Brownian motion on the golden ratio Sierpinski gasket

Abstract

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate.

1. Introduction

The golden ratio Sierpinski gasket $\mathcal {G}$ is a typical example of a self-similar set satisfying the finite type property ([2], see definition 2.1.), which arises in the study of the Hausdorff dimension of self-similar sets with overlaps [26, 31, 32]. Let $q_0=\left (\frac {1}{2},\frac {\sqrt {3}}{2}\right ),q_1=(0,0),q_2=(1,0)$ be the three vertices of an equilateral triangle in $\mathbb {R}^2$, and

\begin{align*} & F_0(x)=\rho^2(x-q_0)+q_0,\\ & F_1(x)=\rho(x-q_1)+q_1,\quad F_2(x)=\rho(x-q_2)+q_2, \end{align*}

with $\rho =\frac {\sqrt {5}-1}{2}$ being the golden ratio. The gasket $\mathcal {G}$ is the invariant set associated with the iterated function system (IFS for short) $\{F_0,F_1,F_2\}$, i.e. $\mathcal {G}$ is the unique non-empty compact set satisfying

\[ \mathcal{G}=\bigcup_{i=0}^2 F_i\mathcal{G}. \]

See figure 1.

Fig. 1. The golden ratio Sierpinski gasket $\mathcal {G}$.

The large overlap $F_1\mathcal {G}\cap F_2\mathcal {G}$ makes $\mathcal {G}$ different from the existing examples of self-similar sets on which Brownian motions are constructed.

First, any effort to disconnect the bottom line of $\mathcal {G}$ requires the removal of infinitely many points, so there is not a finitely ramified cell structure [35] on $\mathcal {G}$. Well-known classes of fractals with finitely ramified cell structures include Lindstr$\phi$m's nested fractals [27], Kigami's post-critically finite (PCF) self-similar sets [20, 21], finitely ramified graph-directed fractals [9, 19], and some Julia sets of polynomials [1, 14, 33] or rational functions [10]. See [8, 16, 24] for pioneering works on the Sierpinski gasket, and also books [3, 22] for systematic discussions.

Second, although there is a graph-directed construction related with $\mathcal {G}$ (see § 2), by dividing $\mathcal {G}$ into blocks of nearly the same size, the graph is much more complicated. As a result, the deep and famous constructions on the Sierpinski carpet [4–6] by Barlow and Bass, and on certain symmetric fractals [25] by Kusuoka and Zhou would be extremely difficult here. See also [7] for a theorem of uniqueness on the Sierpinski carpet.

Instead, thanks to the golden ratio, there is an ‘infinite cell structure’ on $\mathcal {G}$. For the first level, we consider the cell $F_0\mathcal {G}$ and its images under compositions of $F_1,F_2$. The union of these cells covers $\mathcal {G}$ except the bottom line. For each such cell, we can find a finite word $w$, and a contraction map $F_w=F_{w_1}\circ F_{w_2}\circ \cdots \circ F_{w_m}$, so that the cell can be written as $F_w\mathcal {G}$. We name the collection of all such words $W_1$, and construct a resistance form [22] on $\mathcal {G}$, that is self-similar in the sense of the infinite IFS $\{F_w\}_{w\in W_1}$. Roughly speaking, we have the following theorem, see theorems 6.5, 6.6 and 6.8 for detailed and formal results.

Theorem 1.1 There exists a unique strongly local regular resistance form $(\mathcal {E},\mathcal {F})$ on $\mathcal {G}$ such that $f\in \mathcal {F}$ if and only if $f\circ F_w\in \mathcal {F}$ for all $w\in W_1$ and $\sum _{w\in W_1} \rho _w^{-\theta }\mathcal {E}(f\circ F_w,f\circ F_w)<\infty$, where $\rho _w$ is the similarity ratio of $F_w$ and $0<\theta <1$ is a constant. In addition,

\[ \mathcal{E}(f,f)=\sum_{w\in W_1} \rho_w^{-\theta}\mathcal{E}(f\circ F_w,f\circ F_w). \]

Moreover, the form is decimation invariant with respect to the graph-directed construction of $\mathcal {G}$.

The form $(\mathcal {E},\mathcal {F})$ is then a strongly local regular Dirichlet form on $L^2(\mathcal {G},\mu _H)$, where $\mu _H$ is the normalized Hausdorff measure on $\mathcal {G}$. In addition, there is an associated diffusion process on $\mathcal {G}$. (Readers are suggested to refer the book [15] for more explanations.) Although, our construction is based on an infinite IFS, the behaviour of the process is same on each cell before hitting the boundary, up to a time scaling, since any cell can be decomposed in a same manner.

In addition, by following the well-established method of Hambly and Kumagai [18], which is organized in Barlow's book [3], we can obtain a sub-Gaussian heat kernel estimate (see § 7). We refer to [8, 13, 23] for earlier results on transition density estimates on fractals.

Theorem 1.2 There is a symmetric transition density $p(t,x,y)$ associated with the form $(\mathcal {E},\mathcal {F})$ on $\mathcal {G}$. In addition, there are constants $c_1,c_2,c_3,c_4$ so that

\begin{align*} & c_1t^{{-}d_H/\beta}\exp\left({-}c_2\left(\frac{d(x,y)^\beta}{t}\right)^{\frac{1}{\beta-1}}\right)\\ & \quad \leq p(t,x,y)\leq c_3t^{{-}d_H/\beta}\exp\left({-}c_4\left(\frac{d(x,y)^\beta}{t}\right)^{\frac{1}{\beta-1}}\right), \end{align*}

for $0< t\leq 1$, with $\beta =\theta +d_H$, where $d_H\approx 1.6824$ is the Hausdorff dimension of $\mathcal {G}$, and $d$ represents the Euclidean metric.

The main difficulty in proving this result is establishing an estimate for the resistance metric $R$ on $\mathcal {G}$ of the form $c_1\,{\rm d}(x,y)^\theta \leq R(x,y)\leq c_2\,{\rm d}(x,y)^\theta$ for some constant $c_1,c_2>0$.

We organize the structure of the paper as follows. In § 2, we will briefly introduce some facts about the geometry of $\mathcal {G}$. From § 3 to 5, we study the trace of forms on an infinite graph. In § 3, we establish the resistance forms on the graph. In § 4, we study the trace map and a related renormalization map. We will show the joint continuity of the renormalization map. In § 5, we show that there is a unique solution to a renormalization problem. With all these preparations, we construct the resistance form on $\mathcal {G}$ in § 6, and at the same time we derive an upper-bound estimate for the resistance metric. Lastly, we obtain the transition density estimate through a lower bound for the resistance metric in § 7.

Before ending this section, we remark that the result in this paper has a natural extension, by replacing $0<\rho <1$ to be a real root of $x^n-2x+1$ with $n\geq 4$, and taking the contraction ratios corresponding to $F_0, F_1, F_2$ to be $1-\rho, \rho,\rho$. Indeed, by doing so, we obtain a class of gaskets that possess a similar overlapping structure of $\mathcal {G}$, see figure 2.

Fig. 2. A gasket with $0<\rho <1$ being a root of $x^4-2x+1=0$.

2. Preliminary

The golden ratio Sierpinski gasket $\mathcal {G}$ is a typical example of a self-similar set with overlaps but satisfying the finite type property.

Let $K$ be a general self-similar set associated with an IFS $\{F_i\}_{i=0}^{N-1}$ with contraction ratios $\{\rho _i\}_{i=0}^{N-1}$ with respect to the Euclidean metric. For $m\geq 1$, we call $w=w_1w_2\cdots w_m$ with $w_i\in \{0,1,\dots,N-1\}$, a word of length $m$ (denoted by $|w|$), and call $\emptyset$ the empty word. We denote the set of all words by $\tilde {W}_*$. For any word $w\in \tilde {W}_*$, we write $F_w=F_{w_1}\circ F_{w_2}\cdots \circ F_{w_{|w|}}$, and let $F_\emptyset$ be the identity map for consistency. Let $\rho _*=\min \{\rho _i: 0\leq i< N\}$.

Definition 2.1 finite type property

A self-similar set $K$ is of finite type if there are only finitely many maps $h=F_w^{-1}F_v$ with $w,v\in \tilde W_*$ and $F_w K\cap F_v K\neq \emptyset$, and with similarity ratio $\rho _h\in (\rho _*,1/\rho _*)$.

The finite type property of $K$, formulated in algebraic terms, was introduced in [2] by Bandt and Rao. It guarantees the existence of an ‘almost non-overlapping’ graph-directed construction (see [2, 31] for details) of $K$, which is quite useful for calculating the Hausdorff dimension of $K$. See [26, 32] for more flexible variants of the finite type property.

It is easy to verify that $\mathcal {G}$ satisfies the finite type property, noticing that $F_{122}\mathcal {G}=F_{211}\mathcal {G}$. In particular, it has the following graph-directed construction [28].

Definition 2.2 a graph-directed construction of $\mathcal {G}$

1. (a). Let $K_1=\mathcal {G}$ and $K_2=\overline {\mathcal {G}\setminus F_{22}\mathcal {G}}$.

2. (b). Let $\Gamma (S,E)$ be a directed graph with the vertex set $S=\{1,2\}$, and the edge set $E=\{e_i\}_{i=1}^6$, where $e_1=(1,2),e_2=(1,1),e_3=(2,1),e_4=(2,2),e_5=(2,2),e_6=(2,1)$.

3. (c). Define $\psi _{e_1}=Id$, $\psi _{e_2}=F_{22}$, $\psi _{e_3}=F_0$, $\psi _{e_4}=F_1$, $\psi _{e_5}=F_{21}$, $\psi _{e_6}=F_{20}$.

Clearly, we have

\[ K_1=\bigcup_{i=1}^2 \psi_{e_i}K_{e_{i,2}}\text{ and }K_2=\bigcup_{i=3}^6 \psi_{e_i}K_{e_{i,2}}, \]

where we use the notation $e_i=(e_{i,1},e_{i,2})$ for a directed edge. In addition, there exist bounded open sets $O_1$ and $O_2$ such that $\bigcup _{i=1}^2 \psi _{e_i}O_{e_{i,2}}\subset O_1$ and $\bigcup _{i=3}^6 \psi _{e_i}O_{e_{i,2}}\subset O_2$, where the unions are disjoint. See figure 3 for an illustration.

Fig. 3. A graph-directed construction of $\mathcal {G}$.

Then similar to the open set condition situation, one can calculate the exact value of the Hausdorff dimension of $\mathcal {G}$ to be

\[ d_H=\frac{\log\eta}{-2\log\rho}\approx1.6824 \]

with $\eta$ being the largest root of $x^3-6x^2+5x-1$. In addition, the associated Hausdorff measure of $\mathcal {G}$ is positive and finite. See details in [31] by Ngai and Wang.

Throughout this paper, we use $d$ to denote the Euclidean metric, and take $\mu _H$ to be the normalized Hausdorff measure on $\mathcal {G}$ with respect to $d$, i.e. $\mu _H(\mathcal {G})=1.$ For $p,q\in \mathcal {G}$, let

\[ d_g(p,q)=\inf\{|\gamma|:\gamma \text{ is a path connecting }p,q, \text{ and }\gamma\subset\mathcal{G}\}, \]

be the geodesic metric between $p,q$. It is not hard to verify the following lemma.

Lemma 2.3

1. (a). Let $B_s(p)=\{q\in \mathcal {G}:{\rm d}(p,q)< s\}$. There are constants $c_1,c_2>0$ such that

   \[ c_1s^{d_H}\leq\mu_H(B_s(p))\leq c_2s^{d_H}, \quad\forall p\in \mathcal{G},0< s\leq1. \]

2. (b). There exists a constant $c\geq 1$ such that

   \[ {\rm d}(p,q)\leq d_g(p,q)\leq c{\rm d}(p,q),\quad \forall p,q\in\mathcal{G}. \]

The statement (a) is a well-known fact (for example, see [12, corollary 6.4.4]). The proof of (b) relies on the finite type property. The rough idea is to link $p,q$ with a bounded number of cells of diameter approximating to ${\rm d}(p,q)$.

