The"Hot Spots"Conjecture on the Vicsek Set

We prove the Hot Spot conjecture on the Vicsek set. Specifically, we show that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary.


Introduction
The "hot spots" conjecture studies whether a flat piece of metal that is given an initial heat distribution will achieve its highest temperatures on its boundary given enough time. That is, the conjecture claims that in a two-dimensional, bounded, connected domain D, the heat at point x at time t, u(x, t), achieves its maximum value on the boundary of D. The "hot spots" conjecture was first posed by Rauch in 1974. An equivalent formulation of the conjecture is as follows: every eigenfunction of the second eigenvalue of the Neumann Laplacian attains its maximum and minimum on the boundary. The conjecture has been shown to be true for some Euclidean domains [1][2][3][4] including recently for thin curved strips [5], but it has also been shown to fail in others [6,7].
There is now a notion of a Laplacian on many fractals, and the theory of eigenfunctions of the Laplacian is well developed in many cases. Therefore, one can formulate the "hot spot" conjecture on these fractals. We are going to use the theory developed by Kigami [8], see also [9] that applies to the class of post critically finite (p.c.f.) fractals. For many such fractals, eigenvalues and eigenfunctions of the Laplacian can be computed explicitly via a method called spectral decimation [10][11][12][13]. The "hot spots" conjecture has been shown to hold on the Sierpinski gasket and higher dimensional variants [14][15][16] but fail on the hexagasket fractal [17]. The hexagasket fractal is determined by an iterated function system consisting of six contractions. However, the analytic boundary in the sense of Kigami studied in [17] consists of only three of the six fixed points of the iterated function system. On the other hand, the boundary of the Sierpinski gasket and its higher dimensional variants mentioned above consists of all of the fixed points of the iterated function system that determines the gasket. Therefore, one might wonder whether the failure of the "hot spots" conjecture on the hexagasket fractal might be due to the "smaller" boundary considered in [17]. The Vicsek set [18][19][20] is another type of fractal that has been studied heavily. Zhou [21] described the spectral decimation on a family of Vicsek sets, VSn, in terms of Chebyshev polynomials, and the authors of [22] used Zhou's formulas to study the Laplacian and spectral operators on the Vicsek set. We study in this paper the Vicsek set VS 2 that is generated by five contractions; however, its analytic boundary consists of only four of the five fixed points of the iterated function systems. Our main theorem states that, unlike the hexagasket, the "hot spots" conjecture on the Vicsek set is true. The proof of the main theorem is inspired by proofs in [15,16] and [14]. It is, however, more involved and technical.
The organization of the paper is as follows: the second part of the introduction contains a background on the Vicsek set VS 2 , and the energy and the Laplacian on the Vicsek set. In section 2, we review the Neumann Laplacian on VS 2 , and show how to use the spectral decimation to determine the second smallest eigenvalue of the Neumann Laplacian together with a basis of its eigenspace. Section 3 contains our main theorem. The proof of the theorem follows relatively easily from Lemma 3.2 by an argument similar to the one in [15]. The proof of the lemma, however, is very long and technical and occupies the entirety of Section 4. We placed the statement and proof of some formulas used throughout the paper in the Appendix in order to help with the readability of Section 4.

