Explicit Spectral Decimation for a Class of Self-Similar Fractals

Abstract

The method of spectral decimation is applied to an infinite collection of self-similar fractals. The sets considered are a generalization of the Sierpinski Gasket to higher dimensions; they belong to the class of nested fractals and are thus very symmetric. An explicit construction is given to obtain formulas for the eigenvalues of the Laplace operator acting on these fractals.

1. Introduction

In 1989, J. Kigami [1] gave an analytic definition of a Laplace operator acting on the Sierpinski Gasket; a few years later, this definition was extended to include the Laplacians on a large class of self-similar fractal sets [2], known as postcritically finite sets (PCF sets). The method of spectral decimation introduced by Fukushima and Shima in the 1990s provides a way to evaluate the eigenvalues of Kigami’s Laplacian. In general terms, this method consists in finding the eigenvalues of the self-similar fractal set by taking limits of eigenvalues of discrete Laplacians that act on some graphs that approximate the fractal. The spectral decimation method was applied in [3] to the Sierpinski Gasket, in order to give an explicit construction which allows one to obtain the set of eigenvalues. In [4], it was shown that it is possible to apply the spectral decimation method to a large collection of p.c.f. sets, including the family of fractals known as nested fractals that was introduced by T. Lindstrøm in [5]. In addition to the Sierpinski Gasket, the spectral decimation method has been applied in several specific cases of p.c.f. fractals (e.g., [6–10]); also, the method has been proved useful to study the spectrum of particular fractals that are not p.c.f. (e.g., [11, 12]) and of fractafolds modeled on the Sierpinski Gasket [13]. The spectral decimation method has also shown to be a very useful tool for the analysis of the structure of the spectra of the Laplacians of some fractals (e.g., [14–16]).

In the present work, we develop in an explicit way the spectral decimation method for an infinite collection of self-similar sets that we will denote by 𝐏_𝑛 (𝑛≥2 a positive integer). The definition of these sets is given in Definition 1. For the cases 𝑛=2,3, they correspond, respectively, to the unit interval and the Sierpinski Gasket. For larger values of 𝑛 they give a quite natural extension of the Sierpinski Gasket to higher dimensions. The spectral decimation method for the cases 𝑛=2,3 is presented with thorough detail in [17]. Our presentation follows this reference to some extent. However, some technical difficulties arise for 𝑛≥4. This is mainly due to the fact that—even though the fractals considered are very symmetric—the graphs approximating the fractal are not as homogeneous as the ones approximating the Sierpinski Gasket. For instance, if we consider the graph obtained by taking away the boundary points from Γ₁ (see Definition 2 and Figures 1 and 2), then it will be a complete graph only for 𝑛≤3. A consequence of this is the appearance of sets of two types of vertices that have to be dealt with separately and which we denote by 𝐹_𝑟,𝑠 and 𝐺_𝑟,𝑠; for 𝑛≤3, the sets 𝐺_𝑟,𝑠 are empty. We also make the observation that the approximating graphs Γ_𝑘 are nonplanar when 𝑛>3.

Figure 1 

The first approximating graph Γ1 for the fractal 𝐏4.

Figure 2 

The graph Γ1 for the fractal 𝐏4, minus the boundary points.

In Section 2, we present general facts about self similar sets, for the sake of completeness and in order to establish notation. In Section 3, we introduce the sets 𝐏_𝑛 that are the subject of study in this work; at the end of the section, we find the Hausdorff dimension of these fractals when embedded in the Euclidean space. In Section 4, we define the graphs that approximate the self-similar sets 𝐏_𝑛 and fix more notations. Our main result is presented in Section 5 (Theorem 3); it is shown that the eigenvalues and eigenfunctions of the discrete Laplacians of the approximating graphs can be obtained recursively. Finally, in Section 6, it is shown that the eigenvalues of the Laplace operator in 𝐏_𝑛 can be recovered by taking limits of the discrete Laplacians; in order to do this, we solve the so-called renormalization problem for this case (see Theorem 5).

2. Notation and Preliminaries

We denote by 𝐒_𝑛 the shift space with 𝑛 symbols. In this work, we will always consider these 𝑛 symbols to be the numbers 0,1,…,𝑛−1. 𝐒_𝑛 is a compact space (see, e.g., [18]) when equipped with the metric 𝛿𝑎₀𝑎₁𝑎₂𝑎₃…;𝑏₀𝑏₁𝑏₂𝑏₃…=𝑟^𝑘,0<𝑟<1, (1) where 𝑘=min𝑗≥0∣𝑎_𝑗≠𝑏_𝑗. (2) We will use the dot notation 𝑎̇, meaning that the symbol 𝑎 repeats to infinity.

