Sierpi\'nski fractals and the dimension of their Laplacian spectrum

We estabish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated to Sierpi\'nski lattices and infinite Sierpi\'nski gaskets and other post-critically finite self-similar sets.


Introduction
The study of the Laplacian on manifolds has been a very successful area of mathematical analysis for over a century, combining ideas from topology, geometry, probability theory and harmonic analysis. A comparatively new development is the theory of a Laplacian for certain types of naturally occurring fractals, see [29,26,31,9,28,12,21], to name but a few. A particularly well-known example is the following famous set. Definition 1.1. The Sierpiński triangle T ⊂ R 2 (see Figure 1(a)) is the smallest non-empty compact set 1 such that 3 i=1 T i (T) = T where T 1 , T 2 , T 3 : R 2 → R 2 are the affine maps A second object which will play a role is the following infinite graph.
2020 Mathematics subject classification. 28A80, 37F35, 37C30, 65D05 * The authors were partly supported by ERC-Advanced Grant 833802-Resonances. 1 In the literature, this set is also often referred to as the Sierpiński gasket, and denoted SG 2 .
where we use the inverses The definition of V ∞ depends on the choice of ω, however as will be explained below, the relevant results do not, allowing us to omit the dependence in our notation. The points in V ∞ correspond to the vertices of an infinite graph L called a Sierpiński lattice for which the edges correspond to pairs of vertices (v, v ′ ), with v, v ′ ∈ V ∞ such that ∥v − v ′ ∥ 2 = 1 (see Figure  1(b)). Equivalently, L has an edge (v, v ′ ) if and only if v, v ′ ∈ T −1 ω 1 • · · · • T −1 ωn • T in • · · · • T i 1 (V 0 ) for some i 1 , . . . , i n ∈ {1, 2, 3}, n ⩾ 0.
Finally, we will also be interested in infinite Sierpiński gaskets, which can be defined similarly to Sierpiński lattices as follows. Definition 1.3. For a fixed sequence ω = (ω n ) n∈N , we define an infinite Sierpiński gasket to be the unbounded set T ∞ given by which is a countable union of copies of the standard Sierpiński triangle T (see Figure 1(c)).
As for Sierpiński lattices, the definition of T ∞ depends on the choice of ω, but we omit this dependence in our notation as the cited results hold independently of it.
The maps T 1 , T 2 and T 3 are similarities on R 2 with respect to the Euclidean norm, and more precisely ∥T i (x 1 , y 1 ) − T i (x 2 , y 2 )∥ 2 = 1 2 ∥(x 1 , y 1 ) − (x 2 , y 2 )∥ 2 for (x 1 , y 1 ), (x 2 , y 2 ) ∈ R 2 and i = 1, 2, 3, and thus by Moran's theorem the Hausdorff dimension of T has the explicit value dim H (T) = log 3 log 2 [8]. We can easily give the Hausdorff dimensions of the other spaces. It is clear that dim H (L) = 1 and since an infinite Sierpiński gasket T ∞ consists of countably many copies of T it follows that we also have dim H (T ∞ ) = log 3 log 2 . In this note we are concerned with other fractal sets closely associated to the infinite Sierpiński gasket T ∞ and the Sierpiński lattice L, for which the Hausdorff dimensions are significantly more difficult to compute.
In §2 we will describe how to associate to T a Laplacian ∆ T which is a linear operator defined on suitable functions f : T → R. An eigenvalue λ ⩾ 0 for −∆ T on the Sierpiński triangle is then a solution to the basic identity The spectrum σ(−∆ T ) ⊂ R + of −∆ T is a countable set of eigenvalues. In particular, its Hausdorff dimension satisfies dim H (σ(−∆ T )) = 0. A nice account of this theory appears in the survey note of Strichartz [29] and his book [30].
By contrast, in the case of the infinite Sierpiński gasket and the Sierpiński lattice there are associated Laplacians, denoted ∆ T ∞ and ∆ L , respectively, with spectra σ(−∆ T ∞ ) ⊂ R + and σ(−∆ L ) ⊂ R + which are significantly more complicated. In particular, their Hausdorff dimensions are non-zero and therefore their numerical values are of potential interest. However, unlike the case of the dimensions of the original sets T ∞ and L, there is no clear explicit form for this quantity. Fortunately, using thermodynamic methods we can estimate the Hausdorff dimension 2 numerically to very high precision.
A key point in our approach is that we have rigorous bounds, and the value in the above theorem is accurate to the number of decimal places presented. We can actually estimate this Hausdorff dimension to far more decimal places. To illustrate this, in the final section we give an approximation to 100 decimal places.
It may not be immediately obvious what practical information the numerical value of the Hausdorff dimension gives about the sets T ∞ and L but it may have the potential to give an interesting numerical characteristic of the spectra. Beyond pure fractal geometry, the spectra of Laplacians on fractals are also of practical interest, for instance in the study of vibrations in heterogeneous and random media, or the design of so-called fractal antennas [6,10].
We briefly summarize the contents of this note. In §2 we describe some of the background for the Laplacian on the Sierpiński graph. In particular, in §2.3 we recall the basic approach X 0 X 1 X 2 Figure 2: The first three graphs for the Sierpiński triangle.
of decimation which allows σ(∆ T ) to be expressed in terms of a polynomial R T (x). Although we are not directly interested in the zero-dimensional set σ(−∆ T ), the spectra of σ(−∆ T ∞ ) and σ(−∆ L ) actually contain a Cantor set J T ⊂ [0, 5], the so-called Julia set associated to the polynomial R T (x).
As one would expect, other related constructions of fractal sets have similar spectral properties and their dimension can be similarly studied. In §3 we consider higher-dimensional Sierpiński simplices, post-critically finite fractals, and an analogous problem where we consider the spectrum of the Laplacian on infinite graphs (e.g., the Sierpiński graph and the Pascal graph). In §4 we recall the algorithm we used to estimate the dimension and describe its application. This serves to both justify our estimates and also to use them as a way to illustrate a method with wider applications.
2 Spectra of the Laplacians