Background
We begin by reviewing several concepts in analysis on fractals as applied to the Vicsek set. First, an iterated function system on a complete metric space X is a finite set of contraction mappings F i : X → X, i = 1, . . . , n [23,24]. Given such an iterated function system there exists a unique compact invariant set K ⊆ X; that is, K satisfies the following self-similar property, The main object of study in this paper is the second order Vicsek set, VS 2 , which is the unique invariant subset of R 2 of the iterated function system defined by the following five similarities: where p 1 = (0, 1), p 2 = (1, 1), p 3 = (1, 0), p 4 = (0, 0), and p 5 = (1/2, 1/2). A picture of the Vicsek set is provided in the following figure. The Vicsek set VS 2 is an example of what in the literature is called a p.c.f fractal and, thus, Kigami's theory [8] (see also [9]) applies to VS 2 . We describe next how this theory is applied to the Vicsek set in order to define the standard energy and Laplacian on VS 2 . A useful feature of the Vicsek is that it can be approximated by an increasing sequence of graphs, Γm as follows: the level zero graph approximation of the Vicsek set, Γ 0 , shown below, consists of the set of vertices V 0 at q 1 = (0, 1), q 2 = (1, 1), q 3 = (1, 0), and q 4 = (0, 0) connected as in a complete graph. Note that F i (q i ) = q i , where i = 1, 2, 3, 4. The set V 0 is also the boundary of the fractal in the sense of [8,21]. We call Γ 0 the graph 0-cell. The level 1 graph approximation of the Vicsek set, Γ 1 is  constructed as follows: each of the five scaled versions of the level 0 Vicsek set which comprise Γ 1 can be obtained by applying F 1 (Γ 0 ), F 2 (Γ 0 ), etc. Therefore and, in particular, the set of vertices of Γ 1 equals . We call the sets F i (Γ 0 ) graph 1-cells. We note that there are five graph 1-cells. In the general case, for m ≥ 1, the level m + 1 graph approximation Γ m+1 is obtained from the level m graph approximation Γm via The image under the maps F i of the graph m-cells form the graph (m +1)-cells. We note that each graph m-cell contains five graph (m + 1)-cells. That is, when going from level m to level m + 1, a graph m-cell is going to be replaced by five graph (m + 1)-cells. We write Vm as the set of vertices at the level m graph approximation. Inductively, it can be shown that Vm ⊂ V m+1 , and that ⋃︀ ∞ m=0 Vm is a dense subset of VS 2 . Thus, it suffices to study continuous functions on ⋃︀ ∞ m=0 Vm and extend them via continuity to VS 2 . As detailed by [9], we can define graph energy at level m for the Vicsek set VS 2 as follows: where x ∼ y means that x and y are neighboring vertices in Γm (that is, there is an edge in Γm joining x and y). Energy generally requires renormalization with a given renormalization factor. For VS 2 , the renormalization factor is equal to 3 [21]. Thus, for VS 2 , the renormalized graph energy is Em(u) = 3 −m Em(u). One can define the fractal's energy, E(u), via with the domain of the energy, dom E, consisting of all continuous functions u on VS 2 such that E(u) < ∞. As detailed in [8,9,21], the choice of the renormalization constant in the definition of Em guarantees that the domain of the energy is non-trivial. Now, knowing that the energy exists and can be written as above, we know from [8,9] that we can determine a fractal's Laplacian. In domE, E extends via the polarization formula to the bilinear function E(u, v). We consider the standard invariant measure µ on VS 2 ; that is, the unique measure that satisfies the following property (see [24] and [23] for details): Define the weak formulation of the Laplacian as follows [8,9]: we say that a function u ∈ dom E belongs to the domain dom ∆ of the Laplacian if there is a continuous function f on VS 2 such that for all v ∈ dom 0 E := {v ∈ domE : v| V0 = 0}. In this case we write ∆u = f . As proven in [21], there exists an equivalent pointwise formula for the Laplacian on the Vicsek set. This formula is given as the normalized limit of a series of graph approximations and is written as follows for VS 2 : for all x not in the boundary V 0 . Here the graph Laplacian of the graph Γm, denoted by ∆m, is defined by for all x ∈ Vm \ V 0 , where degx represents the number of neighbors of x in Γm. The degree of a vertex x in Γm is either 3 or 6 for all m ≥ 1.

Neumann Laplacian and spectral decimation
We study the Neumann Laplacian in this paper. That is, the Laplacian as defined with Neumann boundary conditions. Neumann boundary conditions behave such that Equation (2) holds for all x ∈ Vm, including the boundary V 0 . The Neumann Laplacian of the level 0 graph approximation Γ 0 for VS 2 is then given by the following matrix: The 1/3 scaling factor outside of the matrix is derived from the 1/degx that appears in front of the summation in the equation for ∆m u(x).