Let 𝐱=𝑥₀𝑥₁𝑥₂… be an element of 𝐒_𝑛  and 𝑎∈{0,…,𝑛−1}. We denote by 𝑇_𝑎 the shift-operator given by 𝑇_𝑎(𝐱)=𝑎𝑥₀𝑥₁𝑥₂…. (3) It is easy to see that 𝛿(𝑇(𝐱);𝑇(𝐲))=𝑟𝛿(𝐱;𝐲), (4) so that 𝑇_𝑎 is a contraction (by factor 𝑟). The space 𝐒_𝑛 is a self-similar set, equal to 𝑛 smaller copies of itself, with {𝑇₀,…,𝑇_𝑛−1} the corresponding contractions. Even more so, it can be proved (Theorem  1.2.3. in [18]) that if 𝐾 is any self-similar set, then it is homeomorphic to a quotient space of the form 𝐒_𝑛/∼ for a suitable equivalence relation.

For 𝐾=𝐒_𝑛/∼ and 𝑎 a word of length 𝑚𝐚=𝑎₀𝑎₁⋯𝑎_𝑚−1, (5) denote by 𝑇_𝐚 the shift operator given by 𝑇_𝐚(𝐱)=𝑎₀⋯𝑎_𝑚−1𝑥₀𝑥₁𝑥₂…. (6) The operator 𝑇_𝐚 is called an 𝑚-contraction, and the sets of the form 𝑇_𝐚(𝐾) are known as the cells of level 𝑚 of the self-similar set 𝐾. We note that, for each choice of 𝑚, 𝐾 is the union of the 𝑛^𝑚 cells of level 𝑚.

3. The Self-Similar Fractals P_𝑛

Here we will introduce the self-similar fractals 𝐏_𝑛 that are the subject of analysis in this work.

Definition 1. For 𝑛∈ℕ, define 𝐏_𝑛 as the quotient space 𝐒_𝑛/∼, with the equivalence relation given by 𝑎₀𝑎₁𝑎₂⋯𝑎_𝑘𝑏𝑐̇∼𝑎₀𝑎₁𝑎₂⋯𝑎_𝑘𝑐𝑏̇, (7) for any choice of symbols 𝑎_𝑗, 𝑏, and 𝑐.

𝐏₁ is a trivial space with only one element, 𝐏₂ is homeomorphic to a compact interval in ℝ, and 𝐏₃ is homeomorphic to the well-known Sierpinski Gasket. For any value of 𝑛, 𝐏_𝑛 can be embedded in the Euclidean space; more precisely, there exists a (quite natural) homeomorphism between 𝐏_𝑛 and a compact self-similar set 𝐾_𝑛⊂𝑅^𝑛−1. Below, we define the sets 𝐾_𝑛; for these representations of 𝐏_𝑛, we will be able to find their Hausdorff dimensions.

Take 𝑛 points 𝑥₀,…,𝑥_𝑛−1∈ℝ^𝑛−1 that do not lie in the same (𝑛−2)-dimensional hyperplane; for 𝑛=3, those points will be the “vertices” of the Sierpinski Gasket. For 𝑛=4, the fractal 𝐾₄ will be some sort of a Sierpinski tetrahedron (see Figure 3), while the four points 𝑥_𝑗 will be the vertices of the tetrahedron.

Figure 3 

A representation of 𝐾4,  generated with Matlab.

Consider the contractions 𝑓_𝑖(𝑥)=𝑥+𝑥_𝑖2,𝑖=1,…,𝑛−1. (8) We note that 𝑓_𝑖 maps each 𝑥∈ℝ^𝑛−1 to the midpoint of 𝑥 and 𝑥_𝑖 (hence, leaving 𝑥_𝑖 fixed). Define 𝐾_𝑛 as the unique compact set such that 𝐾_𝑛=^𝑛−1_𝑖=0𝑓_𝑖𝐾_𝑛. (9) We note that, for 𝑖≠𝑗, the sets 𝑓_𝑖(𝐾_𝑛) and 𝑓_𝑗(𝐾_𝑛) intersect at exactly one point: 𝑓_𝑖(𝑥_𝑗)=𝑓_𝑗(𝑥_𝑖). From this, it follows that the map 𝜋∶𝐏_𝑛→𝐾_𝑛 given by 𝜋𝜔₀𝜔₁𝜔₂…=_𝑚≥0𝑓_𝜔0∘𝑓_𝜔1∘⋯∘𝑓_𝜔𝑚𝐾_𝑛 (10) is a well-defined homeomorphism; also, for every 𝑘=0,…,𝑛−1, the following diagram commutes (cf. Theorem  1.2.3 in [18]): (11)