Energy forms
There are various approaches to defining the Laplacian ∆ T on T. We will use one of the simplest ones, using energy forms.
Following Kigami [12] the definition of the spectrum of the Laplacian for the Sierpiński gasket T involves a natural sequence of finite graphs X n with X 0 ⊂ X 1 ⊂ X 2 ⊂ · · · ⊂ n X n ⊂ n X n =: T, the first three of which are illustrated in Figure 2. To this end, let be the three vertices of X 0 . The vertices of X n can be defined iteratively to be the set of points satisfying We denote by ℓ 2 (V n ) (for n ⩾ 0) the real valued functions f : V n → R (where the ℓ 2 notation is used for consistency with the infinite-dimensional case despite having no special significance for finite sets). Definition 2.1. To each of the finite graphs X n (n ⩾ 0) we can associate bilinear forms E n : ℓ 2 (V n ) × ℓ 2 (V n ) → R called self-similar energy forms given by where x, y ∈ V n are vertices of X n , and x ∼ n y denotes neighbouring edges in X n . In particular, x ∼ n y precisely when there exists x ′ , y ′ ∈ V n−1 and i ∈ {1, 2, 3} such that x = T i (x ′ ) and y = T i (y ′ ). The value c n > 0 denotes a suitable scaling constant. With a slight abuse of notation, we also write E n (f ) := E n (f, f ) for the corresponding quadratic form ℓ 2 (V n ) → R.
To choose the values c n > 0 (for n ⩾ 0) we want the sequence of bilinear forms (E n ) ∞ n=0 to be consistent by asking that for any f n−1 : V n−1 → R (for n ⩾ 1) we have where f n : V n → R denotes an extension which satisfies: The following is shown in [30], for example.
The proof of this lemma is based on solving families of simultaneous equations arising from (a) and (b). We can now define a bilinear form for functions on T using the consistent family of bilinear forms (E n ) ∞ n=0 .

Definition 2.3.
For any continuous function f : T → R we can associate the limit Remark 2.4. We can consider eigenfunctions f ∈ dom(E) which satisfy Dirichlet boundary conditions (i.e., f |V 0 = 0).