In order to study eigenvalues and eigenfunctions on VS 2 , we use the process of spectral decimation as described in [9,21] that we review next. First, there is a local extension algorithm which shows a unique way to extend a function u that satisfies the eigenvalue equation −∆m u = λm u on Vm \ V 0 to a function that we still denote by u that satisfies the eigenvalue equation −∆ m+1 u = λ m+1 u on V m+1 \ V 0 . Moreover there exists a rational function R(λ) such that λm = R(λ m+1 ) if λm is not a forbidden eigenvalue. That is, a singularity of the function R. It is "forbidden" to decimate to such eigenvalues. Because forbidden eigenvalues do not have a predecessor, i.e. there is no λ m−1 corresponding to λm, we say that forbidden eigenvalues are "born" at a level of approximation m. Spectral decimation for VS 2 is performed as follows. First, define where T 2 and U 2 represent the Chebyshev polynomials of the 1st and 2nd kind, i.e. T 1 (λ) = λ, T 2 (λ) = 2λ 2 − 1, U 0 (λ) = 1, and U 1 (x) = 2λ. Therefore, f 2 (λ) = 18λ 2 − 21λ + 4, g 2 (λ) = 6λ − 3, and h 2 (λ) = 6λ − 5. Zhou proved in [21] (see also [9,22]) that the spectral decimation function R is Additionally, the forbidden eigenvalues of VS 2 are 4/3 and the zeroes of f 2 and g 2 , 0, 1/2, and (7 ± 17 0.5 )/12. The extension of eigenfunctions of VS 2 from one level to the next is given by [22]: where Note that a, b, c, d and are functions of λ. Hereafter λ is any number that is not a forbidden eigenvalue of with the degrees of every vertex as its diagonal entries, and M is a diagonal matrix such that M ii = -X ii . Going from Γm to Γ m+1 , this matrix is applied to the vertices of one graph m-cell to obtain the values on the five new graph (m + 1)-subcells. We follow the convention from [22, Figure 4] when it comes to the labeling of the columns of the matrix (5). That is, the first column corresponds to the point q 5 = F 1 (q 2 ), the second column corresponds to the point q 6 = F 1 (q 3 ) = F 5 (q 1 ), the third column corresponds to q 7 = F 1 (q 4 ), the fourth column corresponds to q 8 = F 2 (q 1 ) and so on. To better understand the meaning of the matrix (5), let λ 0 be an eigenvalue of ∆ 0 , and let u be a λ 0 eigenvector on V 0 . To simplify the notation, write u(q i ) = u i , i = 1, 2, 3, 4. Let λ 1 be a solution of R(λ) = λ 0 . Then u is extended to a λ 1 eigenvector on V 1 using (5) as described next. First, u| V0 does not change. The extension of u to q 5 = F 1 (q 2 ) is computed using the first column of (5) as follows: where , a, and c are evaluated at λ 1 . The extension of u to q 6 is computed using the second column of (5) as follows: where , b, and d are evaluated at λ 1 . The computation of the extension of u to the remaining vertices in V 1 is computed based on the corresponding columns in (5). The eigenvalue extension function R(λ) = 36λ 3 − 48λ 2 + 15λ has three local inverses [22]. Let ϕ 1 , ϕ 2 , ϕ 3 denote the inverse functions of R in increasing order; that is, ϕ 1 is the inverse of R(λ) on the interval (0, 8−19 0. 5 18 ), ϕ 2 is the inverse of R(λ) on ( 8−19 0.5 18 , 8+19 0.5 18 ), and ϕ 3 is the inverse of R(λ) on ( 8+19 0.5 18 , 1). Observe ϕ 1 (x) < ϕ 2 (y) < ϕ 3 (z) for all x, y, z in the corresponding domain, and ϕ 2 is decreasing while ϕ 1 and ϕ 3 are increasing. Let ρ 2 = 15 be the renormalization factor for the Laplacian. The Neumann eigenvalues are nonnegative and they accumulate at ∞. Then the rules for spectral decimation in the case of VS 2 are summarized as follows [22]: The existence of the limit is proven in [21]. In the first case, the eigenvalue is in the 0-series, and in the second series the eigenvalue is in the 4/3 series born on level k. 2. All but a finite number of the ωm are equal to 1. 3. For the 0-series, the first ω j with ω j ≠ 1 must be an odd number, and for the 4/3 series, ω 1 must be an odd number but ω 1 ≠ 3. 4. The multiplicity of each eigenvalue in the 0-series is 1, while the multiplicity of each 4/3-series eigenvalue on level k is 2(5) k + 1.