By the compactness of $\mathcal {G}$ (for example, see [3, lemma 2.1.1]), there is always a path admitting the infimum length between $p,q$. So, the metric space $(\mathcal {G},d_g)$ satisfies the so-called midpoint property, i.e. for any $p,q\in \mathcal {G}$, there exists $p'$ so that $d_g(p,p')=d_g(p',q)=\frac {1}{2}d_g(p,q)$. The space $(\mathcal {G},d_g,\mu _H)$ is then a fractional metric space, see [3, definition 3.2].

We will return to look at the geometric properties of $\mathcal {G}$ listed in this section. But first, from § 3 to 5, we will instead consider an infinite IFS and the associated infinite graph.

3. Resistance forms on the infinite graph $V_1$

The golden ratio Sierpinski gasket $\mathcal {G}$ can be realized as an invariant set of an infinite IFS. For convenience, we introduce some notation. For any word $w,w'\in \tilde {W}_*$, we write $ww'$ for the concatenation of $w,w'$. For $w=w_1w_2\cdots w_m$ and $0\leq l\leq m$, we write $[w]_l=w_1w_2\cdots w_l$. The following notation is a little different from the standard ones.

Notation. Choose a set of finite words $W_1\subset \bigcup _{n=0}^\infty \{1,2\}^n\times \{0\}$ so that

1. 1. for any $w\in \bigcup _{n=0}^\infty \{1,2\}^n\times \{0\}$, there exists $w'\in W_1$ such that $F_w=F_{w'}$;

2. 2. for different words $w,w'\in W_1$, we have $F_w\neq F_{w'}$.

In addition, based on $W_1$, we introduce some more notations.

1. (a) For $n\geq 1$, define $W_{1,n}=\{w\in W_1:|w|=n\}$;

2. (b) For $m\geq 2$, define $W_m:=W_1^m=\{w_1w_2\cdots w_m: w_i\in W_1, 1\leq i\leq m\}$;

3. (c) Write $V_0=\{q_i\}_{i=0}^2$ and for $m\geq 1$, $V_m=\bigcup _{w\in W_m}F_wV_0$. Denote $\bar {V}_m$ the closure of $V_m$;

4. (d) For distinct $p,q\in V_1$, we denote $p\sim q$ if and only if $p,q\in F_wV_0$ for some $w\in W_1$, which induce an infinite graph $(V_1,\sim )$. See figure 4 for an illustration.

Fig. 4. The infinite graph $(V_1,\sim )$. (The bottom line equals to $\bar V_1\setminus V_1$.)

Obviously, we have

\[ \mathcal{G}=\overline{\bigcup_{w\in W_1}F_w\mathcal{G}}, \]

and thus $\{F_w\}_{w\in W_1}$ is an infinite i.f.s associated with $\mathcal {G}$. See [30] for more details about infinite IFSs. The advantage of this IFS lies in the fact that

\[ F_w\mathcal{G}\cap F_{w'}\mathcal{G}=F_wV_0\cap F_{w'}V_0, \quad\forall w\neq w'\in W_1. \]

Remark For $n\geq 1$, if we rename the vertices $\{F_w q_0\}_{w\in W_{1,n}}$ to be $\{p_i^{(n)}\}_{i=1}^{N_n}$ with $N_n:=\# W_{1,n}$, so that for each $i$, $p_i^{(n)}$ is on the left of $p_{i+1}^{(n)}$. Then it directly calculates that $d(p_i^{(n)}, p_{i+1}^{(n)})$ is either $\rho ^{n}$ or $\rho ^{n+1}$, and thus $N_n\asymp \rho ^{-n}$.

In addition, for $p,q\in V_1$ with ${\rm d}(p,q)<\rho ^{n+2}$, there always exist $w,w'\in \{1,2\}^{n}$ such that $p\in F_{w}V_1,q\in F_{w'}V_1$ and $F_wV_1\cap F_{w'}V_1\neq \emptyset$. In fact, by a direct observation, $p\notin \bigcup _{n'=1}^{n+1}\{p^{(n')}_i\}_{i=1}^{N_{n'}}$, and so we can find $\tilde {w}\in \{1,2\}^{n+1}$ such that $p\in F_{\tilde {w}}V_1$, and one can then see that $q\in \bigcup \left \{F_{w'}V_1: w,w'\in \{1,2\}^n,\ F_{\tilde {w}}V_1\subset F_wV_1,\ F_{w'}V_1\cap\right. \left. F_wV_1\neq \emptyset \right \}$, since otherwise ${\rm d}(p,q)>\frac {\sqrt {3}}{2}\cdot \rho ^{n+1}\geq \rho ^{n+2}$.

In the rest of this section, we consider a class of resistance forms generated by decimation. For convenience of readers, we recall the general definition of resistance forms in the following. See [22] for more details.

Definition 3.1 Let $X$ be a set, and $l(X)$ be the space of all real-valued functions on $X$. A pair $(\mathcal {E},\mathcal {F})$ is called a (non-degenerate) resistance form on $X$ if it satisfies the following conditions:

1. (RF1) $\mathcal {F}$ is a linear subspace of $l(X)$ containing constants and $\mathcal {E}$ is a nonnegative symmetric quadratic form on $\mathcal {F}$; $\mathcal {E}(f):=\mathcal {E}(f,f)=0$ if and only if $f$ is constant on X.

2. (RF2) Let ‘$\sim$’ be an equivalence relation on $\mathcal {F}$ defined by $f\sim g$ if and only if $f-g$ is constant on X. Then $(\mathcal {F}/\sim, \mathcal {E})$ is a Hilbert space.

3. (RF3) For any finite subset $V\subset X$ and any $u\in l(V)$, there exists a function $f\in \mathcal {F}$ such that $f|_V=u$.

4. (RF4) For any distinct $p,q\in X$, $R(p,q):=\sup \{\frac {|f(p)-f(q)|^2}{\mathcal {E}(f)}:f\in \mathcal {F},\mathcal {E}(f)>0\}$ is finite.

5. (RF5) If $f\in \mathcal {F}$, then $\bar {f}={ \min \{\max \{f,0\}, 1\}}\in \mathcal {F}$ and $\mathcal {E}(\bar {f})\leq \mathcal {E}(f)$.

Sometimes, we write $\mathcal {F}=Dom(\mathcal {E})$, and abbreviate $(\mathcal {E}, \mathcal {F})$ to $\mathcal {E}$ when no confusion occurs. It is well-known ([22]) that $R(p,q)$ defined in (RF3) is a metric on $X$, named the effective resistance metric.

On the finite set $V_0$, a resistance form $\mathcal {D}$ always has the form

(3.1)\begin{equation} \mathcal{D}(f,g)=\frac 1 2\sum_{i,j} a_{i,j}\left(f(q_i)-f(q_j)\right)\left(g(q_i)-g(q_j)\right),\quad \forall f,g\in l(V_0), \end{equation}

where $a_{i,i}=0$ and the $3\times 3$ matrix $(a_{i,j})$ is positive, symmetric and irreducible. For convenience, we write $\mathcal {M}$ for the collection of all resistance forms on $V_0$. We view $\mathcal {M}$ as a subset of $\mathbb {R}^3$, which is not closed with the induced topology.

Given a resistance form $\mathcal {D}$, we define a resistance form on $V_1$ associated with $\mathcal {D}$ in a self-similar manner, respecting the infinite IFS $\{F_w\}_{w\in W_1}$.

Definition 3.2 For $r>0$, $\mathcal {D}\in \mathcal {M}$, we define $\Psi _r\mathcal {D}$ as

\[ \Psi_r\mathcal{D}(f,g)=\sum_{w\in W_1}r^{-|w|+1}\mathcal{D}(f\circ F_w,g\circ F_w), \]

with $Dom(\Psi _r\mathcal {D})=\{f\in l(V_1): \Psi _r\mathcal {D}(f)<\infty \}$.

It is not hard to show that $(\Psi _\lambda \mathcal {D},Dom(\Psi _r\mathcal {D}) )$ is a resistance form on $V_1$. However, to get a good resistance form, we need to restrict the range of $r$.

Proposition 3.3 Let $\mathcal {D}\in \mathcal {M}$ and $r<1$, then $Dom(\Psi _r\mathcal {D})\subset C(\bar {V}_1)$ by a natural identification. In addition, if $\rho < r<1$, then $(\Psi _r\mathcal {D},Dom(\Psi _r\mathcal {D}) )$ extends to a resistance form on $\bar {V}_1$, with the associated resistance metric $R(p,q)$ satisfying the estimate

(3.2)\begin{equation} R(p,q)\leq \frac{4}{r^3(1-r)}R_0^*(\mathcal{D})\,{\rm d}(p,q)^{\frac{\log r}{\log \rho}}, \quad \forall p,q\in \bar{V}_1, \end{equation}

where $R_0^*(\mathcal {D})=\max _{p,q\in V_0}R_0(p,q)$ with $R_0$ being the resistance metric on $V_0$ associated with $\mathcal {D}$.

Proof. By scaling, $R(p,q)\leq R_0^*(\mathcal {D})r^{n-1}$ for any distinct $p,q\in F_wV_0$ with $w\in W_{1,n}$ and $n\geq 1$. For $w\in \{1,2\}^n$, write $[w]_l=w_1w_2\cdots w_l$ and $p_l=F_{[w]_l}(q_0)$, with $0\leq l\leq n$, then

\[ R(p_i,p_j)\leq \sum_{l=i}^{j-1} R(p_l,p_{l+1})\leq R_0^*(\mathcal{D})\sum_{l=i}^{j-1} r^l< R_0^*(\mathcal{D})\frac{r^i}{1-r}, \quad \forall 0\leq i< j\leq n. \]

In particular, this implies that $R(p,q)\leq \frac {2r^n}{1-r}R_0^*(\mathcal {D})$ for any $p,q\in F_wV_1$ and $w\in \{1,2\}^n$. Now, if $p,q\in V_1$ and ${\rm d}(p,q)<\rho ^{n+2}$, then by the remark before definition 3.1, there exist $w,w'\in \{1,2\}^n$ such that $p\in F_{w}V_1, q\in F_{w'}V_1$, and $F_{w}V_1\cap F_{w'}V_1\neq \emptyset$, which implies that $R(p,q)\leq \frac {4r^n}{1-r}R_0^*(\mathcal {D})$. As a consequence, we have

(3.3)\begin{equation} R(p,q)\leq \frac{4}{r^3(1-r)}R_0^*(\mathcal{D})\,{\rm d}(p,q)^{\frac{\log r}{\log \rho}}, \quad \forall p,q\in V_1. \end{equation}

On the other hand, for any $f\in Dom(\Psi _r\mathcal {D})$, by (RF4), we immediately have

(3.4)\begin{equation} |f(p)-f(q)|\leq \left(R(p,q)\Psi_r\mathcal{D}(f)\right)^{1/2}. \end{equation}

Combining (3.3) and (3.4), we then get $Dom(\Psi _r\mathcal {D})\subset C(\bar {V}_1)$ by a natural identification.

