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

TL;DR

This paper constructs an example of a finitely ramified P.C.F. self-similar fractal that does not admit a self-similar energy, resolving a longstanding open problem in fractal analysis.

Contribution

It provides the first known example of a P.C.F. self-similar set lacking a self-similar energy, answering a key open question.

Findings

Existence of P.C.F. self-similar set without self-similar energy

Counterexample to previous assumptions in fractal analysis

Advances understanding of energy structures on fractals

Abstract

A general class of finitely ramified fractals is that of P.C.F. self-similar sets. An important open problem in analysis on fractals was whether there exists a self-similar energy on every P.C.F. self-similar set. In this paper, I solve the problem, showing an example of a P.C.F. self-similar set where there exists no self-similar energy.

Equations

$$V^{(1)}=\bigcup _{\psi \in \Psi }\psi (V^{(0)}).$$

$$E(u)=\sum\limits_{\{j_{1},j_{2}\}\in J}c_{\{j_{1},j_{2}\}}(E)\big{(}u(P_{j_{1}})-u(P_{j_{2}})\big{)}^{2}$$

$${\Cal{L}}_{V;j_{1},j_{2}}=\big{\{}u\in\mathbb{R}^{V}:u(P_{j_{1}})=0,u(P_{j_{2}})=1\big{\}}.$$

$${\tilde{C}}_{\{j_1,j_2\}}(E)=\min \{E(u):u\in {\text{Cal}L}_{V;j_1,j_2}\}=\min \{E(u):u\in {\text{Cal}L}_{V;j_2,j_1}\}.$$

$$\min\big{\{}E(u):u\in\mathbb{R}^{V^{(0)}}:u(P_{j_{1}})=t_{1},u(P_{j_{2}})=t_{2}\big{\}}=(t_{1}-t_{2})^{2}\,\widetilde{C}_{\{j_{1},j_{2}\}}.$$

$$\Lambda_{r}(E)(u)=\inf\Big{\{}S_{1,r}(E)(v),v\in{\Cal{L}}(u)\Big{\}},$$

$$S_{1,r}(E)(v):=\sum\limits_{i=1}^{k}r_{i}E(v\circ\psi_{i}),\quad{\Cal{L}}(u):=\big{\{}v\in\mathbb{R}^{V^{(1)}}:v=u\ \text{on}\ V^{(0)}\big{\}}.$$

$$V_i\cap V_{i+1}=\{{\overline{Q}}_i\},{\overline{Q}}_i=\{\begin{array}{l}{\psi }_i(P_{i+1})={\psi }_{i+1}(P_i)\text{ if }i\text{ odd,}\\ {\psi }_i(P_{i+9})={\psi }_{i+1}(P_{i-8})\text{ if }i\text{ even.}\end{array}$$

$$\Big{(}\sum\limits_{i=1}^{n}b_{i}^{-1}\Big{)}^{-1}=\min\Big{\{}\sum\limits_{i=1}^{n}b_{i}x_{i}^{2}:x\in Y_{n}\Big{\}},\quad Y_{n}:=\Big{\{}x\in\mathbb{R}^{n}:\sum\limits_{i=1}^{n}x_{i}=1\Big{\}}.$$

$$\widetilde{C}_{`\{j_{1},j_{2}\}}(E)=\min\big{\{}S_{1,r}(E)(v):v\in H_{j_{1},j_{2}}\big{\}}\quad\forall\,\{j_{1},j_{2}\}\in J.$$

$$\exists\,\overline{h}=0,...,9:\forall\,d=1,2:r_{2\overline{h}+d}\geq\overline{r},\$$

$$\forall h=0,...,9\exists d=1,2:r_{2h+d}\le \overline{r}.$$

$$S_{1,r}(E)(v)\geq\overline{r}\big{(}E(v\circ\psi_{2\overline{h}+1})+E(v\circ\psi_{2\overline{h}+2})\big{)}.$$

$$v_{1}(P_{2\overline{h}+1})=v({P_{2\overline{h}+1}})=0,\quad v_{1}(P_{2\overline{h}+2})=v\big{(}\psi_{2\overline{h}+1}(P_{2\overline{h}+2})\big{)}=v(\overline{Q}_{2\overline{h}+1})=t,$$

$$v_{2}(P_{2\overline{h}+2})=v({P_{2\overline{h}+2}})=1,\quad v_{2}(P_{2\overline{h}+1})=v\big{(}\psi_{2\overline{h}+2}(P_{2\overline{h}+1})\big{)}=v(\overline{Q}_{2\overline{h}+1})=t.$$