In the remainder of the paper, we use the spectral decimation as applied to the following setup. Let λ 0 = 4/3 be the second smallest eigenvalue of ∆ 0 . Then λ 0 = 4/3 has multiplicity 3 and a basis for its eigenspace on V 0 is given by the following eigenvectors: Extend λ 0 at all levels along ϕ 1 . That is, define and define We extend u 1 , u 2 and u 3 to eigenvectors of λ (2) on the Vicsek set via the spectral decimation. Figure 4 presents the extension of u 1 from V 0 to V 2 using the spectral decimation described above. The following important fact follows from [22, Theorem 2.2]. Proposition 2.1. λ (2) is the 2nd smallest eigenvalue of the Neumann Laplacian on the Vicsek set.

The "hot spots" conjecture for the Vicsek set
We are now able to state the main theorem. The proof of the theorem follows from the main lemma 3.2. The proof of this lemma is long and technical, and it will occupy the entirety of the next section. Our main result states that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary. Our approach is based on techniques from [15] and [16]. Our computations, however, are much more involved due to the complexity of the spectral decimation matrix (5).

Theorem 3.1 (Main Theorem).
Let VS 2 be the Vicsek set and ∆ be the Neumann Laplacian as described in the previous section. Then every eigenfunction of the second smallest eigenvalue λ (2) of ∆ attains its minimum and maximum value on the boundary V 0 .
In order to prove this theorem, we begin by recalling the space of finite words that the five iterated functions (1) of the Vicsek set generate. Let Σ = {1, 2, 3, 4, 5} be the corresponding alphabet and let Σ m = {ω 1 . . . ω j . . . ωm | j ∈ {1, 2, 3, 4, 5} } be the set of words of length m. Note that in order to simplify the notation we do not separate the letters in a word by commas. So, for example, we write 14 for the word of length 2 that is formed with the letters 1 and 4. Define Σ * = ⋃︀ ∞ m=0 Σ m as the set of all finite words. For every ω ∈ Σ m , we write |ω| := m for the length of ω. Let ∅ denote the empty word and |∅| = 0. Furthermore, Σ 0 = {∅}. If ω, ν ∈ Σ * then we write ων for the word obtained by concatenating the two words ω and ν together. In particular, we will create words by adding just one letter. For example, ωi is the word that adds the letter i at the end of ω. If all the letters of ω are the same, say equal to i ∈ {1, 2, 3, 4, 5}, then we write ω as [i] m , where m is the length of ω. Recall Vm forms a dense subset of VS 2 .
Following some ideas from [15], we use the bases that we picked to define a partition of unity on Σ * . Specifically, we define the functions f , g, h, and k with domain Σ * via: Then they satisfy the following crucial lemma.
where δ ij is the Kronecker-Delta function.
As mentioned at the beginning of the section, the proof of the lemma is long, technical, and will occupy the next section of this paper. From the lemma, the theorem is proven as follows.