The sets 𝐾_𝑛 satisfy the Moran-Hutchinson open set condition namely, there exists a bounded nonempty open set 𝑂⊂𝐏_𝑛 such that 𝑓_𝑖(𝑂){},𝑓⊂𝑂,∀𝑖∈0,…,𝑛−1_𝑖(𝑂)∩𝑓_𝑗(𝑂),∀𝑖≠𝑗. (12) Just take 𝑂=𝐾_𝑛⧵{𝑥₀,…,𝑥_𝑛−1}. From this and the fact that 𝐾_𝑛 is equal to 𝑛 contractions of itself (by factor 1/2), it follows from Moran’s theorem (Corollary  1.5.9 in [18]) that the Hausdorff dimension of 𝐾_𝑛, with respect to theEuclidean metric, is equal to log𝑛/log2.

We end this section with two relevant notes.  For some values of 𝑛, it might be possible to embed 𝐾_𝑛 isometrically into the Euclidean space of a dimension 𝑚 smaller than 𝑛−1. Of course, the dimension of the fractal gives a restriction to the minimal value of 𝑚.  The representations 𝐾_𝑛 are somehow useful to visualize the self-similar fractals 𝐏_𝑛. However, this representation and its metric do not play any role in the analysis carried out in the next sections; we will therefore focus on the more abstract definition of 𝐏_𝑛 given at the beginning of this section.

4. Graph Approximations of Self-Similar Sets

In this and the next sections, we consider the self-similar set 𝐏_𝑛 defined above, for an arbitrary but fixed value of 𝑛≥2.

Let 𝑉₀ be the set of points in 𝐏_𝑛 that have the form 𝑘̇ with 𝑘=1,…,𝑛−1. We call 𝑉₀ the boundary of 𝐏_𝑛. Likewise, for 𝑚∈ℕ, let 𝑉_𝑚 be the subset of 𝐏_𝑛 of points of the form 𝑎₀⋯𝑎_𝑚−1𝑘̇. In other words, 𝑥∈𝑉_𝑚 if and only if it belongs to the image of 𝑉₀ under some 𝑚-contraction.

Next, we define the graphs that will approximate 𝐏_𝑛.

Definition 2. Denote by Γ₀ the complete graph of 𝑛 vertices, with 𝑉₀ being its set of vertices. For 𝑚∈ℕ, let Γ_𝑚 be the graph with the set of vertices 𝑉_𝑚 and edge relation established by requiring 𝑥 to be connected with 𝑦 if and only if there exists an 𝑚-contraction 𝑇_𝐚 such that both points 𝑥 and 𝑦 are in 𝑇_𝐚(𝑉₀).

We can see that an equivalent formulation is that two vertices 𝑥 and 𝑦 share an edge in Γ_𝑚 only when their first 𝑚 symbols coincide. It is worth noting that even though 𝑉₀⊂𝑉₁⊂𝑉₂…, the edge relation is never preserved; this follows from the fact that if 𝑥≠𝑦 are connected in Γ_𝑚, then their (𝑚+1)-th symbols cannot be equal, so that they will not be connected in Γ_𝑚+1.

For each 𝑚∈ℕ, let Δ_𝑚 be the graph Laplacian on Γ_𝑚. We consider the Laplacian as acting on a space with boundary. More precisely, for a real-valued function 𝑢 defined on 𝑉_𝑚 and 𝑥 in 𝑉_𝑚⧵𝑉₀, Δ_𝑚𝑢(𝑥)=_𝑦∼𝑥𝑢(𝑥)−𝑢𝑦, (13) with the sum over all vertices 𝑦 that share an edge with 𝑥; the boundary values remain unchanged. Also, 𝑢 is an eigenfunction of Δ₀ with eigenvalue 𝜆, if Δ_𝑚𝑢(𝑥)=𝜆𝑢(𝑥),∀𝑥∈𝑉_𝑚⧵𝑉₀. (14) We denote by 𝐸_𝑚(⋅,⋅) the associated quadratic form (known as the energy product of the graph): 𝐸_𝑚(𝑢,𝑣)=Δ_𝑚𝑢,𝑣=_𝑥∼𝑦𝑢(𝑥)−𝑢𝑦𝑣(𝑥)−𝑣𝑦, (15) for 𝑢 and 𝑣 real-valued functions defined on 𝑉_𝑚 and the sum being taken over the pairs of vertices (𝑥,𝑦) that are connected to each other. Also, we use the abbreviation 𝐸(𝑢)=𝐸(𝑢,𝑢).