$$E(v\circ\psi_{2\overline{h}+1})+E(v\circ\psi_{2\overline{h}+2})\geq\big{(}t^{2}+(1-t)^{2}\big{)}\widetilde{C}_{2\overline{h}+1,2\overline{h}+2}(E)\geq{1\over 2}\widetilde{C}_{2\overline{h}+1,2\overline{h}+2}(E).$$

$$\overline{r}\le 2.$$

$${\tilde{C}}_{\hat{l},\hat{l}+10}(E)\ge {\tilde{C}}_{l,l+10}(E)\forall l=1,...,20.$$

$$u_{i,t_1,t_2}(P_i)=t_1,u_{i,t_1,t_2}(P_{i+10})=t_2,E(u_{i,t_1,t_2})=(t_1-t_2{)}^2{\tilde{C}}_{i,i+10}(E).$$

$$v\circ {\psi }_i=\{\begin{array}{l}0\text{if}i=\hat{l}\\ 1\text{if}i=\hat{l}+10\\ u_{\sigma (i),s(i-\hat{l}-1),s(i-\hat{l})}\text{if}i=\hat{l}+1,...,\hat{l}+9,\\ u_{\sigma (i),s^{\prime }(i-\hat{l}-11),s^{\prime }(i-\hat{l}-10)}\text{if}i=\hat{l}+11,...,\hat{l}+19.\end{array}$$

$$S_{1,r}(E)(v)=\sum\limits_{i=\widehat{l}}^{\widehat{l}+19}r_{i}E(v\circ\psi_{i})=\Big{(}\sum\limits_{i=\widehat{l}+1}^{\widehat{l}+9}r_{i}E(v\circ\psi_{i})+\sum\limits_{i=\widehat{l}+11}^{\widehat{l}+19}r_{i}E(v\circ\psi_{i})\Big{)}$$

$$=\sum\limits_{i=\widehat{l}+1}^{\widehat{l}+9}r_{i}E\big{(}u_{\sigma(i),s(i-\widehat{l}-1),s(i-\widehat{l})}\big{)}+\sum\limits_{i=\widehat{l}+11}^{\widehat{l}+19}r_{i}E\big{(}u_{\sigma(i),s^{\prime}(i-\widehat{l}-11),s^{\prime}(i-\widehat{l}-10)}\big{)}$$

$$=\sum _{i=\hat{l}+1}^{\hat{l}+9}{r}_i{\tilde{C}}_{\sigma (i),\sigma (i+10)}{x}_{i-\hat{l}}^2+\sum _{i=\hat{l}+11}^{\hat{l}+19}{r}_i{\tilde{C}}_{\sigma (i),\sigma (i+10)}(x_{i-\hat{l}-10}^{\prime }{)}^2$$

$$=\sum _{i=1}^9{r}_{i+\hat{l}}{\tilde{C}}_{\sigma (i+\hat{l}),\sigma (i+\hat{l})+10}{x}_i^2+\sum _{i=1}^9{r}_{i+\hat{l}+10}{\tilde{C}}_{\sigma (i+\hat{l}+10),\sigma (i+\hat{l}+10)+10}(x_i^{\prime }{)}^2$$

$$\leq\widetilde{C}_{\widehat{l},\widehat{l}+10}(E)\Big{(}\sum\limits_{i=1}^{9}r_{i+\widehat{l}}\ x_{i}^{2}+\sum\limits_{i=1}^{9}r_{i+\widehat{l}+10}(x^{\prime}_{i})^{2}\Big{)}.$$

$$S_{1,r}(E)(v)\leq\widetilde{C}_{\widehat{l},\widehat{l}+10}(E)\Big{(}\big{(}\sum\limits_{i=1}^{9}(r_{i+\widehat{l}})^{-1}\big{)}^{-1}+\big{(}\sum\limits_{i=1}^{9}(r_{i+\widehat{l}+10})^{-1}\big{)}^{-1}\Big{)}.$$

$${\tilde{C}}_{\hat{l},\hat{l}+10}(E)\le S_{1,r}(E)(v)<\frac{\overline{r}}{2}{\tilde{C}}_{\hat{l},\hat{l}+10}(E).$$

Full text

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: peironemat.uniroma2.it

Abstract A general class of finitely ramified fractals is that of P.C.F. self-similar sets. An important open problem in analysis on fractals was whether there exists a self-similar energy on every P.C.F. self-similar set. In this paper, I solve the problem, showing an example of a P.C.F. self-similar set where there exists no self-similar energy.

1. Introduction.

An important problem in analysis of fractals is the construction of a Laplace operator, or equivalently, an energy, more precisely, a self-similar Dirichlet form. The construction of a self-similar Dirichlet form has been investigated specially on finitely ramified fractals. Roughly speaking, a fractal is finitely ramified if the intersection of each pair of copies 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.