Laplacian for T
To define the Laplacian ∆ T the last ingredient is to consider an inner product defined using the natural measure µ on the Sierpiński triangle T.
Definition 2.5. Let µ be the natural measure on T such that where co(V 0 ) is the convex hull of V 0 , i.e., the filled-in triangle.  Figure 3: The polynomial R T (x) and the contracting inverse branches S −1,T and S +1,T for the Sierpiński triangle T.
In particular, µ is the Hausdorff measure for T, and the unique measure on T for which The subspace dom(E) ⊂ L 2 (T, µ) is a Hilbert space. Using the measure µ and the bilinear form E we recall the definition of the Laplacian ∆ T .
Definition 2.6. For u ∈ dom(E) which vanishes on V 0 we can define the Laplacian to be a continuous function Remark 2.7. For each finite graph X n , the spectrum σ(−∆ Xn ) for the graph Laplacian ∆ Xn will consist of a finite number of solutions of the eigenvalue equation This is easy to see because V n is finite and thus the space ℓ 2 (V n ) is finite-dimensional and so the graph Laplacian can be represented as a matrix. There is then an alternative pointwise formulation of the Laplacian of the form
we can associate local inverses (see Figure 3) The process of spectral decimation (see [30, §3.2] or [9]) describes the eigenvalues of −∆ T as renormalized limits of (certain) eigenvalue sequences of −∆ Xn , n ∈ N. These eigenvalues, essentially, follow the recursive equality λ n+1 = S ±1,T (λ n ), while the corresponding eigenfunctions of −∆ X n+1 are such that their restrictions to V n are eigenfunctions for −∆ Xn . Thus, the eigenvalue problem can be solved inductively, constructing solutions f to the eigenvalue equation (2.2) at level n + 1 from solutions at level n ∈ N. The values of f at vertices in V n+1 \ V n are obtained from solving the additional linear equations that arise from the eigenvalue equation ∆ X n+1 f + λf = 0, which allows for exactly two solutions. The exact limiting process giving rise to eigenvalues of −∆ T is described by the following result. Proposition 2.9 ( [9,21,4]). Every solution λ ∈ R to the eigenvalue equation can be written as for a sequence (λ m ) m⩾m 0 and a positive integer c ∈ N 0 satisfying 1. λ m 0 = 2 and c = 0, or λ m 0 = 5 and c ⩾ 1, or λ m 0 = 3 and c ⩾ 2; for all m ⩾ m 0 ; and 3. the limit (2.6) is finite.
Conversely, the limit of every such sequence gives rise to a solution of (2.5).
We remark that equivalently, the sequence (λ m ) m⩾m 0 could be described recursively as λ m+1 = S ϵm,T (λ m ) where ϵ m ∈ {±1} for m ⩾ m 0 . The finiteness of the limit (2.6) is equivalent to there being an m ′ ⩾ m 0 such that ϵ m = −1 for all m ⩾ m ′ .

Spectrum of the Laplacian for Sierpiński lattices
For a Sierpiński lattice, we define the Laplacian ∆ L by which is a well-defined and bounded operator from ℓ 2 (V ∞ ) to itself (this follows from the fact that each vertex of L has at most 4 neighbours). The operator −∆ L : has a more complicated spectrum which depends on the following definition.
Definition 2.11 (cf. [8]). We define the Julia set associated to R T to be the smallest non-empty closed set J T ⊂ [0, 5] such that This leads to the following description of the spectrum σ(−∆ L ).
This immediately leads to the following.
Thus estimating the Hausdorff dimension of the spectrum σ(−∆ L ) is equivalent to estimating that of the Julia set J T . The following provides a related application.
Example 2.14 (Pascal graph). Consider the Pascal graph P [18], which is an infinite 3-regular graph, see Figure 4. Its edges graph is the Sierpiński lattice L, and as was shown by Quint [18], the spectrum σ(−∆ P ) of its Laplacian −∆ P is the union of a countable set and the Julia set of a certain polynomial (affinely) conjugated to R T . From this we deduce that which we estimate in Theorem 1.4.

Spectrum of the Laplacian for infinite Sierpiński gaskets
We finally turn to the case of an infinite Sierpiński gasket T ∞ . The Laplacian ∆ T ∞ is an operator with a domain in L 2 (T ∞ , µ ∞ ). Here µ ∞ is the natural measure on T ∞ , whose restriction to T equals µ, and such that any two isometric sets are of equal measure (see [31]). Remark 2.10 applies almost identically also to the Sierpiński gasket case: T ∞ depends non-trivially on the choice of a sequence ω in its definition, and different sequences typically give rise to non-isometric gaskets, with the boundary of T ∞ empty if and only if ω is eventually constant [31, Lemma 5.1]. The spectrum σ(−∆ T ∞ ), however, is independent of ω (even if the spectral decomposition is not, see [31,Remarks 5.4] or [26, Proposition 1]). Using the notation we have the following result on the spectrum σ(−∆ T ∞ ).
. A number of generalizations of this result for other unbounded nested fractals have been proved, see e.g. [25,27]. The proposition immediately yields the following corollary.

Related results for other gaskets and lattices
In this section we describe other examples of fractal sets to which the same approach can be applied. In practice the computations may be more complicated, but the same basic method still applies.