5. Spectral Decimation

Let 𝑚>1 and suppose 𝑢 is an eigenfunction of Δ_𝑚−1, with eigenvalue 𝜆_𝑚−1. We will show that it is always possible to extend this function to the domain 𝑉_𝑚 so that it will be an eigenfunction of Δ_𝑚 (with not the same eigenvalue). In order to do this, we will derive necessary conditions for the extension to be an eigenfunction; in the process, it will become clear that those conditions are also sufficient.

Suppose that 𝑢 is an eigenfunction of Δ_𝑚 with eigenvalue 𝜆_𝑚; we aim to write the values of 𝑢_𝑚 in 𝑉_𝑚⧵𝑉_𝑚−1 in terms of its values in 𝑉_𝑚−1. Without loss of generality, we can restrict ourselves to the set 𝑉_𝑚∩𝑇_𝐚(𝐏_𝑛) for a fixed (𝑚−1)-contraction 𝑇_𝐚; this is because the vertices of Γ_𝑚 that belong to the set (𝑉_𝑚⧵𝑉_𝑚−1)∩𝑇_𝐚(𝐏_𝑛) are not connected to any vertices outside the cell 𝑇_𝐚(𝐏_𝑛). Denote the elements of this set by 𝑥_𝑏,𝑐=𝐚𝑏𝑐,̇𝑏,𝑐=0,…,𝑛−1. (16) It is clear that 𝑥_𝑏,𝑐=𝑥_𝑐,𝑏 and also that 𝑥_𝑏,𝑐∈𝑉_𝑚−1 if and only if 𝑏=𝑐. This is shown in Figure 4.

Figure 4 

A cell of level 𝑚−1 of a graph Γ𝑚,  approximating 𝐏4.

For each point 𝑥_𝑟,𝑠∈𝑉_𝑚∩𝑇_𝐚(𝐏_𝑛), define the sets of vertices 𝐹_𝑟,𝑠=𝑥_𝑖,𝑗{}∩,𝐺∣𝑖≠𝑗,𝑟,𝑠𝑖,𝑗≠∅_𝑟,𝑠=𝑥_𝑖,𝑗∣𝑖≠𝑗,{𝑟,𝑠}∩𝑖,𝑗=∅. (17) In other words, 𝐹_𝑟,𝑠 is the set of vertices (not in Γ_𝑚−1) that are connected to the vertex 𝑥_𝑟,𝑠 in Γ_𝑚, and 𝐺_𝑟,𝑠 is the set of vertices (not in Γ_𝑚−1, either) that are not connected to it.

In the case 𝑛=4, the graph Γ_𝑚⧵Γ_𝑚−1 is an octahedron; hence, for each pair {𝑟,𝑠}, the subgraph determined by the vertices in 𝐹_𝑟,𝑠 is a 4-cycle one, while 𝐺_𝑟,𝑠 consists of a single vertex (the one opposite to 𝑥_𝑟,𝑠 in the octahedron). For general 𝐏_𝑛, we can see that the following. The graph Γ_𝑚⧵Γ_𝑚−1 has 𝑛(𝑛−1) vertices, all of them with degree 2(𝑛−2). Each of these vertices is connected to another 2 vertices in Γ_𝑚−1.  The subgraph determined by 𝐹_𝑟,𝑠 consists of two complete graphs, each one with 𝑛−2 vertices. The two complete graphs are joined to each other pairwise, thus forming a “prism,” (a true prism only in the case 𝑛=5, where the base is a 3-cycle one, as shown in Figure 5).  The subgraph determined by 𝐺_𝑟,𝑠 has (𝑛−2)(𝑛−3)/2 vertices, each one of them with degree 2(𝑛−4).  In Γ_𝑚, each vertex that belongs to 𝐺_𝑟,𝑠 is connected to exactly four vertices in 𝐹_𝑟,𝑠. On the other hand, each vertex that belongs to 𝐹_𝑟,𝑠 is connected to 𝑛−2 vertices in 𝐺_𝑟,𝑠.

Figure 5 

The graph determined by 𝐹0,1 in a graph Γ𝑚,  approximating 𝐏5.