More specially, we consider the P.C.F. self similar-sets, a general class of finitely ramified fractals introduced by Kigami in [3]. A general theory with many examples can be found in [4]. On such a class of fractals, the basic tool used to construct a self-similar Dirichlet form is a discrete Dirichlet form defined on a special finite subset $V^{(0)}$ of the fractal. Such discrete Dirichlet forms have to be eigenforms, i.e. the eigenvectors of a special nonlinear operator $\Lambda_{r}$ called renormalization operator, which depends on a set of positive weights $r_{i}$ placed on the cells of the fractal. In [5], [10] and [6] criteria for the existence of an eigenform with prescribed weights are discussed. In particular, in [5], T. Lindstrøm proved that there exists an eigenform on the nested fractals with all weights equal to $1$, C. Sabot in [10] proved a rather general criterion, and V. Metz in [6] improved the results in [10].

In [1], [7], [8] and [9], instead, the problem is considered whether on a given fractal there exists a G-eigenform. By this we mean a form $E$ which is an eigenform of the operator $\Lambda_{r}$ for some set of weights $r$. In such papers the existence of a G-eigenform was proved on 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 $V_{i}$ only intesects $V_{i-1}$ and $V_{i+1}$, and 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 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 ${\Cal{F}}:=(V^{(0)},V^{(1)},\Psi)$ where $V^{(0)}=V$ and $V^{(1)}$ are finite sets with $\#V^{(0)}\geq 2$, and $\Psi$ is a finite set of one-to-one maps from $V^{(0)}$ into $V^{(1)}$ satisfying

[TABLE]

We put V^{(0)}=\big{\{}P_{1},...,P_{N}\big{\}}, and of course $N\geq 2$. A set of the form $\psi(V^{(0)})$ with $\psi\in\Psi$ will be called a cell or a $1$-cell. We require that

a) For each $j=1,...,N$ there exists a (unique) map $\psi_{j}\in\Psi$ such that $\psi_{j}(P_{j})=P_{j}$, and \Psi=\big{\{}\psi_{1},...,\psi_{k}\big{\}}, with $k\geq N$.

b) $P_{j}\notin\psi_{i}(V^{(0)})$ when $i\neq j$ (in other words, if $\psi_{i}(P_{h})=P_{j}$ with $i=1,...,k$, $j,h=1,...,N$, 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 $V^{(0)}\subseteq V^{(1)}$. Let $W=]0,+\infty[^{k}$ and put $V_{i}=\psi_{i}(V^{(0)})$ for each $i=1,...,k$. Let $J(=J(V))=\{\{j_{1},j_{2}\}:j_{1},j_{2}=1,...,N,j_{1}\neq j_{2}\}$. It is well-known that on every fractal triple we can construct a P.C.F.-self-similar set.

We denote by $\Cal{D}({\Cal{F}})$ or simply $\Cal{D}$ the set of the Dirichlet forms on $V$, invariant with respect to an additive constant, i.e., the set of the functionals $E$ from $\mathbb{R}^{V}$ into $\mathbb{R}$ of the form

[TABLE]

with $c_{\{j_{1},j_{2}\}}(E)\geq 0$. I will denote by $\widetilde{\Cal{D}}({\Cal{F}})$ or simply $\widetilde{\Cal{D}}$ the set of the irreducible Dirichlet forms, i.e., $E\in\widetilde{\Cal{D}}$ if $E\in\Cal{D}$ and moreover $E(u)=0$ if and only if $u$ is constant. The numbers $c_{\{j_{1},j_{2}\}}(E)$ are called coefficients of $E$. We also say that $c_{\{j_{1},j_{2}\}}(E)$ is the conductivity between $P_{j_{1}}$ and $P_{j_{2}}$ (with respect to $E$). Next, I recall the notion of effective conductivity. Let $E\in\widetilde{\Cal{D}}$, and let $j_{1},j_{2}=1,...,N$, $j_{1}\neq j_{2}$. Then we put

[TABLE]

It can be easily proved that the minimum $\min\{E(u):u\in{\Cal{L}}_{V;j_{1},j_{2}}\}$ exists, is attained at a unique function, and amounts to $\min\{E(u):u\in{\Cal{L}}_{V;j_{2},j_{1}}\}$. So, for $E\in\widetilde{\Cal{D}}$ and $\{j_{1},j_{2}\}\in J$, we define $\widetilde{C}_{\{j_{1},j_{2}\}}(E)(=\widetilde{C}_{j_{1},j_{2}}(E))$ by

[TABLE]

The value $\widetilde{C}_{\{j_{1},j_{2}\}}(E)$ or short $\widetilde{C}_{\{j_{1},j_{2}\}}$, is called effective conductivity between $P_{j_{1}}$ and $P_{j_{2}}$ (with respect to $E$). Note that $\widetilde{C}_{\{j_{1},j_{2}\}}>0$. The following remark can be easily verified (see Remark 2.9 in [9].