Higher-dimensional infinite Sierpiński gaskets
Let d ⩾ 2 and T i : R d → R d be contractions defined by where e i is the ith coordinate vector. The d-dimensional Sierpiński gasket T d ⊂ R d is the smallest non-empty closed set such that In [21], the analogous results are presented for the spectrum of the Laplacian ∆ T d associated to the corresponding Sierpiński gasket Definition 3.1. For a sequence (ω n ) n∈N ⊂ {1, . . . , d} N we can define an infinite Sierpiński gasket in d dimensions as As before, we can associate a Julia set J T d and consider its Hausdorff dimension dim H (J T d ). More precisely, in each case, we can consider the decimation polynomial R T d : Let J T d ⊂ [0, 3 + d] be the limit set of these two contractions, i.e., the smallest non-empty closed set such that .
Proof. We can write Thus for x ∈ I 1 we have bounds Similarly, we can define In particular, we can then bound

Post-critically finite self-similar sets
The method of spectral decimation used for the Sierpiński gasket by Fukushima and Shima [9], was extended by Shima [28] to post-critically finite self-similar sets and thus provided a method for analyzing the spectra of their Laplacians.
Let T 1 , . . . , T k : R d → R d be contracting similarities and let X be the limit set, i.e., the smallest closed subset with X = k i=1 T i (X). Let π : Σ → X be the natural continuous map defined by π ((w n ) ∞ n=0 ) = lim n→+∞ T w 0 T w 1 · · · T wn (0).
We say that X is post-critically finite if The original Sierpiński triangle T is an example of a limit set which is post-critically finite. So is the following variant on the Sierpiński triangle.
Example 3.6 (SG 3 gasket). We can consider the Sierpiński gasket SG 3 (see Figure 5) which is the smallest non-empty closed set X SG 3 such that for j = 1, . . . , 6, X 0 X 1 Figure 5: The first two graphs for SG 3 (left, centre) and the SG 3 gasket (right).
In this case we can associate the decimation rational function R SG 3 : [0, 6] → [0, 6] given by for which there are four local inverses S j,SG 3 (for j = 1, 2, 3, 4) [7], see Figure 6. The associated Julia set J SG 3 , which is the smallest non-empty closed set such that Using Mathematica with a sufficiently high precision setting (see Example 4.4 for more details) we can numerically compute the Hausdorff dimension of the Julia set J SG 3 associated to the Sierpiński gasket SG 3 to be

Example 3.7 (Vicsek graph). The
Vicsek set X V is the smallest non-empty closed set such that In this case, studied in [20,Example 6.3], one has that R V : [−1, 0] → R is given by with three inverse branches S 1 , S 2 , S 3 : [−1, 0] → [−1, 0] given by where t(x) = 9 · (81x 2 + 56x − 75) 1/2 + 81x + 28 1/3 . The associated Julia set J V is the smallest non-empty closed set such that J V = 3 j=1 S j,V (J V ). The following theorem is proved similarly to Theorem 1.4, as described in Section 4.   Other examples to which the same method could be applied include the modified Koch curve (see [19], [15]) for which the associated rational function is More families of such examples can be found in [32].
Remark 3.11. The spectral decimation method can also apply to some non-post-critically finite examples, such as the diamond fractal [14], for which the associated polynomial is R(x) = 2x(2 + x). On the other hand, there are symmetric fractal sets which do not admit spectral decimation, such as the pentagasket, as studied in [1].

Dimension estimate algorithm for Theorem 1.4
This section is dedicated to finishing the proof of Theorem 1.4, by describing an algorithm yielding estimates (with rigorous error bounds) for the values of the Hausdorff dimension. By the above discussion we have reduced the estimation of the Hausdorff dimensions of σ(−∆ L ) and σ(−∆ T ∞ ) to that of dim H (J T ) for the limit set J T associated to S ±1,T from (2.4) (and similarly for the other examples). Unfortunately, since the maps S ±1,T are non-linear it is not possible to give an explicit closed form for the value dim H (σ(−∆ T )) = dim H (J T ).
Recently developed simple methods make the numerical estimation of this value relatively easy to implement, which we summarize in the following subsections.

A functional characterization of dimension
where f ∈ B and x ∈ I, and S ±,T are as in (2.4).
It is well known that the transfer operator L t (for t ⩾ 0) is a well-defined positive bounded operator from B to itself. To make use of the results in the previous sections, we employ the following "min-max method" result: 17]). Given choices of 0 < t 0 < t 1 < 1 and strictly positive continuous functions f, g : Proof. We briefly recall the proof. We require the following standard properties.
1. For any t ⩾ 0 the operator L t has a maximal positive simple eigenvalue e P (t) (with positive eigenfunction), where P is the pressure function [23,16].
By property 1. and the first inequality in (4.1) we can deduce that By property 1. and the second inequality in (4.1) we can deduce that Comparing properties 2. and 3. with (4.2) and (4.3), the result follows.