Now, having noted all that, we proceed with the calculations. For every 𝑟≠𝑠 we have 2(𝑛−1)−𝜆_𝑚𝑢𝑥_𝑟,𝑠=𝑢𝑥_𝑟,𝑟+𝑢𝑥_𝑠,𝑠+_{𝐹𝑟,𝑠}𝑢𝑥_𝑖,𝑗. (18)

Adding this up over all the possible values of 𝑟 and 𝑠 and rearranging terms yields 2−𝜆_{𝑚𝑟≠𝑠}𝑢𝑥_𝑟,𝑠=(𝑛−1)^𝑛−1_𝑗=0𝑢𝑥_𝑗,𝑗, (19) which for any fixed 𝑎≠𝑏 can also be written in the form 2−𝜆_𝑚𝑢𝑥_𝑎,𝑏+_{𝐹𝑎,𝑏}𝑢𝑥_𝑖,𝑗+_{𝐺𝑎,𝑏}𝑢𝑥_𝑖,𝑗=(𝑛−1)^𝑛−1_𝑗=0𝑢𝑥_𝑗,𝑗. (20) This, together with (18), allows us to express the sum of the values in 𝐺_𝑎,𝑏 in terms of 𝑢(𝑥_𝑎,𝑏) and the values at points in 𝑉_𝑚−1; namely, provided 𝜆_𝑚≠2, we have that _{𝐺𝑎,𝑏}𝑢𝑥_𝑖,𝑗=𝑢𝑥_𝑎,𝑎+𝑢𝑥_𝑏,𝑏−2𝑛−1−𝜆_𝑚𝑢𝑥_𝑎,𝑏+(𝑛−1)2−𝜆_𝑚^𝑛−1_𝑗=0𝑢𝑥_𝑗,𝑗. (21)

Next, we will take the sum of the same terms, but only over the 𝑥_𝑖,𝑗 inside the set 𝐹_𝑎,𝑏 for fixed values 𝑎≠𝑏. Since 𝐹_𝑎,𝑏 contains two complete graphs with 𝑛−1 vertices and these complete graphs are pairwise connected to each other, it is clear that each 𝑥_𝑖,𝑗∈𝐹_𝑎,𝑏 is connected to other 𝑛−1 vertices in 𝐹_𝑎,𝑏. Also, recall that each 𝑥_𝑖,𝑗∈𝐺_𝑎,𝑏 is connected to exactly four vertices in 𝐹_𝑎,𝑏. For the vertices in 𝑉_𝑚−1, we note that 𝑥_𝑗,𝑗 is connected to 𝑛−2 vertices in 𝑥_𝑖,𝑗∈𝐹_𝑎,𝑏 if 𝑗=𝑎,𝑏 and to only two vertices otherwise.

From the preceding discussion, it follows the equality 𝑛−𝜆_{𝑚𝐹𝑎,𝑏}𝑢𝑥_𝑖,𝑗=4_{𝐺𝑎,𝑏}𝑢𝑥_𝑖,𝑗+(𝑛−2)𝑢𝑥_𝑎,𝑎+𝑢𝑥_𝑏,𝑏+2(𝑛−2)𝑢𝑥_𝑎,𝑏+2_{𝑗≠𝑎,𝑏}𝑢𝑥_𝑗,𝑗. (22)

Consider the expression given by (18) for {𝑎,𝑏}={𝑟,𝑠}, multiply it by 𝑛−𝜆_𝑚, and substitute equality (22) into it; this gives after arranging terms 𝜆²_𝑚−(3𝑛−2)𝜆_𝑚+2𝑛²−2𝑛+2𝑢𝑥_𝑎,𝑏=4_{𝐺𝑎,𝑏}𝑢𝑥_𝑖,𝑗+2_{𝑗≠𝑎,𝑏}𝑢𝑥_𝑗,𝑗+2(𝑛−1)−𝜆_𝑚𝑢𝑥_𝑎,𝑎+𝑢𝑥_𝑏,𝑏. (23) We want to get rid of the terms corresponding to 𝐺_𝑎,𝑏, so we replace it by (21). After straightforward computations, we can see that for 𝜆_𝑚≠2𝜆²_𝑚−(3𝑛+2)𝜆_𝑚+2𝑛(𝑛+2)𝑢𝑥_𝑎,𝑏=𝜆²_𝑚−2(𝑛+2)+8𝑛𝑢𝑥_𝑎,𝑎+𝑢𝑥_𝑏,𝑏+22𝑛−𝜆_{𝑚𝑗≠𝑎,𝑏}𝑢𝑥_𝑗,𝑗2−𝜆_𝑚⁻¹. (24) The quadratic equation for 𝜆_𝑚 in the left-hand side has roots 𝑛+2 and 2𝑛. The one in the right-hand side has roots 4 and 2𝑛. This gives us the following expression for 𝑢(𝑥_𝑎,𝑏) in terms of the values of 𝑢 in 𝑉_𝑚−1: 𝑢𝑥_𝑟,𝑠=4−𝜆_𝑚𝑢𝑥_𝑟,𝑟+𝑢𝑥_𝑠,𝑠+2∑_{𝑗≠𝑟,𝑠}𝑢𝑥_𝑗,𝑗2−𝜆_𝑚(𝑛+2)−𝜆_𝑚, (25) valid for any eigenvalue 𝜆_𝑚≠2,𝑛+2,2𝑛.

