Table Of Content

A P.C.F. Self-Similar Set with no Self-Similar Energy Roberto Peirone Università di Roma ”Tor Vergata”, Dipartimento di Matematica via della Ricerca Scientifica, 00133, Roma, Italy e.mail: [email protected] 7 1 0 Abstract A general class of finitely ramified fractals is that of P.C.F. self-similar sets. 2 An important open problem in analysis on fractals was whether there exists a self-similar n energy on every P.C.F. self-similar set. In this paper, I solve the problem, showing an a J example of a P.C.F. self-similar set where there exists no self-similar energy. 6 2 1. Introduction. ] 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 [ Snowflake are finitely ramified fractals, while the Sierpinski Carpet is not. 1 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 8 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 1 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 2 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 3 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 4 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 5 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) 9 (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 References [1] B.M. Hambly, V. Metz, A. Teplyaev, Self-similar energies on post-critically finite self- similar fractals, J. London Math Soc. (2) 74, pp. 93-112, 2006 [2] K.Hattori, T. Hattori, H. Watanabe, Gaussian field theories on general networks and the spectral dimension, Progr. Theoret. Phys. Suppl. 92, pp. 108-143, 1987 [3] J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335, pp.721-755, 1993 [4] J. Kigami, Analysis on fractals, Cambridge University Press, 2001 [5] T. Lindstro¨m, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 No. 420, 1990 [6] V. Metz, The short-cut test, J. Funct. Anal. 220, pp. 118-156, 2005 [7] R. Peirone, Existence of eigenforms on fractals with three vertices, Proc. Royal Soc. Edinburgh Sect. A 137, 2007 [8] R. Peirone, Existence of eigenforms on nicely separated fractals, in Analysis of graphs and its applications, Amer. Math. Soc., Providence, pp. 231-241, 2008 [9] R. Peirone, Existence of self-similar energies on finitely ramified fractals, Journal d’Analyse Mathmatique Volume 123 Issue 1, pp. 35-94, 2014 [10] C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Sci. Ećole Norm. Sup. (4) 30, pp. 605-673, 1997 [11] R.S. Strichartz, Differential equations on fractals: a tutorial, Princeton University Press, 2006 6