
Xi in [10] proved a rather general criterion, and V. Metz in [6] improved the results in [10].
r In [1], [7], [8] and [9], instead, the problem is considered whether on a given fractal there
a
exists a G-eigenform. By this we mean a form E which is an eigenform of the operator
Λ for some set of weights r. In such papers the existence of a G-eigenform was proved on
r
some classes of P.C.F. self-similar sets. In fact, the following open problem is well-known
Does a G-eigenform exist on every P.C.F.self-similar set?
In this paper, I solve such a problem showing an example of P.C.F.self-similar set with no
G-eigenform. Here, I consider a very general class of P.C.F. self-similar sets, as usually in
my previous papers on this topic (see Section 2 for the details), but also in other papers
(for example it is considered in [11]. The example is constructed in Section 3. It is a
variant of the N-Gaskets, in the sense that every cell V only intesects V and V , and
i i−1 i+1
has twenty vertices. The ptoof is based on the evaluation of the effective conductivities on
pairs of close vertices and of far vertices. Note that in [9] the existence of a G-eigenform
is proved on every fractal (of the class considered here) but only if we consider the fractal
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generated by a set of similarities which is not necessarily the given set of similarities (see
[9] for the details).
2. Definitions and Notation.
I will now define the fractal setting, which is based on that in [9]. This kind of approach
was firstly given in [2]. We define a fractal by giving a fractal triple, i.e., a triple F :=
(V(0),V(1),Ψ) where V(0) = V and V(1) are finite sets with #V(0) ≥ 2, and Ψ is a finite
set of one-to-one maps from V(0) into V(1) satisfying
V(1) = ψ(V(0)).
ψ[∈Ψ
We put V(0) = P ,...,P , and of course N ≥ 2. A set of the form ψ(V(0)) with ψ ∈ Ψ
1 N
will be called a(cid:8)cell or a 1-c(cid:9)ell. We require that
a) For each j = 1,...,N there exists a (unique) map ψ ∈ Ψ such that ψ (P ) = P , and
j j j j
Ψ = ψ ,...,ψ , with k ≥ N.
1 k
(cid:8) (cid:9)
b) P ∈/ ψ (V(0)) when i 6= j (in other words, if ψ (P ) = P with i = 1,...,k, j,h =
j i i h j
1,...,N, then i = j = h).
c) Any two points in V(1) can be connected by a path any edge of which belongs to a
1-cell, depending of the edge.
Of course, it immediately follows V(0) ⊆ V(1). Let W =]0,+∞[k and put V = ψ (V(0)) for
i i
each i = 1,...,k. Let J(= J(V)) = {{j ,j } : j ,j = 1,...,N,j 6= j }. It is well-known
1 2 1 2 1 2
that on every fractal triple we can construct a P.C.F.-self-similar set.
We denote by D(F) or simply D the set of the Dirichlet forms on V, invariant with
respect to an additive constant, i.e., the set of the functionals E from RV into R of the
form
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E(u) = c (E) u(P )−u(P )
{j1,j2} j1 j2
{j1X,j2}∈J (cid:0) (cid:1)
with c (E) ≥ 0. I will denote by D(F) or simply D the set of the irreducible Dirichlet
{j1,j2}
forms, i.e.,E ∈ D ifE ∈ D andmoreoverE(u) = 0ifandonlyifuisconstant. Thenumbers
e e
c (E) are called coefficients of E. We also say that c (E) is the conductivity
{j1,j2} e {j1,j2}
between P and P (with respect to E). Next, I recall the notion of effective conductivity.
j j
1 2
Let E ∈ D, and let j ,j = 1,...,N, j 6= j . Then we put
1 2 1 2
e
L = u ∈ RV : u(P ) = 0,u(P ) = 1 .
V;j ,j j j
1 2 1 2
(cid:8) (cid:9)
It can be easily proved that the minimum min{E(u) : u ∈ L } exists, is attained at a
V;j ,j
1 2
unique function, and amounts to min{E(u) : u ∈ L }. So, for E ∈ D and {j ,j } ∈ J,
V;j ,j 1 2
2 1
we define C (E)(= C (E)) by
{j1,j2} j1,j2 e
e e
C (E) = min{E(u) : u ∈ L } = min{E(u) : u ∈ L }.
{j1,j2} V;j1,j2 V;j2,j1
e
2
The value C (E) or short C , is called effective conductivity between P and P
{j1,j2} {j1,j2} j1 j2
(with respect to E). Note that C > 0. The following remark can be easily verified
e e {j1,j2}
(see Remark 2.9 in [9].
e
Remark 2.1. If j ,j = 1,...,N, j 6= j , and E ∈ D, then
1 2 1 2
e
min E(u) : u ∈ RV(0) : u(P ) = t ,u(P ) = t = (t −t )2C .
j1 1 j2 2 1 2 {j1,j2}
(cid:8) (cid:9)
e
Recall that for every r ∈ W :=]0,+∞[k, (r := r(i)) the renormalization operator is
i
defined as follows: for every E ∈ D and every u ∈ RV(0),
e
Λ (E)(u) = inf S (E)(v),v ∈ L(u) ,
r 1,r
n o
k
S (E)(v) := r E(v◦ψ ), L(u) := v ∈ RV(1) : v = u on V(0) .