For 𝜆_𝑚=0, this reduces to 𝑢𝑥_𝑟,𝑠=2𝑢𝑥𝑛+2^𝑟,𝑟+𝑢𝑥_𝑠,𝑠+1𝑛+2_{𝑗≠𝑟,𝑠}𝑢𝑥_𝑗,𝑗. (26)

It is clear from the construction that if 𝑢(𝑥_𝑎,𝑏) is defined by (25) and 𝜆_𝑚 is given by (35), then we have that Δ_𝑚𝑢𝑥_𝑎,𝑏=𝜆_𝑚𝑢𝑥_𝑎,𝑏(𝑎≠𝑏). (27)

It remains to verify that this is valid as well in 𝑉_𝑚−1. Of course, this cannot be true for arbitrary values of 𝜆_𝑚 but only at most for specific values depending on 𝜆_𝑚−1; we will find those values in what follows.

Take a point in 𝑉_𝑚−1, say 𝑥_𝑝,𝑝=𝐚𝑝,̇𝐚=𝑎₀⋯𝑎_𝑚−1. (28) Suppose that 𝑎_𝑘=𝑞 is the last symbol in 𝐚 that is different from 𝑝; we can assume that such symbol exists, since otherwise 𝑥_𝑝,𝑝 would be in the boundary 𝑉₀. With this, the point 𝑥_𝑝,𝑝 can also be written in the form 𝑥_𝑝,𝑝=𝐚^′𝑞,̇𝐚^′=𝑎₀⋯𝑎_𝑘−1𝑝𝑞⋯𝑞, (29) with the necessary number of 𝑞’s to make 𝐚^′ a word of length 𝑚−1. Hence, 𝑥_𝑝,𝑝 is in exactly two different (𝑚−1)-cells: 𝑇_𝐚(𝐏_𝑛) and 𝑇_𝐚′(𝐏_𝑛), corresponding to each one of its two representations.

Denote by 𝑥^′_𝑟,𝑠 the points in 𝑇_𝐚′(𝐏_𝑛)∩𝑉_𝑚, defined as in (16) for the points in 𝑇_𝐚(𝐏_𝑛)∩𝑉_𝑚; in particular, 𝑥_𝑝,𝑝=𝑥^′_𝑞,𝑞 (see Figure 6). The value of 𝑢 in the points 𝑥^′_𝑟,𝑠 is given by the analogue of (25). The vertex 𝑥_𝑝,𝑝 is connected in Γ_𝑚 to the 2(𝑛−1) points of the form 𝑥_𝑝,𝑗 and 𝑥^′_𝑞,𝑗, from which it follows that 𝑢 is an eigenfunction of Δ_𝑚 with eigenvalue 𝜆_𝑚, if and only if (25) holds for all 𝑥_𝑟,𝑠∈𝑉_𝑚⧵𝑉_𝑚−1 and the following equality holds for all 𝑥_𝑝,𝑝∈𝑉_𝑚−1: 2(𝑛−1)−𝜆_𝑚𝑢𝑥_𝑝,𝑝=_{𝑖≠𝑝,𝑗≠𝑞}𝑢𝑥_𝑝,𝑗+𝑢𝑥^′_𝑞,𝑗. (30)

Figure 6 

Two cells of level 𝑚−1 intersecting in a vertex of Γ𝑚.

On the other hand, since we know that 𝑢 is an eigenfunction of Δ_𝑚−1 with eigenvalue 𝜆_𝑚−1, we also have that 2(𝑛−1)−𝜆_𝑚−1𝑢𝑥_𝑝,𝑝=_{𝑖≠𝑝,𝑗≠𝑞}𝑢𝑥_𝑖,𝑖+𝑢𝑥^′_𝑗,𝑗. (31)

Replacing each term in the right-hand side of (30) by its expression given by (25), we can see that 2(𝑛−1)−𝜆_𝑚𝑢𝑥_𝑝,𝑝=2(𝑛−1)4−𝜆_𝑚𝑢𝑥_𝑝,𝑝+2𝑛−𝜆_𝑚2−𝜆_𝑚⁻¹×(𝑛+2)−𝜆_𝑚⁻¹, (32) and, using (31), this gives 𝑢𝑥_𝑝,𝑝=2(𝑛−1)4−𝜆_𝑚+2𝑛−𝜆_𝑚2(𝑛−1)−𝜆_𝑚×𝑢𝑥_𝑝,𝑝2(𝑛−1)−𝜆_𝑚⁻¹2−𝜆_𝑚⁻¹×(𝑛+2)−𝜆_𝑚⁻¹. (33)

Taking 𝑢(𝑥_𝑝,𝑝)≠0 and cancelling out, after computations, the above equality reduces to the quadratic 𝜆²_𝑚−(𝑛+2)𝜆_𝑚+𝜆_𝑚−1=0, (34) which in turn gives the following recursive characterization of the eigenvalues: 𝜆_𝑚=(𝑛+2)±(𝑛+2)²−4𝜆_𝑚−12. (35)

Since this procedure can be reversed, we have proved the following result.