To show the second assertion, we let $(X,R)$ be the completion of $(V_1,R)$, and recall [22, theorem 2.3.10] to get that $(\Psi _\lambda \mathcal {D},Dom(\Psi _r\mathcal {D}) )$ extends to be a resistance form on $X$. It suffices to show that the identity map $Id:V_1\to V_1$ extends to an homeomorphism from $(\bar {V}_1,d)$ to $(X,R)$ under the assumption $\rho < r<1$. First, by (3.3), $Id$ is continuous from $(V_1,d)$ to $(V_1,R)$. Next, let $f\in l(V_1)$ be a restriction of a linear function on $\mathbb {R}^2$. We have

\begin{align*} \Psi_r\mathcal{D}(f)& =\sum_{w\in W_1}r^{-|w|+1}\mathcal{D}(f\circ F_w)=\sum_{n=1}^\infty\sum_{w\in W_{1,n}}r^{{-}n+1}\mathcal{D}(f\circ F_w)\\ & =\sum_{n=1}^\infty \#W_{1,n}r^{{-}n+1}\rho^{2(n+1)}\mathcal{D}(f|_{V_0}), \end{align*}

where the last equality follows from the fact that $f$ is linear. Since $\#W_{1,n}\asymp \rho ^{-n}$, we have $\Psi _r\mathcal {D}(f)<\infty$ when $r>\rho$, so that $f\in Dom(\Psi _r\mathcal {D})$. Noticing that for any points $p\neq q\in \bar {V}_1$, we can find a linear function $f$ such that $f(p)\neq f(q)$, we have $Id$ is injective. Finally, due to the fact that $(\bar {V}_1,d)$ is a compact Hausdorff space and $(X,R)$ is the completion of $(V_1,R)$, we then have that $Id$ is an homeomorphism from $(\bar {V}_1,d)$ to $(X,R)$. This implies that $(\Psi _r\mathcal {D},Dom(\Psi _r\mathcal {D}))$ is a resistance form on $\bar {V}_1$, and (3.2) follows immediately from (3.3).

Remark The restriction $\rho < r<1$ is sharp. If $r\leq \rho$, there is no $f\in Dom(\Psi _r\mathcal {D})$ such that $f(q_1)=0$ and $f(q_2)=1$. In fact, for any $f\in C(\bar {V}_1)$ with $f(q_1)=0$ and $f(q_2)=1$, by the remark before definition 3.1, the total energy of $f$ on the union of the cells $F_wV_0,w\in W_{1,n}\cup W_{1,n+1}$ (noticing that this union will induce a connected subgraph in $(V_1,\sim )$, and the resistance between $F_1^nq_0$ and $F_2^nq_0$ is about $r^n\rho ^{-n}$) will be bounded away from $0$ as $n\to \infty$.

4. A renormalization map

In proposition 3.3, we have shown that $\Psi _r\mathcal {D}$ extends to be a resistance form on $\bar {V}_1$ when $\rho < r<1$. It is natural to trace it back to $V_0$, noticing that $V_0\subset \bar V _1$.

Definition 4.1 Let $(\mathcal {D}_1,\mathcal {F}_1)$ be a resistance form on $\bar V_1$, we write

\[ [\mathcal{D}_1]_{V_0}(u)=\inf\{\mathcal{D}_1(f):f|_{V_0}=u, f\in \mathcal{F}_1\}, \quad\forall u\in l(V_0). \]

Note that by a standard electric network theory, there exists a unique function $f$ so that $\mathcal {D}_1(f)$ attains the infimum above; also $[\mathcal {D}_1]_{V_0}$ induces a resistance form on $V_0$ by defining

\[ [\mathcal{D}_1]_{V_0}(u,v):=\frac 14 \left([\mathcal{D}_1]_{V_0}(u+v)-[\mathcal{D}_1]_{V_0}(u-v)\right). \]

For $\rho < r<1$ and $\mathcal {D}\in \mathcal {M}$, we define $\mathcal {R}_r\mathcal {D}=[\Psi _r\mathcal {D}]_{V_0}$, and call $\mathcal {R}_r$ the renormalization map. Sometimes, we also write $\mathcal {R}(r,\mathcal {D}):=\mathcal {R}_r(\mathcal {D})$.

The main purpose of this section is to show the continuity of the map $\mathcal {R}(r,\mathcal {D})$.

Theorem 4.2 The map $\mathcal {R}(r,\mathcal {D})$ is jointly continuous from $(\rho,1)\times \mathcal {M}$ to $\mathcal {M}$.

To prove theorem 4.2, we need a study on the regularity of the resistance form $\Psi _r\mathcal {D}$.

Proposition 4.3 Let $\mathcal {D}\in \mathcal {M}$ and $\rho < r_1< r_2<1$. Then

1. (a) $Dom(\Psi _{r_1}\mathcal {D})$ depends only on $r_1$, and we have $Dom(\Psi _{r_1}\mathcal {D})\subset Dom(\Psi _{r_2}\mathcal {D})$.

2. (b) $Dom(\Psi _{r_1}\mathcal {D})$ is dense in $Dom(\Psi _{r_2}\mathcal {D})$ in the sense that for any $f\in Dom(\Psi _{r_2}\mathcal {D})$ and $\varepsilon >0$, there exists $g\in Dom(\Psi _{r_1}\mathcal {D})$ such that

   \[ \Psi_{r_2}\mathcal{D}(f-g)<\varepsilon,\text{ and }f|_{V_0}=g|_{V_0}. \]
   Moreover, $Dom(\Psi _{r_1}\mathcal {D})$ is dense in $C(V_1)$ so that the resistance form is regular.

Proof. (a) is obvious since all $\mathcal {D}\in \mathcal {M}$ are comparable up to multiplicative constants, we only need to prove (b). Let $f\in Dom(\Psi _{r_2}\mathcal {D})$, and choose $n$ large enough so that

(4.1)\begin{equation} \sum_{l=n}^\infty \sum_{w\in W_{1,l}}r_2^{{-}l+1}\mathcal{D}(f\circ F_w)<\varepsilon. \end{equation}

For convenience, we rename the vertices $\{F_wq_0\}_{w\in W_{1,n}}$ to be $\{p_i\}_{i=1}^{N}$ with $N=\# W_{1,n}$, so that for each $i$, $p_i$ is on the left of $p_{i+1}$. Then, noticing that the effective resistance between $q_1$ and $p_1$ (symmetrically, $q_2$ and $p_N$) is bounded above by a multiple of $r^{-n}$, by (RF4), it is not hard to see

\begin{align*} & r_2^{{-}n}\left(\sum_{i=1}^{N-1}\left(f(p_i)-f(p_{i+1})\right)^2+\left(f(q_1)-f(p_{1})\right)^2+\left(f(q_2)-f(p_N)\right)^2\right)\\ & \leq c_1\left(\sum_{l=n}^\infty \sum_{w\in W_{1,l}}r_2^{{-}l+1}\mathcal{D}(f\circ F_w)\right)< c_1\varepsilon, \end{align*}

where $c_1$ is a constant depending on $\mathcal {D}$ and $r_2$, but not on $n$.

Write $x_i$ for the $x$-coordinate of $p_i$, so we have $0< x_1< x_2<\cdots < x_N<1$. We introduce a piecewise linear function $u$ on $\mathbb {R}^2$ such that

1. 1. $u(x,y)$ depends only on $x$;

2. 2. $u(q_1)=f(q_1)$, $u(q_2)=f(q_2)$, and $u(p_i)=f(p_i)$, $1\leq i\leq N$;

3. 3. $u(x,0)$ is linear on each interval $(0,x_1)$, $(x_N,1)$ and $(x_i,x_{i+1})$, $1\leq i\leq N-1$.

We define $g\in l(V_1)$ as

\[ g(p)=\begin{cases} f(p),\text{ if }p\in \bigcup_{l=1}^{n-1}\bigcup_{w\in W_{1,l}}\{F_wq_0\},\\ u(p),\text{ if }p\in \bigcup_{l=n}^{\infty}\bigcup_{w\in W_{1,l}}\{F_wq_0\}. \end{cases} \]

By a similar estimate to that applied in the proof of proposition 3.3, one can check that $g\in Dom(\Psi _{r_1}\mathcal {D})$, and

\begin{align*} & \sum_{l=n}^\infty \sum_{w\in W_{1,l}}r_2^{{-}l+1}\mathcal{D}(g\circ F_w)\\ & \quad \leq c_2r_2^{{-}n}\left(\sum_{i=1}^{N-1}\left(f(p_i)-f(p_{i+1})\right)^2+\left(f(q_1)-f(p_{1})\right)^2+\left(f(q_2)-f(p_N)\right)^2\right), \end{align*}

where $c_2$ depends only on $\mathcal {D}$ and $r_2$. So we have $\Psi _{r_2}\mathcal {D}(f-g)\leq c_3\varepsilon$ for some constant $c_3$. Since $\varepsilon$ can be arbitrarily small, we have that $Dom(\Psi _{r_1}\mathcal {D})$ is dense in $Dom(\Psi _{r_2}\mathcal {D})$. Finally, the claim that $Dom(\Psi _{r_1}\mathcal {D})$ is dense in $C(\bar {V}_1)$ follows from the same argument.

Proof of theorem 4.2. Let $r_n\to r\in (\rho,1)$ and $\mathcal {D}_n\to \mathcal {D}\in \mathcal {M}$. Also, let $u\in l(V_0)$. First, we show that

(4.2)\begin{equation} \limsup_{n\to\infty}\mathcal{R}(r_n,\mathcal{D}_n)(u)\leq \mathcal{R}(r,\mathcal{D})(u). \end{equation}

We define $f$ to be the unique function in $Dom(\Psi _r\mathcal {D})$ such that $f|_{V_0}=u$ and

\[ \mathcal{R}(r,\mathcal{D})(u)=\Psi_r\mathcal{D}(f). \]

By proposition 4.3, for any $\varepsilon >0$, there is $f_\varepsilon$ such that $f_\varepsilon |_{V_0}=u$, $f_\varepsilon \in Dom(\Psi _{r_n}\mathcal {D}_n)$ for any $n\geq 1$, and

\[ \Psi_r\mathcal{D}(f_\varepsilon)\leq\Psi_r\mathcal{D}(f)+\varepsilon. \]

As a consequence, we have

\[ \limsup_{n\to\infty}\mathcal{R}(r_n,\mathcal{D}_n)(u)\leq \lim_{n\to\infty}\Psi_{r_n}\mathcal{D}_n(f_\varepsilon)=\Psi_r\mathcal{D}(f_\varepsilon)\leq \mathcal{R}(r,\mathcal{D})(u)+\varepsilon, \]

where the equality is due to the dominated convergence theorem. Since $\varepsilon$ can be arbitrarily chosen, we get (4.2).

Next, for each $n$, let $f_n$ be the unique function in $Dom(\Psi _{r_n}\mathcal {D}_n)$ such that $f_n|_{V_0}=u$ and

\[ \mathcal{R}(r_n,\mathcal{D}_n)(u)=\Psi_{r_n}\mathcal{D}_n(f_n). \]

Then $\{f_n\}_{n\geq 1}$ is uniformly bounded by the Markov property (RF5). In addition, $\Psi _{r_n}\mathcal {D}_n(f_n)\leq \mathcal {R}(r_*,\mathcal {D}_n)(u)$ with $r_*=\inf _{n\geq 1} r_n$, so $\{\Psi _{r_n}\mathcal {D}_n(f_n)\}_{n\geq 1}$ is a bounded sequence. By estimates (3.2) and (3.4), we have

\[ |f_n(p)-f_n(q)|\leq c\left({\rm d}(p,q)^{\frac{\log r^*}{\log \rho}}\sup_{n\geq 1}\Psi_{r_n}\mathcal{D}_n(f_n)\right)^{1/2},\quad\forall n\geq 1,\forall p,q\in \bar{V}_1, \]

where $r^*=\sup _{n\geq 1}r_n$ and $c^2=\sup _{n\geq 1}\{\frac {4}{{r_n^{3}}(1-r_n)}R_0^*(\mathcal {D}_n)\}$, and so $\{f_n\}_{n\geq 1}$ is also equicontinuous. Thus, there is a subsequence $\{f_{n_k}\}_{k\geq 1}$ such that $f_{n_k}$ converges uniformly to a function $f\in C(\bar {V}_1)$. Clearly, $f$ is an extension of $u$. By Fatou's lemma,

\[ \mathcal{R}(r,\mathcal{D})(u)\leq\Psi_r\mathcal{D}(f)\leq \liminf_{k\to\infty}\Psi_{r_{n_k}}\mathcal{D}_{n_k}(f_{n_k})=\liminf_{k\to\infty}\mathcal{R}(r_{n_k},\mathcal{D}_{n_k})(u). \]

Combining this with (4.2), we see that

\[ \mathcal{R}(r,\mathcal{D})(u)=\lim_{k\to\infty}\mathcal{R}(r_{n_k},\mathcal{D}_{n_k})(u). \]

Since the argument works for any sequence $(r'_n,\mathcal {D}'_n)\to (r,\mathcal {D})$, we actually have

\[ \mathcal{R}(r,\mathcal{D})(u)=\lim_{n\to\infty}\mathcal{R}(r_n,\mathcal{D}_n)(u). \]

The theorem follows immediately since $u$ can be any function in $l(V_0)$.