Proof of Theorem 3.1. Notice that the functions f , g, h, and k are related as follows: Let u ∈ EF 2 . Then there exist constants c 1 , c 2 , c 3 such that u(x) = c 1 u 1 (x) + c 2 u 2 (x) + c 3 u 3 (x) for all x ∈ VS 2 because u 1 , u 2 , u 3 form a basis for EF 2 . It follows that

Proof of Lemma 3.2
The proof of Lemma 3.2 will occupy the rest of this paper. We provide first a short summary of the proof in order to help the reader navigate through the many lemmas that follow. First, an easy proof shows that f (∅, i) = δ 1i (Lemma 4.1). We observe that it suffices to prove the statement for f for the words ω that begin only with the letters 1, 2, and 5, because f restricted to words that begin with 3 and 4 equals a "rotation" of f restricted to words that begin with 2 (see Remark 4.4). The crucial Lemma 4.2 proves recursive formulas for f (ω, j) when one increases the length of ω by 1. Using these formulas we are able to compute explicitly f (ω, j) for words of the form ω = [i] m = ii . . . i (m-times). Using these explicit formulas, we prove in Proposition 4.8 that the minimum of f is 0 and the maximum of f is 1. We finish the proof of Lemma 3.2 by describing how to recover the same results for g, h, and k from the results proved for f . To improve the readability of this section, we leave the statements and proofs of some useful formulas until the Appendix (see Lemma A.1).
We continue to use the notation described in the previous section. In particular, f , g, h and k are the functions defined in (9), (10), (11), and (12). Recall that we do not separate the letters in a word ω ∈ Σ * . Therefore, 123 is the word of length 3 that consists of the letters 1, 2 and 3 (we read this word "one, two, three", as opposed to "one hundred twenty three"). This word will correspond to the composition F 1 ∘ F 2 ∘ F 3 .
We break the proof of the main Lemma 3.2 into a series of lemmas. We begin with the easier part.
Hence, f (∅, i) = δ 1,i . Similar statements hold for g and h by symmetry, and similar computation shows that the result also holds for k.
The hard part that remains is to prove that each of these functions is between 0 and 1. First, we begin by proving that f (ωi, i) = f (ω, i) and describe recursive relations satisfied by f .
As an immediate consequence of Lemma 4.2, we obtain the following.
We call R 1 and R 2 "rotations" since, if we view f defined on VS 2 via the projection π : Σ * → VS 2 defined by π(ω, i) := q ω,i , then R 1 flips ∪ m≥0 Vm around the diagonal going from q 1 to q 3 , and R 2 rotates the 2-cell F 2 (∪ m≥0 Vm) by 90 ∘ and moves it into the 2-cell F 3 (∪ m≥0 Vm).
There is a permutation that rotates F 2 (∪ m≥0 Vm) and moves it into the 2-cell F 4 (∪ m≥0 Vm). As a consequence, in the following we will only consider words ω ∈ Σ * whose first letter is either 1, 2 or 5.
Next  Proof. We prove the lemma by induction. First, consider ω = ∅ in (18) and (22), and, hence, m = 1. We obtain Recall from (16) and A.1 that Then notice from Lemma A.1 that . Therefore, Factoring out 3 4 λm and the denominator we obtain We now use the following relationship Therefore, . Therefore, The proof that f ([2] m+1 , 1) = f ([2] m+1 , 3) = 0 follows as below: The induction is now complete and the lemma is proved.
If i = 2, the possible values of f (2ω, 1), f (2ω, 2), f (2ω, 3) and f (2ω, 4) are: three of them equal 0, and one equals 3λm /4; three of them equal 3λm /4 and the fourth one is between 3λm /4 and 3λ m+1 /4; and one of the values is 3λ i /4 for some 2 ≤ i ≤ m −1, and the other three values are between 3λm /4 and 3λ i /4. Therefore, the arguments given above for i = 2 and |ω| = 1 can be easily adapted to these cases to prove the inductive step.
If i = 5 then the possible values of f (5ω, 1), f (5ω, 2), f (5ω, 3) and f (5ω, 4): all four are given by Lemma 4.7 and, in particular, three of them are equal; and three of them are equal and all of them are in between the values provided by Lemma 4.7. Then the above argument for i = 5 and |ω| = 1 can also be adapted for the induction step on the length of ω.

Appendix: A few facts used in the proofs
We now collect several relationships and formulas satisfied by α, β, χ and δ that we used throughout the paper. The proofs of the following statements are straight computations and/or easy calculus problems.