Theorem 3. Let 𝜆_𝑚≠2,𝑛+2, and 2𝑛, and let 𝜆_𝑚−1 be given by (34). Suppose 𝑢 is an eigenfunction of Δ_𝑚−1 with eigenvalue 𝜆_𝑚−1. Extend 𝑢 to 𝑉_𝑚 by (25). Then 𝑢 is an eigenfunction of Δ_𝑚 with eigenvalue 𝜆_𝑚. Conversely, if 𝑢 is an eigenfunction of Δ_𝑚 with eigenvalue 𝜆_𝑚≠2,𝑛+2, and 2𝑛, then its restriction to 𝑉_𝑚−1 is an eigenfunction of Δ_𝑚−1 with eigenvalue 𝜆_𝑚−1.

6. The Laplacian on the Self-Similar Fractals

In order to define the Laplace operator of a p.c.f. fractal by means of graph approximations, it is required to solve the so-called renormalization problem for the fractal (e.g., [17], Chapter 4); roughly, this consists in normalizing the graph energies in Γ_𝑚 in order to obtain a self-similar energy in the fractal by taking the limit. This can be achieved if the energies are such that they remain constant for each harmonic extension from Γ_𝑚 to Γ_𝑚+1. Below, we do this for the 𝐏_𝑛 sets.

Definition 4. For a given function 𝑢 with domain 𝑉_𝑚−1, we call the extension of 𝑢 to 𝑉_𝑚 given by (26) its harmonic extension.

The next result gives the explicit solution of the renormalization problem for 𝐏_𝑛.

Theorem 5. Let 𝑢∶𝑉_𝑚−1→ℝ be arbitrary, and let 𝑢^′∶𝑉_𝑚→ℝ be its harmonic extension. Then 𝐸_𝑚𝑢^′=𝑛𝐸𝑛+2^𝑚−1(𝑢). (36)