1,r i i
Xi=1 (cid:8) (cid:9)
It is well known that Λ (E) ∈ D and that the infimum is attained at a unique function
r
v := H (u). When r ∈ W, an element E of D is said to be an r-eigenform with
1,E;r e
eigenvalue ρ > 0 if Λr(E) = ρE. As this amounts to Λr(E) = E, we could also require
e ρ
ρ = 1. The problem discussed in the present paper is that of the existence of a G-eigenform
in D, in other words, the existence of E ∈ D such that Λ (E) = ρE for some ρ > 0 and
r
r ∈ W. In next section, I will describe an example of a fractal triple where there exists no
e e
G-eigenform. To this aim, it will be useful the following standard lemma (see e.g., Lemma
3.3 in [9].
Lemma 2.2 For every E ∈ D and {j ,j } ∈ J we have
1 2
C{j1,j2} eΛr(E) = min S1,r(E)(v) : v ∈ Hj1,j2 ,
where H(cid:0) = (cid:1)v ∈ RV((cid:8)1) : v(P ) = 0,v(P ) =(cid:9)1 .
e j ,j j j
1 2 1 2
(cid:8) (cid:9)
3. The Example.
Let F = (V(0),V(1),Ψ) be a fractal triple so defined. Let N = 2N′ be a positive even
number. Let V(0) = {P ,...,P }, and we fix N′ = 10 so N = 20. Let Ψ = {ψ ,...,ψ }.
1 N 1 20
Here, thus, N = k = 20. In the following, the indices of the points and of the maps will
be meant to be mod 20. For example, i + 9 = 2 if i = 13. Suppose Vi ∩ Vi′ = ∅ if
i′ ∈/ {i,i−1,i+1}, and
ψ (P ) = ψ (P ) if i odd,
i i+1 i+1 i
V ∩V = {Q }, Q =
i i+1 i i
(cid:26) ψ (P ) = ψ (P ) if i even.
i i+9 i+1 i−8
i−1 if i even,
In this way Q = ψ (P ), Q = ψ (P ), with σ(i) = Thus the
i−1 i σ(i) i i σ(i)+10 (cid:26) i−9 if i odd.
points Q and Q are opposite in V . Here we say that P and P are opposite in
i i−1 i h h+10
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V(0) and that ψ (P ) and ψ (P ) are opposite in V . In order to prove Theorem 3.2, we
i h i h+10 i
could use arguments based on effective resistances in series, but, in order to avoid some
slightly technical points, I prefer to give a direct proof. We need the following well known
lemma.
Lemma 3.1. For every positive integer n and every b > 0, i = 1,...,n, we have
i
n n n
−1
b−1 = min b x2 : x ∈ Y , Y := x ∈ Rn : x = 1 .
i i i n n i
(cid:16)Xi=1 (cid:17) nXi=1 o n Xi=1 o
n
Proof. Let f : Rn → R be defined by f(x) = b x2. Since f is continuous and
i i
i=1
P
f(x)−→ +∞, f attains a minimum m on the closed set Y at some point x. We
x→∞ n
find x using the Lagrange multiplier rule. We have b x = λ for some λ ∈ R and ev-
i i
n n n −1
ery i = 1,...,n. Thus, 1 = x = λ b−1, and x = λb−1 = b−1 b−1 . Since
i i i i i j
i=1 i=1 (cid:16)j=1 (cid:17)
P P P
m = f(x), a simple calculation completes the proof.
Theorem 3.2. On F there exists no G-eigenform.
Proof. Suppose by contradiction there exist E ∈ D and r ∈ W such that Λ (E) = E. Of
r
course, in view of Lemma 2.2, this implies
e
C (E) = min S (E)(v) : v ∈ H ∀{j ,j } ∈ J. (3.1)
‘{j1,j2} 1,r j1,j2 1 2
(cid:8) (cid:9)
e
Now, let r = max min{r ,r } : h = 0,...,9 . Thus,
2h+1 2h+2
n o
∃h = 0,...,9 : ∀d = 1,2 : r ≥ r, (3.2)
2h+d
∀h = 0,...,9 ∃d = 1,2 : r ≤ r. (3.3)
2h+d
Next,weevaluateC (E)using(3.1). Letv ∈ H . Then, sincebydefinition
2h+1,2h+2 2h+1,2h+2
20 2
S (E)(v) = reE(v◦ψ ) ≥ r E(v◦ψ ), in view of (3.2) we have
1,r i i 2h+d 2h+d
iP=1 dP=1
S (E)(v) ≥ r E(v◦ψ )+E(v◦ψ ) . (3.4)
1,r 2h+1 2h+2
(cid:0) (cid:1)
Let t = v(Q ). Let v = v ◦ψ , v = v ◦ψ . Then we have
2h+1 1 2h+1 2 2h+2
v (P ) = v(P ) = 0, v (P ) = v ψ (P ) = v(Q ) = t,
1 2h+1 2h+1 1 2h+2 2h+1 2h+2 2h+1
(cid:0) (cid:1)
v (P ) = v(P ) = 1, v (P ) = v ψ (P ) = v(Q ) = t.