Rigorous verification of minmax inequalities
Next, we explain how we rigorously verify the conditions of Lemma 4.2 for a function f : I → R + , that is, In order to obtain rigorous results we make use of the arbitrary precision ball arithmetic library Arb [11], which for a given interval [c−r, c+r] and function f outputs an interval Clearly, the smaller the size of the input interval, the tighter the bounds on its image. Thus, in order to verify the above conditions we partition the interval I adaptively using a bisection method up to depth k ∈ N 0 into at most 2 k subintervals, and check these conditions on each subinterval. While the first condition is often immediately satisfied for chosen test functions f on the whole interval I, the second condition is much harder to check as h is very close to 1 and would require very large depth k.
To counteract the exponential growth of the number of required subintervals, we use tighter bounds on the image of h. Clearly for x ∈ [c − r, c + r] with c ∈ R and r > 0, we have that |h(x) − h(c)| ⩽ sup y∈[c−r,c+r] |h ′ (y)|r by the mean value theorem. More generally, we obtain for p ∈ N that This allows to achieve substantially tighter bounds on h([c − r, c + r]) while using a moderate number of subintervals, at the cost of additionally computing the first p derivatives of h.

Choice of f and g via an interpolation method
Here we explain how to choose suitable functions f and g for use in Lemma 4.2, so that given candidate values t 0 < t 1 we can confirm that t 0 < dim H (J T ) < t 1 . Clearly, if f and g are eigenfunctions of L t 0 and L t 1 for the eigenvalues λ t 0 and λ t 1 , respectively, then condition (4.2) is easy to check. As these eigenfunctions are not known explicitly, we will use the Lagrange-Chebyshev interpolation method to approximate the respective transfer operators by finite-rank operators of rank m, and thus obtain approximations f (m) and g (m) of f and g. If the choices f = f (m) and g = g (m) satisfy the hypotheses of Lemma 4.2 (which can be checked rigorously with the method in the previous section) then we proceed to the next step. If they do not, we increase m and try again.
Conclusion. When the hypothesis of Lemma 4.2 holds then its assertion confirms that t 0 < dim H (J T ) < t 1 .
It remains to iteratively make the best possible choices of t 0 < t 1 using the following approach.

The bisection method
Fix ϵ > 0. We can combine the above method of choosing f and g with a bisection method to improve given lower and upper bounds t 0 and t 1 until the latter are ϵ-close: Initial choice. First we can set t 1 is trivially true.
Iterative step. Given n ⩾ 0 we assume that we have chosen t     Applying this algorithm yields the proof of Theorem 1.4 (and with the obvious modifications also those of Theorems 3.2 and 3.8). Specifically, we computed the value of dim H (J T ) efficiently to the 14 decimal places as stated with the above method, by setting ϵ = 10 −15 , using finite-rank approximation up to rank m = 30, running interval bisections for rigorous minmax inequality verification up to depth k = 18, i.e. using up to 2 18 subintervals, and using p = 2 derivatives. There are of course many ways to improve accuracy further, e.g., with more computation or the use of higher derivatives.

Conclusion
In this note, we have leveraged the existing theory on Laplacians associated to Sierpiński lattices, infinite Sierpiński gaskets and other post-critically finite self-similar sets, in order to establish the Hausdorff dimensions of their respective spectra. We used the insight that, by virtue of the iterative description of these spectra, these dimensions coincide with those of the Julia sets of certain rational functions. Since the contractive local inverse branches of these functions are non-linear, the values of the Hausdorff dimensions are not available in an explicit closed form, in contrast to the dimensions of the (infinite) Sierpiński gaskets themselves, or other self-similar fractals constructing using contracting similarities and satisfying an open set condition. Therefore we use the fact that the Hausdorff dimension can be expressed implicitly as the unique zero of a so-called pressure function, which itself corresponds to the maximal positive simple eigenvalue of a family of positive transfer operators. Using a min-max method combined with the Lagrange-Chebyshev interpolation scheme we can rigorously estimate the leading eigenvalues for every operator in this family. Combined with a bisection method we then accurately and efficiently estimate the zeros of the respective pressure functions, yielding rigorous and effective bounds on the Hausdorff dimensions of the spectra of the relevant Laplacians.