5. A fixed point problem

In this section, analogous to the case of PCF self-similar sets (see [22, 34]), we consider the renormalization equation

(5.1)\begin{equation} \mathcal{R}_r\mathcal{D}=\lambda\mathcal{D}, \end{equation}

with $\lambda >0$. We will prove that for each given $\rho < r<1$, there always exists a positive $\lambda$ such that (5.1) has a solution $\mathcal {D}$ in $\mathcal {M}$. Nevertheless, this is not enough for the construction of a satisfying resistance form on $\mathcal {G}$ for our later purposes. In order that cells of same size are assigned with the same renormalization factors, we will in addition require $\lambda =r^2$, i.e.

(5.2)\begin{equation} \mathcal{R}_r\mathcal{D}=r^2\mathcal{D}. \end{equation}

The existence and uniqueness of such a solution is the main purpose of this section.

It is natural to consider resistance forms on $\mathcal {G}$ that are symmetric with respect to the reflection symmetry of $\mathcal {G}$. So we look at the resistance forms on $V_0$ which are symmetric in the sense that $a_{0,1}=a_{0,2}$ in (3.1). We denote $\mathcal {M}_S$ for the set of all such resistance forms.

Theorem 5.1

1. (a). For each $\rho < r<1$, there exists a unique pair of $\lambda (r)$ and $\mathcal {D}(r)\in \mathcal {M}$ (up to constants) satisfying (5.1), where $\lambda (r)$ is decreasing and continuous in $r$, and $\mathcal {D}(r)$ is in $\mathcal {M}_S$.

2. (b). There exists a unique $\rho < r<1$ such that (5.2) has a unique (up to constants) solution $\mathcal {D}\in \mathcal {M}$.

We will first prove that for each $r$, there exist a unique $\lambda (r)$ such that (5.1) has a solution $\mathcal {D}(r)$ in $\mathcal {M}_S$, then prove that $\mathcal {D}(r)$ is indeed a unique solution (up to constants) in $\mathcal {M}$. The existence and uniqueness of a solution to (5.2) will follow from the properties of $\lambda (r)$. We divide these into two subsections.

5.1 The existence of a symmetric solution

We begin with some simple observations.

Lemma 5.2 Let $\rho < r<1$ be fixed, and suppose that there is a solution to (5.1). Then the constant $\lambda$ depends only on $r$.

Proof. This follows from a standard argument like the finite graph case [29]. Suppose that $\mathcal {D},\mathcal {D}'$ are two solutions to (5.1) with $\lambda,\lambda '$ being the corresponding constant. Let $u\in l(V_0)$ so that $\frac {\mathcal {D}'(u)}{\mathcal {D}(u)}=\sup _{v\neq constants}\frac {\mathcal {D}'(v)}{\mathcal {D}(v)}:=\theta$, and let $f$ be the harmonic extension of $u$ with respect to $\Psi _r\mathcal {D}$. Then

\[ \lambda'\mathcal{D}'(u)=\mathcal{R}_r\mathcal{D}'(u)\leq \Psi_r\mathcal{D}'(f)\leq \theta\Psi_r\mathcal{D}(f)=\theta\mathcal{R}_r\mathcal{D}(u)=\theta \lambda\mathcal{D}(u). \]

This implies that $\lambda '\leq \lambda$. A same argument also shows that $\lambda \leq \lambda '$.

Inspired by lemma 5.2, we can view the constant $\lambda$ in (5.1) as a function of $r$. On the other hand, the problem of solvability of (5.1) can be transferred to a fixed point problem.

Definition 5.3

1. (a) Define

   \begin{align*} & \tilde{\mathcal{M}}_S=\left\{\mathcal{D}\in \mathcal{M}:\mathcal{D}(f)=a\left(f(q_0)-f(q_1)\right)^2+a\left(f(q_0)-f(q_2)\right)^2\right.\\ & \left.\quad +\,(1-a)\left(f(q_1)-f(q_2)\right)^2, 0< a\leq 1\right\}, \end{align*}
   and for $0< s\leq 1$,
   \begin{align*} & \tilde{\mathcal{M}}_S^{[s,1]}=\left\{\mathcal{D}\in \mathcal{M}:\mathcal{D}(f)=a\left(f(q_0)-f(q_1)\right)^2+a\left(f(q_0)-f(q_2)\right)^2\right.\\ & \left.\quad +\,(1-a)\left(f(q_1)-f(q_2)\right)^2, s\leq a\leq 1\right\}. \end{align*}

2. (b) For each $\mathcal {D}\in \mathcal {M}_S$, there is a unique constant $c$ such that $c\mathcal {D}\in \tilde {\mathcal {M}}_S$, and we denote the resulting form $T\mathcal {D}$. We define $\tilde {\mathcal {R}}_r:\mathcal {M}_S\to \tilde {\mathcal {M}}_S$ to be the map given by $\tilde {\mathcal {R}}_r=T\circ \mathcal {R}_r$. As before, we write $\tilde {\mathcal {R}}(r,\mathcal {D})=\tilde {\mathcal {R}}_r(\mathcal {D})$.

The following lemma will play an essential role.

Lemma 5.4 For $\rho < r_0< r_1<1$, there exists $0< s\leq 1$ such that $\tilde{\mathcal {R}}:[r_0,r_1]\times \mathcal {M}_S\to \tilde {\mathcal {M}}_S^{[s,1]}$.

Proof. Let $\mathcal {D}\in \mathcal {M}_S$, $r_0\leq r\leq r_1$ and $R$ be the resistance metric on $V_1$ associated with $\Psi _r\mathcal {D}$. For convenience, we write $\mathcal {D}(f)=a(f(q_0)-f(q_1))^2+a(f(q_0)-f(q_2))^2+b(f(q_1)-f(q_2))^2$, with $a>0,b\geq 0$.

First, by the series law for resistances, for any $f\in l(V_1)$, we have

\[ \Psi_{r}\mathcal{D}(f)\geq \sum_{n=0}^\infty a{r}^{{-}n}\left(f(F_1^nq_0)-f(F_1^{n+1}q_0)\right)^2\geq a(1-r)\left(f(q_0)-f(q_1)\right)^2, \]

so we have $R(q_0,q_1)\leq \frac {1}{a(1-r)}\leq \frac {1}{a(1-r_1)}$.

Next, let $f$ be the linear function on $\mathbb {R}^2$ such that $f(q_1)=0,f(q_2)=1$ and $f(q_0)=\frac {1}{2}$, so $f$ only depends on the $x$-coordinate. We introduce a ‘horizontal’ edge relation ‘$\sim _h$’ on $V_1$: for distinct $p,q\in V_1$, denote

\[ p\sim_h q\text{ if there exists }w\in W_1\text{ so that }p,q\in \{F_wq_1,F_wq_2\}. \]

For each $p\in V_1$, we write

\[ [p]_h=\{q\in V_1: q\sim_h p, \text{ or } q\sim_h q', q'\sim_h p\text{ for some } q'\in V_1\}. \]

Then we modify $f$ on $V_1$ into a function $g\in l(V_1)$ as

\[ g(p)=\frac{\sum_{q\in[p]_h}f(q)}{\#[p]_h},\quad\forall p\in V_1. \]

By doing this we have

1. 1. $g(p)=g(q)$ if $[p]_h=[q]_h$;

2. 2. $|g(p)-g(q)|\leq c_1\rho ^{n}$ if $p,q\in F_wV_0$ with $w\in W_{1,n}$.

Thus, we have

\begin{align*} \Psi_r\mathcal{D}(g)& =\sum_{l=1}^\infty\sum_{w\in W_{1,l}}r^{{-}l+1}\mathcal{D}(g\circ F_w)\\ & \leq 2c_1^2 a\sum_{l=1}^\infty r^{{-}l+1}\rho^{2l}\#W_{1,l}\\ & \leq c_2a\sum_{l=1}^\infty r^{{-}l}\rho^{l}=\frac{c_2\rho}{r-\rho}a\leq \frac{c_2\rho}{r_0-\rho}a, \end{align*}

where we use the estimate $\#W_{1,l}\asymp \rho ^{-l}$. Thus, $g$ extends to $g\in C(\bar {V}_1)$ by proposition 3.3, and it is direct to check that $g|_{V_0}=f|_{V_0}$. As a consequence, we get $R(q_1,q_2)\geq \frac {r_0-\rho }{c_2\rho }a^{-1}$.

Due to the above two estimates, there exists $c_3>0$ independent of $\mathcal {D}$ such that

\[ \frac{R(q_0,q_1)}{R(q_1,q_2)}\leq c_3. \]

Then an effective resistance calculation gives that $\tilde {\mathcal {R}}(r,\mathcal {D})\in \tilde {\mathcal {M}}_S^{[\frac {1}{2c_3},1]}$. The lemma follows.

By using lemmas 5.2 and 5.4 and theorem 4.2, we can easily prove the following proposition.

Proposition 5.5 Let $\rho < r<1$, there always exists a solution to (5.1) in $\mathcal {M}_S$, with $\lambda$ uniquely determined by $r$. In addition, regarding $\lambda$ as a function of $r$, $\lambda (r)$ is decreasing and continuous in $r$.

Proof. First, we have $\tilde {\mathcal {R}}_r:\tilde {\mathcal {M}}_S^{[s,1]}\to \tilde {\mathcal {M}}_S^{[s,1]}$ for some $s>0$ by lemma 5.4. Together with theorem 4.2, the existence of a fixed point of $\tilde {\mathcal {R}}_r$ is then an immediate consequence.

Next, let $r_1< r_2$, and assume that $\mathcal {R}_{r_1}\mathcal {D}_1=\lambda (r_1)\mathcal {D}_1$ and $\mathcal {R}_{r_2}\mathcal {D}_2=\lambda (r_2)\mathcal {D}_2$. Let $u\in l(V_0)$ so that $\frac {\mathcal {D}_2(u)}{\mathcal {D}_1(u)}=\sup _{v\neq constants}\frac {\mathcal {D}_2(v)}{\mathcal {D}_1(v)}:=\theta$, and let $f$ be the harmonic extension of $u$ with respect to $\Psi _{r_1}\mathcal {D}_1$, then we have $\lambda (r_2)\mathcal {D}_2(u)\leq \Psi _{r_2}\mathcal {D}_2(f)\leq \theta \Psi _{r_1}\mathcal {D}_1(f)=\theta \lambda (r_1)\mathcal {D}_1(u)$. So we get $\lambda (r_2){\leq } \lambda (r_1)$.