2 2h+2 2h+2 2 2h+1 2h+2 2h+1 2h+1
(cid:0) (cid:1)
By Remark 2.1 and Lemma 3.1 with n = 2, we thus have
1
E(v◦ψ )+E(v◦ψ ) ≥ t2 +(1−t)2 C (E) ≥ C (E).
2h+1 2h+2 2h+1,2h+2 2 2h+1,2h+2
(cid:0) (cid:1)
e e
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r
By (3.4) and (3.1) we have C (E) ≥ C (E), thus
2h+1,2h+2 2 2h+1,2h+2
e e
r ≤ 2. (3.5)
Let now l = 1,...,20 be so that
b
C (E) ≥ C (E) ∀l = 1,...,20. (3.6)
l,l+10 l,l+10
e e
Note that, in view of Rembabrk 2.1, for every i = 1,...,20 and every t ,t ∈ R there exists
1 2
u ∈ RV(0) such that
i,t ,t
1 2
u (P ) = t , u (P ) = t , E(u ) = (t −t )2C (E). (3.7)
i,t ,t i 1 i,t ,t i+10 2 i,t ,t 1 2 i,i+10
1 2 1 2 1 2
e n
Next, define v ∈ H , in terms of x,x′ ∈ Y . Given such x,x′ let s(n) = x ,
l,l+10 9 i
iP=1
n
s′(n) = 1− x′ forbnb = 0,...,9. Let v ∈ RV(1) be so that
i
i=1
P
0 if i = l
 1 if i = l+10
b
v ◦ψ = 
i  uσ(i),s(i−l−1),s(i−l) if i =bl+1,...,l+9,
u if i =bl+11,...b,l+19.
 σ(i),s′(i−bl−11),s′(bi−l−10)



b b
We easily see that the definition ofbv is corrbect, i.e., the definition of v at the points Q
i
(the only points lying in different cells) is independent of the two representations of Q ,
i
that is, Q = ψ (P ) and Q = ψ (P ). Moreover, as P = ψ (P ), and P =
i i σ(i)+10 i i+1 σ(i+1) l l l l+10
ψ (P ). we immediately see that v ∈ H . Since E(v ◦ ψ ) = E(v ◦ ψ ) = 0,
l+10 l+10 l,l+10 l l+10
we have b b b b
b b bb b b
l+19 l+9 l+19
S (E)(v) = r E(v◦ψ ) = r E(v◦ψ )+ r E(v◦ψ )
1,r i i i i i i
bX (cid:16) bX bX (cid:17)
i=l i=l+1 i=l+11
b b b
l+9 l+19
= r E u + r E u
b i σ(i),s(i−l−1),s(i−l) b i σ(i),s′(i−l−11),s′(i−l−10)
X (cid:0) (cid:1) X (cid:0) (cid:1)
i=l+1 i=l+11
b b b b
l+b9 l+19 b
= r C x2 + r C (x′ )2 by (3.7)
i σ(i),σ(i+10) i σ(i),σ(i+10)
b i−l b i−l−10
X X
i=l+1 e i=l+11 e
b b
9 9
b b
= r C x2 + r C (x′)2
i+l σ(i+l),σ(i+l)+10 i i+l+10 σ(i+l+10),σ(i+l+10)+10 i
Xi=1 Xi=1
e e
b b b b b b
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9 9
≤ C (E) r x2 + r (x′)2 . by (3.6)
l,l+10 i+l i i+l+10 i
(cid:16)Xi=1 Xi=1 (cid:17)
e
So far we have taken arbibtbrary x,x′ ∈ Y .bNow take thbose x,x′ ∈ Y that minimize the
9 9
sums in previous formula. By Lemma 3.1, with this v we have
9 9
S (E)(v) ≤ C (E) (r )−1 −1 + (r )−1 −1 . (3.8)
1,r
l,l+10 i+l i+l+10
(cid:16)(cid:0)Xi=1 (cid:1) (cid:0)Xi=1 (cid:1) (cid:17)
e
bb b b
9
By (3.3), for at least four i = 1,...,9 we have r ≤ r, so (r )−1 −1 < r. Similarly,
i+l i+l 4
(cid:0)iP=1 (cid:1)
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(r )−1 −1 < r. Since v ∈ H , by (3b.8) and (3.1) we hbave
i+l+10 4 l,l+10
(cid:0)iP=1 (cid:1)
b bb
r
C (E) ≤ S (E)(v) < C (E).
1,r
l,l+10 2 l,l+10
e e
Thus, r > 2, which contradbibcts (3.5). bb
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