Remark 2.1. If $j_{1},j_{2}=1,...,N$, $j_{1}\neq j_{2}$, and $E\in\widetilde{\Cal{D}}$, then

[TABLE]

Recall that for every $r\in W:=]0,+\infty[^{k}$, ($r_{i}:=r(i))$ the renormalization operator is defined as follows: for every $E\in\widetilde{\Cal{D}}$ and every $u\in\mathbb{R}^{V^{(0)}}$,

[TABLE]

[TABLE]

It is well known that $\Lambda_{r}(E)\in\widetilde{\Cal{D}}$ and that the infimum is attained at a unique function $v:=H_{1,E;r}(u)$. When $r\in W$, an element $E$ of $\widetilde{\Cal{D}}$ is said to be an $r$-eigenform with eigenvalue $\rho>0$ if $\Lambda_{r}(E)=\rho E$. As this amounts to $\Lambda_{{r\over\rho}}(E)=E$, we could also require $\rho=1$. The problem discussed in the present paper is that of the existence of a G-eigenform in $\widetilde{\Cal{D}}$, in other words, the existence of $E\in\widetilde{\Cal{D}}$ such that $\Lambda_{r}(E)=\rho E$ for some $\rho>0$ and $r\in W$. In next section, I will describe an example of a fractal triple where there exists no 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 $E\in\widetilde{\Cal{D}}$ and $\{j_{1},j_{2}\}\in J$ we have

\widetilde{C}_{\{j_{1},j_{2}\}}\big{(}\Lambda_{r}(E)\big{)}=\min\big{\{}S_{1,r}(E)(v):v\in H_{j_{1},j_{2}}\big{\}},

\text{where}\ \ H_{j_{1},j_{2}}=\big{\{}v\in\mathbb{R}^{V^{(1)}}:v(P_{j_{1}})=0,v(P_{j_{2}})=1\big{\}}\,.

3. The Example.

Let $\overline{\Cal{F}}=(V^{(0)},V^{(1)},\Psi)$ be a fractal triple so defined. Let $N=2N^{\prime}$ be a positive even number. Let $V^{(0)}=\{P_{1},...,P_{N}\}$, and we fix $N^{\prime}=10$ so $N=20$. Let $\Psi=\{\psi_{1},...,\psi_{20}\}$. 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 $V_{i}\cap V_{i^{\prime}}=\varnothing$ if $i^{\prime}\notin\{i,i-1,i+1\}$, and

[TABLE]

In this way $\overline{Q}_{i-1}=\psi_{i}(P_{\sigma(i)})$, $\overline{Q}_{i}=\psi_{i}(P_{\sigma(i)+10})$, with $\sigma(i)=\begin{cases}i-1\text{\ if}\ i\text{\ even,}\\ i-9\text{\ if\ }i\text{\ odd}.\end{cases}$ Thus the points $\overline{Q}_{i}$ and $\overline{Q}_{i-1}$ are opposite in $V_{i}$. Here we say that $P_{h}$ and $P_{h+10}$ are opposite in $V^{(0)}$ and that $\psi_{i}(P_{h})$ and $\psi_{i}(P_{h+10})$ are opposite in $V_{i}$. In order to prove Theorem 3.2, we 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 $b_{i}>0$, $i=1,...,n$, we have

[TABLE]

Proof. Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be defined by $f(x)=\sum\limits_{i=1}^{n}b_{i}x_{i}^{2}$. Since $f$ is continuous and $f(x)\mathop{\longrightarrow}_{x\to\infty}+\infty$, $f$ attains a minimum $m$ on the closed set $Y_{n}$ at some point $\overline{x}$. We find $\overline{x}$ using the Lagrange multiplier rule. We have $b_{i}\overline{x}_{i}=\lambda$ for some $\lambda\in\mathbb{R}$ and every $i=1,...,n$. Thus, $1=\sum\limits_{i=1}^{n}\overline{x}_{i}=\lambda\sum\limits_{i=1}^{n}b_{i}^{-1},$ and \overline{x}_{i}=\lambda b_{i}^{-1}=b_{i}^{-1}\Big{(}\sum\limits_{j=1}^{n}b_{j}^{-1}\Big{)}^{-1}. Since $m=f(\overline{x})$, a simple calculation completes the proof.