Finally, let $r_n\to r$, and let $\mathcal {D}_n\in \tilde {\mathcal {M}}_S$ be a sequence of solutions to $\mathcal {R}_{r_n}\mathcal {D}_n=\lambda (r_n)\mathcal {D}_n$. Clearly, we have $\rho <\inf _{n\geq 1}r_n<\sup _{n\geq 1}r_n<1$, so $\{\mathcal {D}_n\}_{n\geq 1}\subset \tilde {\mathcal {M}}_S^{[s,1]}$ for some $s>0$ by lemma 5.4. Thus, there exists a subsequence $\{n_k\}_{k\geq 1}$ such that $\mathcal {D}_{n_k}$ converges to some $\mathcal {D}\in \tilde {\mathcal {M}}_S$ and $\lambda (r_{n_k})$ converges. By theorem 4.2, we conclude that $\mathcal {R}_r\mathcal {D}=(\lim _{k\to \infty }\lambda ({r_{n_k}}))\mathcal {D}$. So $\lambda (r)=\lim _{k\to \infty }\lambda (r_{n_k})$. Since the argument works for any sequence $r_n\rightarrow r$, $\lambda (r)$ is continuous in $r$.

We have an easy estimate of $\lambda (r)$.

Lemma 5.6 For $\rho < r<1$, we have $\left (\frac {1}{1-r}-\frac {r}{2+2r+2r^2}\right )^{-1}\leq \lambda (r)\leq \frac {2}{2+r}$.

Proof. We consider a function $u\in l(V_0)$ with $u(q_0)=0$ and $u(q_1)=u(q_2)=1$. Without loss of generality, we assume the solution $\mathcal {D}\in \mathcal {M}_S$ has $\mathcal {D}(u)=2$. To get an upper bound for $\lambda (r)$, we construct an extension $f\in l(V_1)$ of $u$ by setting

\[ f(p)=\begin{cases}0, & \text{ if }p=q_0,\\ \frac{2}{2+r}, & \text{ if }p\in \{F_1q_0,F_2q_0\},\\ 1, & \text{ if }p\in F_1 V_1\cup F_2 V_1\setminus\{F_1 q_0, F_2 q_0\}. \end{cases} \]

Then the upper bound follows easily from the following estimate:

\[ \mathcal{R}_\lambda\mathcal{D}(u)\leq \Psi_r\mathcal{D}(f)=\left(\left(\frac{2}{2+r}\right)^2+2r^{{-}1}\left(1-\frac{2}{2+r}\right)^2\right)\mathcal{D} (u)=\frac{2}{2+r}\mathcal{D}(u). \]

To get the lower bound, we look at a subgraph in $(V_1,\sim )$, whose vertices are $\{F_i^lq_0\}_{i,l}$ with $i\in \{1,2\}$ and $l\geq 0$, together with

\begin{align*} & p_{i,0}=F_iq_0,\quad p_{i,1}=F_iF_jq_0,\quad p_{i,2}=F_iF_jF_iq_0,\\ & p_{i,3}=F_iF_jF_i^2q_0, \quad p_{i,4}=F_i^2F_jq_0,\quad p_{i,5}=F_i^2q_0, \end{align*}

with $i,j\in \{1,2\}$ and $j\neq i$, and edges inherited from $(V_1,\sim )$ (with horizontal edges deleted), see figure 5. Let $f\in l(V_1)$ be the harmonic extension of $u$, denote $c_l=r^{-l-1}$ for $l\in \{0,1,2\}$, and $c_l=r^{l-6}$ for $l\in \{3,4\}$, then the lower bound follows from the estimate that

\begin{align*} \mathcal{R}_r\mathcal{D}(u)=\Psi_r\mathcal{D}(f)& \geq \sum_{i=1,2}\left(\sum_{l=0}^\infty r^{{-}l}\left(f(F_i^lq_0)-f(F_i^{l+1}q_0)\right)^2+\sum_{l=0}^4 c_l\left(f(p_{i,l})-f(p_{i,l+1})\right)^2\right)\\ & \geq 2\left(\frac{1}{1-r}-\frac{r}{2+2r+2r^2}\right)^{{-}1}=\left(\frac{1}{1-r}-\frac{r}{2+2r+2r^2}\right)^{{-}1}\mathcal{D}(u), \end{align*}

where the last inequality can be done by an easy computation of the effective resistances on the subgraph (for $i\in \{1,2\}$, firstly connecting the resistors along $\{p_{i,l}\}_{l=0}^5$ in series, secondly connecting the resulting effective resistor with the resistor between $p_{i,0}$ and $p_{i,5}$ in parallel; lastly connecting all the resistors in series).

Fig. 5. The subgraph of $(V_1,\sim )$ constructed in lemma 5.6.

Using proposition 5.5 and lemma 5.6, we arrive at the main result of this subsection, concerning the solvability of (5.2).

Theorem 5.7 There exists a unique $\rho < r<1$ such that (5.2) has a solution $\mathcal {D}\in \mathcal {M}_S$.

Proof. By proposition 5.5, we see that there is a continuous function $\lambda (r)$ so that $\mathcal {R}_\lambda (\mathcal {D})=\lambda (r)\mathcal {D}$ has a solution. Noticing that

\[ \lambda(\rho)\geq\left(\frac{1}{1-\rho}-\frac{\rho}{2+2\rho+2\rho^2}\right)^{{-}1}>1-\rho=\rho^2,\text{ and } \lambda(1)\leq \frac{2}{3}<1, \]

there exists $\rho < r<1$ such that $\lambda (r)=r^2$ by lemma 5.6. The uniqueness follows from the fact that $\lambda (r)$ is decreasing in $r$, while $r^2$ is strictly increasing.

Remark We can see the uniqueness of $r$ from another point of view. Let $\theta =\frac {\log r}{\log \rho }$, we will see in § 7 that $\theta +d_H$ is the walk dimension of the resulting diffusion process on the metric measure space $(\mathcal {G},d,\mu _H)$, whose uniqueness is shown in [17, theorem 4.6] under some weak conditions on the heat kernel.

5.2 The uniqueness

In this subsection, we consider the uniqueness of the solution to (5.1) or (5.2). The proof is inspired by Sabot's work [34].

Theorem 5.8 Let $\rho < r<1$ and $\mathcal {D}\in \mathcal {M}_S$ be a symmetric solution to (5.1). Then $\mathcal {D}$ is the unique solution in $\mathcal {M}$ to (5.1).

For fixed $\rho < r<1$ and $\mathcal {D}\in \mathcal {M}_S$ satisfying (5.1), for convenience, we always write

1. 1. $h_s$ for the harmonic function with $h_s(q_0)=0$, $h_s(q_1)=h_s(q_2)=1$, and denote $E_s=\{f\in l(V_0): f(q_0)=0,f(q_1)=f(q_2)=c, c\in \mathbb {R}\}$;

2. 2. $h_a$ for the harmonic function with $h_a(q_0)=0$, $h_a(q_1)=-h_a(q_2)=1$, and denote $E_a=\{f\in l(V_0): f(q_0)=0,f(q_1)=-f(q_2)=c, c\in \mathbb {R}\}$.

Both $h_s,h_a$ are harmonic with respect to $\Psi _r\mathcal {D}$ on $\bar {V}_1\setminus V_0$, i.e. $\Psi _r\mathcal {D}(h_s,f)=\Psi _r\mathcal {D}(h_a,f)=0$ for any $f\in Dom(\Psi _r\mathcal {D})$ such that $f|_{V_0}=0$.

Lemma 5.9 For $r$, $\mathcal {D}$ as above, we have

\[ h_s(F_1q_0)=h_s(F_2q_0)=\lambda(r),\quad h_a(F_1q_0)={-}h_a(F_2q_0)=\eta, \]

for some $|\eta |<\lambda (r)$.

Proof. For convenience, we write $\mathcal {D}$ in the form $\mathcal {D}(f)=a(f(q_0)-f(q_1))^2+a(f(q_0)-f(q_2))^2+b(f(q_1)-f(q_2))^2$, with $a>0,b\geq 0$.

First, let $h=1-h_s$, we have $\mathcal {R}_r\mathcal {D}(h_s,h)=-2a\lambda (r)$. On the other hand, let $f\in l(V_1)$ be defined as $f(p)=\delta _{q_0,p}$, then clearly $f\in Dom(\Psi _r\mathcal {D})$, and $f|_{V_0}=h|_{V_0}$. Since $h_s$ is harmonic,

\[ \Psi_r\mathcal{D}(h_s,h)=\Psi_r\mathcal{D}(h_s,f)={-}ah_s(F_1q_0)-ah_s(F_2q_0). \]

This shows the first assertion since $\mathcal {R}_r\mathcal {D}(h_s,h)=\Psi _r\mathcal {D}(h_s,h)$.

Next, by the symmetry of $\mathcal {D}$, there exists a number $\eta$ such that $h_a(F_1q_0)=-h_a(F_2q_0)=\eta$. We need to show that $|\eta |<\lambda (r)$. We consider the matrix $M$ such that

\[ \left(h(F_1q_0),h(F_2q_0)\right)^t=M\left(h(q_1),h(q_2)\right)^t, \]

holds for any harmonic function $h$ with $h(q_0)=0$. Due to the Perron–Frobenius theorem, it suffices to show that each entry of $M$ is positive. This can be deduced by proving the harmonic function $h_1$ with boundary value $h_1(q_1)=1,h_1(q_0)=h_1(q_2)=0$ is positive on $V_1\setminus V_0$. To see this, we assume there exists $p\in V_1\setminus V_0$ such that $h_1(p)=0$. Let $\psi _p\in Dom(\Psi _r\mathcal {D})$ be defined as $\psi _p(q)=\delta _{p,q}$, then $\Psi _r\mathcal {D}(\psi _p,h_1)=0$, so $h_1(p)$ is the weighted average of its neighbours. Thus, $h_1$ is zero on the neighbours of $p$. Repeating the argument, we see that $h_1|_{V_1}=0$. A contradiction.

Proof of theorem 5.8. Assume there is another solution $\mathcal {D}'\in \mathcal {M}$ to (5.1).

Firstly, we will show that $\mathcal {D}'$ is also symmetric. By diagonalizing $\mathcal {D}'$ with respect to $\mathcal {D}$, we have two $1$-dimensional non-constant subspaces $L_1,L_2$ of $l(V_0)$ such that

1. 1. $L_1,L_2$ are orthogonal with respect to both $\mathcal {D}$ and $\mathcal {D}'$;

2. 2. $\mathcal {D}'|_{L_1}=\kappa _1 \mathcal {D}|_{L_1}$ and $\mathcal {D}'|_{L_2}=\kappa _2\mathcal {D}|_{L_2}$, with $0<\kappa _1<\kappa _2$.

Let $u\in L_2$ and $h_u$ be the harmonic extension of $u$ with respect to $\Psi _r\mathcal {D}$. Then

\begin{align*} \lambda(r)\mathcal{D}'(u)& =\kappa_2 \lambda(r)\mathcal{D}(u)=\kappa_2\Psi_r\mathcal{D}(h_u)=\sum_{w\in W_{1}} r^{-|w|+1}\kappa_2\mathcal{D}(h_u\circ F_w)\\ & \geq \sum_{w\in W_{1}} r^{-|w|+1}\mathcal{D}'(h_u\circ F_w)=\Psi_r\mathcal{D}'(h_u)\geq \lambda(r)\mathcal{D}'(u). \end{align*}

This implies that for each $w\in W_1$, $\mathcal {D}'(h_u\circ F_w)=\kappa _2\mathcal {D}(h_u\circ F_w)$, and thus $h_u\circ F_w\in L_2+constants$. In particular, we have $h_u\circ F_0\in L_2+constants$, which means $L_2+constants$ is an invariant space under the mapping $u$ to $h_u\circ F_0$. By lemma 5.9, we see that $L_2+constants$ is either $E_s+constants$ or $E_a+constants$. Thus, we have $\mathcal {D}'\in \mathcal {M}_S$.

Secondly, from the above argument, it is not hard to see that $h_s\circ F_w\in E_s+constants$ and $h_a\circ F_w\in E_a+constants$, for any $w\in W_1$.