Proof. Note that the energy at level 𝑘 of a given function equals the sum of the energies at all the 𝑘^′-cells for any 𝑘^′≤𝑘, since different cells share no edges. This allows us to restrict ourselves to one fixed 𝑚−1-cell both while considering 𝐸_𝑚(𝑢^′) and 𝐸_𝑚−1(𝑢). We use the notation of Section 4 for the vertices of Γ_𝑚 in this cell and write 𝐸 for the energy restricted to this cell. We can readily see that 𝐸_𝑚−1(𝑢)=_𝑖≠𝑗𝑢𝑥_𝑖,𝑖−𝑢𝑥_𝑗,𝑗².=(𝑛−1)^𝑛−1_𝑖=0𝑢²𝑥_𝑖,𝑖−2_𝑖≠𝑗𝑢𝑥_𝑖,𝑖𝑢𝑥_𝑖,𝑗. (37) In order to evaluate the energy 𝐸_𝑚, we consider first the edges joining vertices in 𝑉_𝑚−1 with vertices in 𝑉_𝑚⧵𝑉_𝑚−1: the edge joining the vertex 𝑥_𝑎,𝑎 with the vertex 𝑥_𝑎,𝑘 contributes to the energy by 𝑢𝑥_𝑎,𝑎−𝑢𝑥_𝑎,𝑘²=1(𝑛+2)²𝑛𝑢𝑥_𝑎,𝑎−2𝑢𝑥_𝑘,𝑘−_{𝑗≠𝑎,𝑘}𝑢𝑥_𝑗,𝑗². (38) When adding up this over all possible pairs 𝑎≠𝑘, each 𝑥_𝑟,𝑟 will appear 𝑛−1 times as the 𝑥_𝑎,𝑎, another 𝑛−1 times as the 𝑥_𝑘,𝑘, and (𝑛−1)(𝑛−2) as one of the 𝑥_𝑗,𝑗’s. Each pair double product 2𝑥_𝑟,𝑟𝑥_𝑠,𝑠 will appear twice for {𝑟,𝑠}={𝑎,𝑘},  2(𝑛−2) times for {𝑟,𝑠}={𝑎,𝑗} for some 𝑗, also 2(𝑛−2) times for {𝑟,𝑠}={𝑘,𝑗} for some 𝑗, and finally (𝑛−2)(𝑛−3) times when both 𝑟 and 𝑠 are one of the 𝑗’s. All this implies that the contribution to the energy from these edges is, after simplification, _𝑎≠𝑘𝑢𝑥_𝑎,𝑎−𝑢𝑥_𝑎,𝑘²=(𝑛−1)𝑛²+𝑛+2(𝑛+2)^2𝑛−1_𝑖=0𝑢²𝑥_𝑖,𝑖−2𝑛²+𝑛+2(𝑛+2)²_𝑖≠𝑗𝑥_𝑖,𝑖𝑥_𝑗,𝑗. (39) On the other hand, the contribution from the edge that joins the vertices 𝑥_𝑎,𝑏 and 𝑥_𝑎,𝑐 (in 𝑉_𝑚⧵𝑉_𝑚−1) equals 𝑢𝑥_𝑎,𝑏−𝑢𝑥_𝑎,𝑐²=1(𝑛+2)²𝑢𝑥_𝑏,𝑏−𝑢𝑥_𝑐,𝑐². (40) Taking the sum over all of the vertices in 𝑉_𝑚⧵𝑉_𝑚−1 yields _{𝑎≠𝑏≠𝑐}𝑢𝑥_𝑎,𝑏−𝑢𝑥_𝑎,𝑐²=(𝑛−1)(𝑛−2)(𝑛+2)^2𝑛−1_𝑖=0𝑢²𝑥_𝑖,𝑖−2(𝑛−2)(𝑛+2)²_𝑖≠𝑗𝑢𝑥_𝑖,𝑖𝑢𝑥_𝑗,𝑗. (41) From (39) and (41), it follows that the total energy of the cell is 𝐸_𝑚𝑢^′=𝑛(𝑛−1)𝑛+2^𝑛−1_𝑖=0𝑢²𝑥_𝑖,𝑖−2𝑛(𝑛+2). (42) This, together with (37), gives 𝐸_𝑚𝑢^′=𝑛𝐸𝑛+2^𝑚−1(𝑢). (43) Taking this result for all the 𝑚−1-cells concludes the proof.

Definition 6. The energy in 𝐏_𝑛 is given by 𝐸(𝑢)=lim_𝑚→∞𝑛+2𝑛^𝑚𝐸_𝑚(𝑢). (44) The domain of 𝐸(⋅) being the space 𝐷^(𝑛) of functions such that the energy is finite. Write 𝐷₀^(𝑛) for the subspace of 𝐷^(𝑛) of functions that vanish on the boundary. The energy product 𝐸(𝑢,𝑣) can be recovered by the polarization identity.

Let 𝜇 be a self-similar measure in 𝐏_𝑛; the Laplacian Δ_𝜇 is given by the following:

Definition 7 (Kigami’s Laplacian). With 𝜇 and Δ_𝜇 as above, one says that 𝑢 is in the domain of Δ_𝜇 if there exists a continuous function 𝑓 such that 𝐸(𝑢,𝑣)=−_𝑃𝑛𝑓𝑣𝑑𝜇,∀𝑣∈𝐷₀^(𝑛). (45) In such case, we define Δ_𝜇𝑢=𝑓.

Aside from the above weak representation, a pointwise formula can be obtained for Δ_𝑚𝑢, proceeding exactly in the same way as in [17] (Theorem  2.2.1). In the case where 𝜇 is the standard measure in 𝑃_𝑛 (i.e., the only Borel regular measure such that the measure of every 𝑚-cell is equal to 𝑛^−𝑚), the pointwise formula is Δ_𝜇𝑢(𝑥)=𝑛2lim_𝑚→∞(𝑛+2)^𝑚Δ_𝑚𝑢(𝑥). (46)

This leads to the following: if a sequence {𝜆_𝑚} is defined recursively by (35) (assuming that 𝜆_𝑚 is never equal to 𝑛,𝑛+2, or 2𝑛) and 𝑢_𝑚 is given by relation (25), then 𝑛𝜆=2lim_𝑚→∞(𝑛+2)^𝑚𝜆_𝑚 (47) is an eigenvalue of Δ_𝜇 with eigenfunction 𝑢 given by the limit 𝑢_𝑚→𝑢. The limit above exists provided that the sign in relation (35) is chosen to be “+” for at most a finite number of times.

Acknowledgments

The authors are very grateful to Alejandro Butanda and Yolanda Ortega for their valuable assistance with the figures and to the referee for the many useful comments.