Theorem 3.2. On $\overline{\Cal{F}}$ there exists no $G$-eigenform.

Proof. Suppose by contradiction there exist $E\in\widetilde{\Cal{D}}$ and $r\in W$ such that $\Lambda_{r}(E)=E$. Of course, in view of Lemma 2.2, this implies

[TABLE]

Now, let \overline{r}=\max\Big{\{}\min\{r_{2h+1},r_{2h+2}\}:h=0,...,9\Big{\}}. Thus,

[TABLE]

[TABLE]

Next, we evaluate $\widetilde{C}_{2\overline{h}+1,2\overline{h}+2}(E)$ using (3.1). Let $v\in H_{2\overline{h}+1,2\overline{h}+2}$. Then, since by definition $S_{1,r}(E)(v)=\sum\limits_{i=1}^{20}r_{i}E(v\circ\psi_{i})\geq\sum\limits_{d=1}^{2}r_{2\overline{h}+d}E(v\circ\psi_{2\overline{h}+d})$, in view of (3.2) we have

[TABLE]

Let $t=v(\overline{Q}_{2\overline{h}+1})$. Let $v_{1}=v\circ\psi_{2\overline{h}+1}$, $v_{2}=v\circ\psi_{2\overline{h}+2}$. Then we have

[TABLE]

[TABLE]

By Remark 2.1 and Lemma 3.1 with $n=2$, we thus have

[TABLE]

By (3.4) and (3.1) we have $\widetilde{C}_{2\overline{h}+1,2\overline{h}+2}(E)\geq\displaystyle{{\overline{r}\over 2}}\widetilde{C}_{2\overline{h}+1,2\overline{h}+2}(E)$, thus

[TABLE]

Let now $\widehat{l}=1,...,20$ be so that

[TABLE]

Note that, in view of Remark 2.1, for every $i=1,...,20$ and every $t_{1},t_{2}\in\mathbb{R}$ there exists $u_{i,t_{1},t_{2}}\in\mathbb{R}^{V^{(0)}}$ such that

[TABLE]

Next, define $v\in H_{\widehat{l},\widehat{l}+10}$, in terms of $x,x^{\prime}\in Y_{9}$. Given such $x,x^{\prime}$ let $s(n)=\sum\limits_{i=1}^{n}x_{i}$, $s^{\prime}(n)=1-\sum\limits_{i=1}^{n}x^{\prime}_{i}$ for $n=0,...,9$. Let $v\in\mathbb{R}^{V^{(1)}}$ be so that

[TABLE]

We easily see that the definition of $v$ is correct, i.e., the definition of $v$ at the points $\overline{Q}_{i}$ (the only points lying in different cells) is independent of the two representations of $\overline{Q}_{i}$, that is, $\overline{Q}_{i}=\psi_{i}(P_{\sigma(i)+10})$ and $\overline{Q}_{i}=\psi_{i+1}(P_{\sigma(i+1)})$. Moreover, as $P_{\widehat{l}}=\psi_{\widehat{l}}(P_{\widehat{l}})$, and $P_{\widehat{l}+10}=\psi_{\widehat{l}+10}(P_{\widehat{l}+10})$. we immediately see that $v\in H_{\widehat{l},\widehat{l}+10}$. Since $E(v\circ\psi_{\widehat{l}})=E(v\circ\psi_{\widehat{l}+10})=0$, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So far we have taken arbitrary $x,x^{\prime}\in Y_{9}$. Now take those $x,x^{\prime}\in Y_{9}$ that minimize the sums in previous formula. By Lemma 3.1, with this $v$ we have

[TABLE]

By (3.3), for at least four $i=1,...,9$ we have $r_{i+\widehat{l}}\leq\overline{r}$, so \big{(}\sum\limits_{i=1}^{9}(r_{i+\widehat{l}})^{-1}\big{)}^{-1}<{\overline{r}\over 4}. Similarly, \big{(}\sum\limits_{i=1}^{9}(r_{i+\widehat{l}+10})^{-1}\big{)}^{-1}<{\overline{r}\over 4}. Since $v\in H_{\widehat{l},\widehat{l}+10}$, by (3.8) and (3.1) we have

[TABLE]

Thus, $\overline{r}>2$, which contradicts (3.5).

References

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[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

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[5] T. Lindstr$\ddot{o}$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 Mathématique Volume 123 Issue 1, pp. 35-94, 2014

[10] C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Sci. École Norm. Sup. (4) 30, pp. 605-673, 1997

[11] R.S. Strichartz, Differential equations on fractals: a tutorial, Princeton University Press, 2006