Lastly, arbitrarily pick a $\tilde {\mathcal {D}}\in \mathcal {M}_S$, we will prove that $\tilde {\mathcal {D}}$ must also solve (5.1). However, this will make $\tilde {\mathcal {R}}_r(\mathcal {M}_S)=\tilde {\mathcal {M}}_S$, which obviously contradicts Lemma 5.4. To achieve this purpose, let $\tilde {h}_s$ and $\tilde {h}_a$ be the harmonic functions with respect to $\Psi _r\tilde {\mathcal {D}}$, with the same boundary value on $V_0$ as $h_s,h_a$. By following the same argument as [34, lemma 5.9] due to Sabot, we can see that $\tilde {h}_s=h_s$ and $\tilde {h}_a=h_a$. For convenience of readers, we reproduce the proof here. Write $g=h_s-\tilde {h}_s$. Also, for each $w\in W_1$, let $g_{w,s}\in E_s+constants, g_{w,a}\in E_a+constants$ such that $g_w=:g\circ F_w=g_{w,s}+g_{w,a}$. Then, denoting $h_{w,s}=: h_s\circ F_w$, we can see that

\begin{align*} \Psi_r\tilde{\mathcal{D}}(g)& =\Psi_r\tilde{\mathcal{D}}(h_s,g)=\sum_{w\in W_{1}}r^{-|w|+1}\tilde{\mathcal{D}}(h_{w,s},g_w)=\sum_{w\in W_{1}}r^{-|w|+1}\tilde{\mathcal{D}}(h_{w,s},g_{w,s})\\ & =c\sum_{w\in W_{1}}r^{-|w|+1}\mathcal{D}(h_{w,s},g_{w,s})\\ & =c\sum_{w\in W_{1}}r^{-|w|+1}\mathcal{D}(h_{w,s},g_w)=c\Psi_r\mathcal{D}(h_s,g)=0, \end{align*}

for some constant $c$, with $h_{s,w}=: h_s\circ F_w$, where the first equality is due to the fact that $g|_{V_0}=0$. Thus, $g=0$ as desired. As a consequence, we can easily see that, $\tilde {\mathcal {D}}$ is a solution to (5.1), so we arrive at the desired contradiction.

Finally, theorem 5.1 immediately follows from proposition 5.5, theorems 5.7 and 5.8.

6. Construction of the Dirichlet form on $\mathcal {G}$

We will construct a resistance form on the golden ratio Sierpinski gasket $\mathcal {G}$ in this section. Let $\rho < r<1$, $\mathcal {D}$ be the unique solution to (5.2), i.e. $\mathcal {R}_r\mathcal {D}=r^2 \mathcal {D}$. We will focus on this standard form in most contents. For short, we write

\[ \theta=\frac{\log r}{\log \rho},\quad \rho_w=\prod_{n=1}^{|w|}\rho_{w_n},\quad r_w=\rho_w^{\theta}, \]

with $\rho _{0}=\rho ^2$ and $\rho _1=\rho _2=\rho$. Obviously, $\rho _w$ is the contraction ratio of $F_w$.

The following definition is similar to the construction in [22], though we use the infinite graphs at each level.

Definition 6.1

1. (a) For $m\geq 0$ and $f\in C(\bar {V}_m)$, we write $\mathcal {D}^{(m)}(f)=\sum _{w\in W_m}r_w^{-1} \mathcal {D}(f\circ F_w)$, and $\mathcal {F}^{(m)}=\{f\in C(\bar {V}_m):\mathcal {D}^{(m)}(f)<\infty \}$. In addition, for $f,g\in \mathcal {F}^{(m)}$, we define

   \[ \mathcal{D}^{(m)}(f,g)=\sum_{w\in W_m}r_w^{{-}1}\mathcal{D}(f\circ F_w,g\circ F_w). \]

2. (b) Define $\mathcal {F}=\{f\in C(\mathcal {G}):\lim _{m\to \infty }\mathcal {D}^{(m)}(f)<\infty \}$. For $f,g\in \mathcal {F}$, define

   \[ \mathcal{E}(f,g)=\lim_{m\to\infty}\mathcal{D}^{(m)}(f,g). \]

It follows from the definition of $\Psi _r\mathcal {D}$, $\mathcal {D}^{(1)}=r^{-2}\Psi _r\mathcal {D}$. The limit in (b) always exists due to fact that

\begin{align*} \mathcal{D}^{(m+1)}(f)& =\sum_{w\in W_{m+1}}r_w^{{-}1}\mathcal{D}(f\circ F_w)=\sum_{w\in W_m}r_w^{{-}1}r^{{-}2}\Psi_r\mathcal{D}(f\circ F_w)\\ & \quad \geq \sum_{w\in W_m}r_w^{{-}1}\mathcal{D}(f\circ F_w)=\mathcal{D}^{(m)}(f). \end{align*}

In the rest of this section, we will show that $(\mathcal {E},\mathcal {F})$ is a good form.

Lemma 6.2 For $m\geq 0$, $(\mathcal {D}^{(m)},\mathcal {F}^{(m)})$ is a resistance form on $\bar V_m$. In addition, let

\[ R_m(p,q)=\sup_{f\in \mathcal{F}^{(m)}}\frac{|f(p)-f(q)|^2}{\mathcal{D}^{(m)}(f)}, \]

then we have $R_n(p,q)=R_m(p,q)$ if $p,q\in \bar {V}_m$ and $n\geq m$.

Proof. (RF1) and (RF5) are trivial. We only need to verify (RF2)–(RF4). For convenience, we focus on $(\mathcal {D}^{(2)},\mathcal {F}^{(2)})$ only, while for larger $m$, the same proof works inductively. (RF2). Let $\{f_k\}_{k\geq 1}$ be a Cauchy sequence in $\mathcal {F}^{(2)}$. Then, $f_k|_{\bar {V}_1}$ converges in $\mathcal {F}^{(1)}$ to some $\tilde {f}$ in $\mathcal {F}^{(1)}$, since $(\mathcal {D}^{(1)},\mathcal {F}^{(1)})$ is a resistance form. Also, for each $w\in W_1$, $f_k\circ F_w$ converges in $\mathcal {F}^{(1)}$ to a function $\tilde {f}_w$. Now, define $f\in l(\bar V_2)$ such that $f\circ F_w=\tilde {f}_w$ and $f|_{\bar {V_1}\setminus V_1}=\tilde {f}$. We show that $f\in C(\bar V_2)$. It suffices to prove that $f$ is continuous at any point $p\in \bar {V}_1\setminus V_1$. In fact, for any $\varepsilon$, there exists $\delta$ and $N$ such that 1. for $q\in B_{\delta }(p)\cap \bar {V}_1$, we have $|f(p)-f(q)|<\varepsilon$; 2. for $w\in \bigcup _{n=N}^\infty W_{1,n}$ and $q,q'\in F_w\bar {V}_1$, we have $|f(q)-f(q')|<\varepsilon$. This follows from the fact that $\mathcal {D}^{(1)}(f\circ F_w)\leq r_w\sup _{k\geq 1}\mathcal {D}^{(2)}(f_k)$. The continuity of $f$ follows immediately. Lastly, by using Fatou's lemma, we can directly check that $f_k$ converges to $f$ in $\mathcal {F}^{(2)}$. (RF3). First, we observe that the minimal energy extension of $f\in \mathcal {F}^{(1)}$ to $l(V_2)$ is continuous by a same argument as in (RF2). Let $V$ be a finite set and $u\in l(V)$. First, we always have $f_1\in \mathcal {F}^{(1)}$ such that $f_1|_{V\cap \bar {V}_1}=u|_{V\cap \bar {V}_1}$. Then we can extend $f_1$ to be a desired function in $\mathcal {F}^{(2)}$. (RF4). Let $p,q\in \bar {V}_2$ and $f\in \mathcal {F}^{(2)}$. If $p\in \bar {V}_1$, we let $p'=p$; otherwise we choose $p'\in V_1$ so that $p,p'\in F_w\bar {V_1}$ for some $w\in W_1$, and thus

\[ \mathcal{D}^{(2)}(f)\geq r_w^{{-}1}\mathcal{D}^{(1)}(f\circ F_w)\geq c_1\left(f(p)-f(p')\right)^2, \]

for some $c_1>0$. Also, we define $q'$ in the same manner. Note that $\mathcal {D}^{(2)}(f)\geq \mathcal {D}^{(1)}(f)\geq c_2(f(p')-f(q'))^2$ for some $c_2>0$, it then follows that

\begin{align*} & \mathcal{D}^{(2)}(f)\geq \min\{c_1,c_2\}\left(\left(f(p)-f(p')\right)^2+\left(f(p')-f(q')\right)^2+\left(f(q')-f(q)\right)^2\right)\\ & \quad \geq c_3\left(f(p)-f(q)\right)^2, \end{align*}

for some $c_3>0$. (RF4) follows immediately. Thus, we have proved that $(\mathcal {D}^{(2)},\mathcal {F}^{(2)})$ is a resistance form on $\bar V_2$. The claim that $R_2(p,q)=R_1(p,q)$ for $p,q\in \bar V_1$ is obvious. The same arguments can be used inductively for $m\geq 3$.

In some situations, it is convenient to involve words in $\tilde {W}_*$.

Lemma 6.3 Let $w\in \tilde {W}_*$ and $m$ be the number of $0'$s in $w$. Then we have

\[ \mathcal{D}^{(1)}(f\circ F_w)\leq r_w\mathcal{D}^{(m+1)}(f), \]

for any $f\in \mathcal {F}^{(m+1)}$. As a consequence, there is a constant $c>0$ such that, for any $p,q\in F_w\bar {V}_1$, we have

\[ R_{m+1}(p,q)\leq c\,{\rm d}(p,q)^\theta. \]

Proof. Noticing that $\{w\tau :\tau \in W_1\}\subset {W}_{m+1}$, the first statement follows. The second statement follows from the first statement and proposition 3.3: for any $p,q\in F_w\bar {V}_1$,

\[ R_{m+1}(p,q)\leq r_w R_1(F_w^{{-}1}p, F_w^{{-}1}q)\leq cr_wd(F_w^{{-}1}p, F_w^{{-}1}q)^\theta=c\,{\rm d}(p,q)^\theta, \]

holds for some constant $c>0$, where the first inequality follows from the first statement, and the second inequality follows from proposition 3.3.

Using lemmas 6.2 and 6.3, we have the following estimate of the resistance metric.

Lemma 6.4 For $m\geq 0$ and $p,q\in \bar {V}_m$, define $\tilde {R}(p,q)=R_m(p,q)$. Then $\tilde {R}(p,q)$ is well defined on $(\bigcup _{m\geq 0}\bar {V}_m)\times (\bigcup _{m\geq 0}\bar {V}_m)$, and we have $\tilde {R}(p,q)\leq c\,{\rm d}(p,q)^\theta$ for some $c>0$.

Proof. First, we claim that there is a constant $c_1>0$ such that

\[ \tilde{R}(p,q)\leq c_1\rho_w^\theta,\quad \forall w\in \tilde{W}_*, \forall p,q\in F_w \mathcal{G}\cap (\bigcup_{m\geq 0}\bar{V}_m). \]

We first consider the case $q\in F_w\bar {V}_1$. Assume that $p\in F_w\bar {V}_n$ for some $n\geq 1$, then we can find $\tau \in W_{n-1}$ such that $p\in F_wF_\tau \bar {V}_1$. We can then find a sequence

\[ q=p_0,p_1,\cdots,p_{|\tau|+1}=p, \]

such that $p_i\in F_wF_{[\tau ]_{i-1}}\bar {V}_1\cap F_wF_{[\tau ]_{i}}\bar {V}_1$ for $1\leq i\leq |\tau |$. As a consequence, by using lemma 6.3, we see that

\[ \tilde{R}(p,q)\leq \sum_{i=0}^{|\tau|} c_2\,{\rm d}(p_i,p_{i+1})^\theta\leq \sum_{i=0}^{|\tau|} c_2(\rho_w\rho^i)^\theta\leq \frac {c_2}{1-\rho^\theta}\rho_w^\theta, \]

where $c_2$ is the same constant in lemma 6.3. For general $q$, we only need to set $c_1=\frac {2c_2}{1-\rho ^\theta }\rho _w^\theta$. Now, let $p,q\in \bigcup _{m=0}^\infty \bar {V}_m$. We choose $w,w'\in \tilde W_*$ such that $p\in F_w\mathcal {G}$, $q\in F_{w'}\mathcal {G}$ and

\[ \rho {\rm d}(p,q)\leq\rho_w,\rho_{w'}<\rho^{{-}1}\,{\rm d}(p,q). \]

In addition, we can find a chain

\[ w=w^{(0)},w^{(1)},\cdots,w^{(k)}=w' \]

such that $\min \{\rho _w,\rho _{w'}\}\leq \rho _{w^{(i)}}<\rho ^{-2}\min \{\rho _w,\rho _{w'}\}$ of length at most $c_3$, where $c_3$ is a constant independent of $p,q$. By choosing a sequence $p=p_0,p_1,\cdots,p_{k+1}=q$ such that $p_i\in F_{w^{(i-1)}}\bar {V}_1\cap F_{w^{(i)}}\bar {V}_1$, $1\leq i\leq k$, we get the desired estimate as above.

Now, we can show that $(\mathcal {E},\mathcal {F})$ is a good form.

Theorem 6.5 $(\mathcal {E},\mathcal {F})$ defined in definition 6.1 is a strongly local regular resistance form on $\mathcal {G}$.

Proof. First, we claim that $(\mathcal {E},\mathcal {F})$ is a resistance form on $\bigcup _{m\geq 0}\bar {V}_m$. (RF1) and (RF5) are obvious given lemma 6.2. Observing that by iterating the minimal energy extension, we can extend any $f\in \mathcal {F}^{(m)}$ to $f\in \mathcal {F}$ thanks to the upper-bound estimate of the resistance metric in lemma 6.4. (RF2), (RF3) and (RF4) are then easy to show with lemma 6.2. In addition, we see that

\[ \tilde{R}(p,q)=R(p,q):=\sup_{f\in \mathcal{F}}\frac{|f(p)-f(q)|^2}{\mathcal{E}(f)},\quad \forall p,q\in \bigcup_{m\geq 0}\bar{V}_m. \]

Next, to prove that $(\mathcal {E},\mathcal {F})$ is a resistance form on $\mathcal {G}$, we need to show that $\mathcal {F}$ separates points in $\mathcal {G}$, just like in proposition 3.3. It suffices to prove that $\mathcal {F}$ is dense in $C(\mathcal {G})$. Let $u\in C(\mathcal {G})$, we fix $N$ large enough so that $|u(x)-u(y)|<\varepsilon$ if $x,y \in F_wK$ and $|w|\geq N$. We can apply proposition 4.3 to create $f\in \mathcal {F}$ such that $\|f-u\|_{L^\infty (\mathcal {G})}< 2\varepsilon$. First, we find $f_1\in \mathcal {F}^{(1)}$ such that 1. $\|f_1-u|_{\bar {V_1}}\|_{L^\infty }<\varepsilon$; 2. $f_1(p)=u(p)$ for any $p\in \bigcup _{n=1}^N\bigcup _{w\in W_{1,n}} F_wV_0$. Then we apply harmonic extension to $f_1$ on $\bar {V}_2\setminus \bigcup _{n=1}^N\bigcup _{w\in W_{1,n}} F_w\bar {V}_1$. On the cells $F_w\bar {V}_1$ with $|w|< N$, we apply the same construction to get $f_2$, but with $N-2$ replacing $N$ this time. After $k=[N/2]+1$ times, we get $f_{k}\in \mathcal {F}^{(k)}$ such that $\|f_k-u|_{\bar {V}_k}\|_{L^\infty }<2\varepsilon$. Since all cells have size smaller than $\rho ^N$, by harmonically extending, we get $f\in \mathcal {F}$ such that $\|f-u\|_{L^\infty (\mathcal {G})}<2\varepsilon$. Thus, $(\mathcal {E},\mathcal {F})$ is regular resistance form on $\mathcal {G}$. It remains to show that the form is strongly local. Let $f,g\in \mathcal {F}$ with $supp(f)\cap supp(g)=\emptyset$, then there exists $\varepsilon >0$ such that $d(supp(f),supp(g))>\varepsilon$. Thus, we have $\mathcal {D}^{(n)}(f,g)=0$ for large $n$, because the supports of $f$ and $g$ are suitably separated by $n$-cells for large $n$. By taking the limit, we see that $\mathcal {E}(f,g)=0$. Clearly $1\in \mathcal {F}$ with $\mathcal {E}(1)=0$, and it follows that the form is strongly local.

In the remaining part of this section, we would like to characterize $(\mathcal {E},\mathcal {F})$ as the unique self-similar form associated with the infinite IFS $\{F_w\}_{w\in W_1}$.

Theorem 6.6 The resistance form $(\mathcal {E},\mathcal {F})$ satisfies the following properties:

1. (a) $\mathcal {F}\subset C(\mathcal {G})$.

2. (b) For each $f\in \mathcal {F}$, we have $f\circ F_w\in \mathcal {F}$ for all $w\in W_1$, and in addition,

   \[ \mathcal{E}(f)=\sum_{w\in W_1}\rho_w^{-\theta}\mathcal{E}(f\circ F_w). \]

3. (c) Conversely, let $f\in C(\mathcal {G})$, if $f\circ F_w\in \mathcal {F}$ for all $w\in W_{1}$, and $\sum _{w\in W_1}\rho _w^{- \theta }\mathcal {E}(f\circ F_w)<\infty$, then $f\in \mathcal {F}$.

Moreover, $(\mathcal {E},\mathcal {F})$ (up to constants) and $\theta$ are uniquely determined by the above properties.

Proof. The claimed properties of $(\mathcal {E},\mathcal {F})$ are immediate consequences of the construction.

The uniqueness follows by a well-known argument, but in the infinite graph version. Let $(\mathcal {E}',\mathcal {F}')$ be another form satisfying the above properties with $\theta '$ replacing $\theta$. Define $\mathcal {D}'$ to be the trace of $\mathcal {E}'$ onto $V_0$, and write $r'_w=\rho _w^{\theta '}$, $r'=\rho ^{\theta '}$. For any $u\in l(V_0)$, let $h_u$ be the harmonic extension of $u$ to $\mathcal {F}'$, then we can see that

\begin{align*} \mathcal{D}'(u)& =\mathcal{E}'(h_u)=\sum_{w\in W_1} {r'}_w^{{-}1}\mathcal{E}'(h_u\circ F_w)\geq \sum_{w\in W_1}{r'}_w^{{-}1}\mathcal{D}'\left((h_u\circ F_w)|_{V_0}\right)\\ & \quad \geq{r'}^{{-}2}\mathcal{R}_{r'}\mathcal{D}'(u), \end{align*}

where $\mathcal {R}_{r'}$ is the renormalization map introduced in definition 4.1, and we use properties (a) and (b) in the inequalities.

On the other hand, we can perform the harmonic extension of $u$ in two steps: first, we extend $u$ to $f_1\in C(\bar {V}_1)$ so that $f_1$ minimizes $\Psi _{r'}\mathcal {D}'$, then we take harmonic extension of $f_1$ on each cell $F_w\mathcal {G},w\in W_1$, to $f\in C(\mathcal {G})$, by using property (a) and the Markov property (RF5). In addition, $f\in \mathcal {F}'$ by the property (c). Then, by property (b),

\[ {r'}^{{-}2}\mathcal{R}_{r'}\mathcal{D}'(u)={r'}^{{-}2}\Psi_{r'}\mathcal{D}'(f_1)=\sum_{w\in W_1}{r'}_w^{{-}1}\mathcal{E}'(f\circ F_w)=\mathcal{E}'(f)\geq \mathcal{D}'(u). \]

Thus, we get $\mathcal {R}_{r'}\mathcal {D}'={r'}^2\mathcal {D}'$, which implies that $\mathcal {D}'=\mathcal {D}$ and $\theta '=\theta$ by theorem 5.1. Finally, by a similar argument, one can easily find that the restriction of $\mathcal {E}'$ to $\bar {V}_m$ is $\mathcal {D}^{(m)}$, and the claim that $\mathcal {E}'=\mathcal {E}$ follows immediately by taking the limit.

Finally, the form $(\mathcal {E},\mathcal {F})$ is decimation invariant with respect to the graph-directed construction in definition 2.2.

Definition 6.7 Using the same notation as in definition 2.2, let $(\mathcal {E}_1,\mathcal {F}_1)=(\mathcal {E},\mathcal {F})$, and define $(\mathcal {E}_2,\mathcal {F}_2)$ as follows:

\[ \begin{cases} \mathcal{E}_2(f,g)=\sum_{w\in W_1,F_w\mathcal{G}\subset K_2} \rho_w^{-\theta}\mathcal{E}(f\circ F_w, g\circ F_w),\\ \mathcal{F}_2=\{f\in C(K_2): \text{ }f\circ F_w\in \mathcal{F}, \forall w\in W_1 \text{ such that }F_w\mathcal{G}\subset K_2, \text{ }\mathcal{E}_2(f)<\infty\}. \end{cases} \]

It is not hard to verify that $(\mathcal {E}_2,\mathcal {F}_2)$ is a resistance form on $K_2$. Moreover, we have

Theorem 6.8 Recall the notation of definition 2.2, and write $\rho _{e_j}$ for the similarity ratio of $\psi _{e_j}$, $1\leq j\leq 6$. Let $(\mathcal {E}_i,\mathcal {F}_i),i=1,2$ be defined as in definition 6.7. Then, for $f_i\in \mathcal {F}_i$, $i=1,2$, we have $f_{e_{j,1}}\circ \psi _{e_j}\in \mathcal {F}_{e_{j,2}}$ for $1\leq j\leq 6$ and

\[ \mathcal{E}_1(f_1)=\sum_{j=1}^2\rho^{-\theta}_{e_j}\mathcal{E}_{e_{j,2}}(f_1\circ \psi_{e_j}),\quad \mathcal{E}_2(f_2)=\sum_{j=3}^6\rho^{-\theta}_{e_j}\mathcal{E}_{e_{j,2}}(f_2\circ \psi_{e_j}). \]

Conversely, let $f_1\in C(K_1)$, if $f_1\circ \psi _{e_j}\in \mathcal {F}_{e_{j,2}}$ for $j=1,2$, then $f_1\in \mathcal {F}_1$. The same holds for $(\mathcal {E}_2,\mathcal {F}_2)$.

Remark At the end of this section, we remark that a same construction can be applied to get some non-standard self-similar forms on $\mathcal {G}$ with respect to the infinite IFS $\{F_w\}_{w\in W_1}$, by starting with any solution $R_{r'}\mathcal {D}'=\lambda (r')\mathcal {D}'$. Theorems 6.5 and 6.8 still hold for the forms, with slight changes of the renormalization factors. Nevertheless, the good heat kernel estimate (theorem 7.4) will not hold, but it is possible to get a heat kernel estimate in the form of Hambly and Kumagai's on PCF self-similar sets [18].

7. Transition density estimate

Let $\mu _H$ be the normalized Hausdorff measure on $\mathcal {G}$. $(\mathcal {E},\mathcal {F})$ becomes a local regular Dirichlet form on $L^2(\mathcal {G},\mu _H)$ ($L^2(\mathcal {G})$ for short) in a standard way (see [22, theorem 2.4.1]). By the celebrated result [15, theorem 7.2.1], there is a Hunt process $X=(\mathbb {P}^x,x\in \mathcal {G},X_t,t\geq 0)$ associated with $(\mathcal {E},\mathcal {F})$ such that

\[ \mathbb{E}^x[f(X_t)]=P_tf(x),\quad \text{ a.e. } x\in\mathcal{G}, \]

where $(P_t)_{t\geq 0}$ is the associated semigroup. In this last section, we will show that $X$ is a fractional diffusion. We recall from Barlow's book [3, § 3], for the definition of this fractional diffusion.

Definition 7.1 A Markov process $X=(\mathbb {P}^x,x\in \mathcal {G},X_t,t\geq 0)$ is a fractional diffusion on the fractional metric space $(\mathcal {G},d_g,\mu _H)$ (see § 2) if (a). $X$ is a conservative Feller diffusion with state space $\mathcal {G}$; (b). $X$ is $\mu _H$-symmetric; (c). $X$ has a symmetric transition density $p(t,x,y)=p(t,y,x)$, $t>0$, $x,y\in \mathcal {G}$, which satisfies the Chapman–Kolmogorov equations and is jointly continuous for $t>0$; (d). There exist a constant $\beta$ and $c_1$–$c_4>0$, such that for $0< t\leq 1$,

\begin{align*} & c_1t^{{-}d_H/\beta}\exp\left({-}c_2\left(\frac{d_g(x,y)^\beta}{t}\right)^{\frac{1}{\beta-1}}\right)\leq p(t,x,y)\\ & \quad \leq c_3t^{{-}d_H/\beta}\exp\left({-}c_4\left(\frac{d_g(x,y)^\beta}{t}\right)^{\frac{1}{\beta-1}}\right), \end{align*}

where $d_H$ is the Hausdorff dimension of $\mathcal {G}$.

Since $d_g\asymp d$ by lemma 2.3, it suffices to consider the Euclidean metric $d$ in the following.

We will closely follow Barlow's book [3] and Hambly and Kumagai's paper [18]. We only provide some essential estimates, including a Nash inequality and an estimate of the resistance metric $R$.

For convenience, for $0< s<1$, we write $\tilde {W}_s=\{w\in \tilde {W}_*:\rho _w\leq s< \rho _{([w]_{|w|-1})}\}$, and by identifying words representing the same cells, we get a quotient class $\hat {W}_s$.

Proposition 7.2 Nash inequality

Let $d_S=\frac {2d_H}{d_H+\theta }$ with $\theta =\frac {\log r}{\log \rho }$, and $f\in \mathcal {F}$, we have

\[ \|f\|_{L^2(\mathcal{G})}^{2+4/d_S}\leq c\left(\mathcal{E}(f)+\|f\|_{L^2(\mathcal{G})}^2\right)\|f\|_{L^1(\mathcal{G})}^{4/d_S}, \]

for some constant $c>0$ independent of $f$.

Proof. The proof is essentially the same as that for PCF self-similar sets [18]. We reproduce it here for convenience of readers. First, we claim that for any $f\in \mathcal {F}$, $\|f\|^2_{L^2(\mathcal {G})}\leq c_1(\mathcal {E}(f)+\|f\|^2_{L^1(\mathcal {G})})$ for some constant $c_1>0$ independent of $f$. In fact, let $\bar f=\int _{\mathcal {G}}f\,{\rm d}\mu _H$ and $g=f-\bar f$, it suffices to check that $\|g\|_{L^2(\mathcal {G})}^2\leq c_2\mathcal {E}(f)$ for some $c_2>0$, which follows from

\begin{align*} \|g\|^2_{L^2(\mathcal{G})}& =\frac 12 \int_{\mathcal{G}}\int_{\mathcal{G}}\left(g(x)-g(y)\right)^2\,{\rm d}\mu_H(x)\,{\rm d}\mu_H(y)\\ & =\frac 12 \int_{\mathcal{G}}\int_{\mathcal{G}}\left(f(x)-f(y)\right)^2\,{\rm d}\mu_H(x)\,{\rm d}\mu_H(y)\leq c_2 \mathcal{E}(f), \end{align*}

where the last inequality is due to (RF4) and proposition 3.3. Next, write $f_w=f\circ F_w$ for $w\in \tilde W_*$ for short. Then for $0< s<1$,

\begin{align*} \|f\|^2_{L^2(\mathcal{G})}& \leq \sum_{w\in \hat{W}_s}\rho_w^{d_H}\|f_w\|^2_{L^2(\mathcal{G})}\leq c_1\sum_{w\in \hat{W}_s}\rho_w^{d_H}\left(\mathcal{E}(f_w)+\|f_w\|^2_{L^1(\mathcal{G})}\right)\\ & \leq c_3s^{d_H+\theta}\sum_{w\in \hat{W}_s}\rho_w^{-\theta}\mathcal{E}(f_w)+c_4s^{{-}d_H}\sum_{w\in \hat{W}_s}(\rho_w^{d_H}\|f_w\|_{L^1(\mathcal{G})})^2\\ & \leq c_5\left(s^{d_H+\theta}\mathcal{E}(f)+s^{{-}d_H}\|f\|_{L^1(\mathcal{G})}^2\right), \end{align*}

for some $c_3-c_5>0$, where in the last inequality, we use the observation that $\sum _{w\in \hat {W}_s}\rho _w^{-\theta }\mathcal {E}(f_w)\leq c'\mathcal {E}(f)$ for some $c'\geq 1$. In the case that $\mathcal {E}(f)>\|f\|^2_{L^1(\mathcal {G})}$, we choose $s$ such that $s^{2d_H+\theta }\mathcal {E}(f)=\|f\|^2_{L^1(\mathcal {G})}$, then $\|f\|^2_{L^2(\mathcal {G})}\leq 2c_5\mathcal {E}(f)^{\frac {d_H}{2d_H+\theta }}\|f\|_{L^1(\mathcal {G})}^{\frac {2d_H+2\theta }{2d_H+\theta }}$, and so the desired result follows immediately. In the case that $\mathcal {E}(f)\leq \|f\|^2_{L^1(\mathcal {G})}$, we have $\|f\|^2_{L^2(\mathcal {G})}\leq c_1(\mathcal {E}(f)+\|f\|_{L^1(\mathcal {G})}^2)\leq 2c_1\|f\|^2_{L^1(\mathcal {G})}$, and the result still follows.

The Nash inequality provides an upper-bound estimate $p(t,x,y)\leq c_1t^{-d_S/2}$. In addition, $|p(t,x,y)-p(t,x,y')|\leq c_2t^{-1-d_S/2}R(y,y'), \forall 0< t\leq 1, x,y,y'\in \mathcal {G}$. See [11] for a proof.

Proposition 7.3 Let $R(\cdot,\cdot )$ be the resistance metric associated with $(\mathcal {E},\mathcal {F})$ on $\mathcal {G}$. Then there exist $c_1, c_2>0$ such that

\[ c_1\,{\rm d}(p,q)^\theta\leq R(p,q)\leq c_2\,{\rm d}(p,q)^\theta,\quad \forall p,q\in\mathcal{G}. \]

In addition, for $p\in \mathcal {G}$ and $A\subset \mathcal {G}$, define $R(p,A)=\sup \{\mathcal {E}(f)^{-1}:f\in \mathcal {F}, f(p)=1,f|_A=0\}$. Then there exists $c_3,c_4>0$ such that

\[ c_3s^\theta\leq R\left(p,B^c_s(p)\right)\leq c_4s^\theta, \]

where $B_s(p)=\{q\in \mathcal {G}:{\rm d}(p,q)< s\}$ with $p\in \mathcal {G}$ and $0< s<1$, and $B_s^c(p)$ is the complement of $B_s(p)$ in $\mathcal {G}$.

Proof. We already have the estimate $R(p,q)\leq c_2\,{\rm d}(p,q)^\theta$ from lemma 6.4 and theorem 6.5. Now we show $R(p,B^c_s(p))\geq c_3s^\theta$ for $p\in \mathcal {G}$ and $0< s<1$. Define

\begin{align*} & U_{p,s,0}=\bigcup_{w\in \hat{W}_{p,s,0}} F_w\mathcal{G} \text{ with }\hat{W}_{p,s,0}=\{w\in \hat{W}_{s\rho^2}:p\in F_w\mathcal{G}\},\\ & U_{p,s,1}=\bigcup_{w\in \hat{W}_{p,s,1}} F_w\mathcal{G} \text{ with }\hat{W}_{p,s,1}=\{w\in \hat{W}_{s\rho^2}:F_w\mathcal{G}\cap U_{p,s,0}\neq \emptyset\}. \end{align*}

Clearly, we have $U_{p,s,0}\subset U_{p,s,1}\subset B_{s}(p)$. Since $(\mathcal {E},\mathcal {F})$ is regular, there exists $f_{p,s}\in \mathcal {F}$ so that $f_{p,s}|_{{U}^c_{p,s,1}}=0$ and $f_{p,s}|_{U_{p,s,0}}=1$. As $\mathcal {G}$ satisfies the finite type property, there exists a finite class $\{(p_i,s_i)\}_{i=1}^N$ such that for any $p\in \mathcal {G}$ and $0< s<1$, there exists $1\leq i\leq N$ and an affine map $\psi$ such that $\psi :U_{p,s,l}\to U_{p_i,s_i,l}$ for $l=0,1$, which maps cells corresponding to $\hat W_{p,s,l}$ to those corresponding to $\hat W_{p_i,s_i,l}$. In addition, we require that $\psi$ maps the boundary of $U_{p,s,l}$ to the boundary of $U_{p_i,s_i,l}$, which only depend on how the outside cells of approximately same size intersect $U_{p_i,s_i,1}$. Thus, we can assume that

\[ f_{p,s}(q)=\begin{cases} f_{p_i,s_i}\circ\psi (q), & \text{ if }q\in U_{p,s,1},\\ 0, & \text{ if }q\in U^c_{p,s,1}.\end{cases} \]

By a similar observation as in lemma 6.3, there exists $m\in \mathbb {Z}$ such that

\[ \mathcal{D}^{(n)}(f_{p_i,s_i})\leq \rho_\psi^{\theta}\mathcal{D}^{(n+m)}(f_{p,s}), \]

where $\rho _\psi$ is the similarity ratio of $\psi$. So we have $\mathcal {E}(f_{p,s})=\rho _{\psi }^{-\theta }\mathcal {E}(f_{p_i,s_i})\leq c_3^{-1}s^{-\theta }$ for some constant $c_3$ independent of $p,s,i$. Since $f_{p,s}|_{B_s^c(p)}=0$ and $f_{p,s}(p)=1$, we get the estimate $R(p,B_s^c(p))\geq c_3s^\theta$.

Finally, the estimates $R(p,q)\geq c_1\,{\rm d}(p,q)^\theta$ follows from the fact that $R(p,q)\geq R(p,B_{{\rm d}(p,q)}^c(p))\geq c_3{{\rm d}(p,q)}^\theta$, and $R(p,B_s^c(p))\leq c_4s^\theta$ follows from the fact that $R(p,B_s^c(p))\leq R(p,q)\leq c_2s^\theta$ for some $q\in \mathcal {G}$ satisfying ${\rm d}(p,q)=s$.

By the resistance metric estimate in proposition 7.3, the Ahlfors regularity of the measure $\mu _H$ (lemma 2.3) and the resulted estimates from the Nash inequality, there exist a lower-bound estimate $p(t,x,y)\geq c_3t^{-d_S/2}$ and an estimate of the hitting time $c_4s^{\theta +d_H}\leq \mathbb {E}^x\tau (x,s)\leq c_5s^{\theta +d_H}$, where $\tau (x,s)=\inf \{t\geq 0:X_t\notin B_s(x)\}$. See [3, § 8] for details. Finally, by [3, theorem 3.1.1] of Barlow or by following [18], we can finally find that our diffusion is a fractional diffusion.

Theorem 7.4 The Hunt process $X=(\mathbb {P}^x,x\in \mathcal {G},X_t,t\geq 0)$ associated with the form $(\mathcal {E},\mathcal {F})$ on $L^2(\mathcal {G},d_H)$ is a fractional diffusion, with $\beta =\theta +d_H$, in the sense of definition 7.1.