Laplacian, on the Arrowhead Curve


In terms of analysis on fractals, the Sierpinski gasket stands out as one of the most studied example. 
The underlying aim of those studies is to determine a differential operator equivalent to the classic Laplacian. 
The classically adopted approach is a bidimensional one, through a sequence of so-called prefractals, i.e. a sequence of graphs that converges towards the considered domain. 
The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on the prefractals. 
 
It turns out that the gasket is also the image of a Peano curve, the so-called Arrowhead one, obtained by means of similarities from a starting point which is the unit line. 
This raises a question that appears of interest. Dirichlet forms solely depend on the topology of the domain, and not of its geometry. 
Which means that, if one aims at building a Laplacian on a fractal domain as the aforementioned curve, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account its specific geometry. 
 
Another difference due to the geometry, is encountered may one want to build a specific measure. 
For memory, the sub-cells of the Kigami and Strichartz approach are triangular and closed: the similarities at stake in the building of the Curve called for semi-closed trapezoids. 
As far as we know, and until now, such an approach is not a common one, and does not appear in such a context. 
 
It intererestingly happens that the measure we choose corresponds, in a sense, to the natural counting measure on the curve. 
Also, it is in perfect accordance with the one used in the Kigami and Strichartz approach. 
In doing so, we make the comparison -- and the link -- between three different approaches, that enable one to obtain the Laplacian on the arrowhead curve: the natural method; the Kigami and Strichartz approach, using decimation; the Mosco approach. 
  
 
 


INTRODUCTION
In [11,12], Jun Kigami has laid the foundations of what is now known as differential calculus on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpiński gasket stands out of the best known example. It has, since then, been taken up, developed and popularized by Robert S. Strichartz [20,21].
The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on a sequence of graphs that converges towards the considered domain. It is these Dirichlet forms that enable one to obtain energy forms on this domain.
It is interesting to note that the existing results mainly deal with nested fractals, generated by an iterated function system with at least two fixed points, connected, symmetric (a classical example is the Vicsek snowflake), or post-critically finite (p.c.f.) ones. Both categories have in common to be self-similar.
A step forward has been made by Uta Freiberg and Maria Rosaria Lancia [3], where the authors build an energy form on non-self similar closed fractal curves, by integrating the Lagrangian on these curves, following the works of Umberto Mosco [15].
Yet, things are not that simple, and interesting questions arise on the scene. One may wonder what happens when a fractal set can be obtained by two different processes, first, by means of an iterated function system, second, by means of a L-system where the same self-similarity no longer seems to hold, as it is the case of the Sierpiński gasket?
The Sierpiński gasket is, in fact, the limit of the so-called Sierpiński arrowhead curve (see [14, plate 141] and [16, page 346]). First, the curve has a completely different topology, second, contrary to the triangle, it can be thought to belong to the category of Moran fractals (we refer to [2,9]), since, at first sight (only), it is not self-similar as the gasket. Yet, in so far as the limit curve reaches each point of the gasket, and as explained by Benoît Mandelbrot, by "fusing together four appropriately neighboring tiles, one gets a tile increased in the ratio of 2", it appears legitimate to consider the limit curve as self-similar. There does not seem to be other references to the arrowhead curve in the literature, and none dealing with the building of a Laplacian on such a curve.
This brings us to a question that appears of interest. Dirichlet forms solely depend on the topology of the domain, and not of its geometry. Which means that, if one aims at building a Laplacian on a fractal domain as the aforementioned curve, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account such a specific geometry.
Building a Laplacian on the arrowhead curve has thus led us to use methods that are not so standard. May one want to follow the work of J. Kigami and R. S. Strichartz, it requires a measure. J. Kigami and R. S. Strichartz choose to build measures that were more like the Riemann or Cauchy ones, rather than the Lebesgue one. To this purpose, they introduced, first, a regular probability measure, where each subcell of level m P N ‹ , a triangular one, has measure 1 3 m . This measure happened to coincide, up to a constant, to the Hausdorff measure of the triangle. It is worth noting that the considered integral is thus a two dimensional one.
The Sierpiński gasket can be considered as a curve: it is, as evoked in the above, the limit of the so-called Sierpiński arrowhead curve. The point is that, generally, the Hausdorff measure is not suited for integration along fractal curves, due to dissimilarities between its properties, and the one of integrals along curves, as it is well explained in the paper by Boris Kats [10]: for a fractal set F of the complex plane C, the Hausdorff dimension of which is α P t1, 2u, the related Hausdorff measure m α is such that the function is continuous on the whole complex plane, whereas the integral has unit gaps on F. For whom aims at studying diffusion in such media, it appears as more appropriate to consider the integral over the whole area delimited by the curve, rather than the one along the curve.
To this purpose, we found it all the more natural to introduce a sequence of trapezes (T mj ) mPN , 1 ď j ď 3 m´1 , delimitating a trapezoidal domain infinitely close to the arrowhead curve SG C , D(SG C ). In the literature, one can already find approximating polygons, for instance in the case of the Peano curve, as introduced by W. Wunderlich [24]. Such a notion was then adopted by H. Sagan [18,19].
It also seemed natural, in the sense that it enables one to take into account the trapezoidal replicated patterns that appear thanks to the curve, patterns, the measure of which plays the part of a pound. This joins the seminal work of J. Harrison et al. [7,8,6], in the spirit of the one by H. Withney [23], and the remark of U. Mosco [15], where the author suggests, in the case of the Sierpiński curve, to "fill each small simplex (...) not only with its edges, but with the whole portion of the limit curve which it encompasses." Again, one encounters a difference due to the geometry, since the sub-cells of the Kigami and Strichartz approach are triangular and closed, whereas ours are semi-closed trapezoids. As far as we know, and until now, such an approach is not a common one, and does not appear in such a context.
It interestingly happens that the measure we choose corresponds, in a sense, to the natural counting measure on the curve. Also, it is in perfect accordance with the one used in the Kigami and Strichartz approach, as we will see it further. In doing so, we make the comparison -and the linkbetween three different approaches, that enable one to obtain the Laplacian on the arrowhead curve: the natural method; the Kigami and Strichartz approach [13,21], using decimation; the Mosco approach [15].
Thus, we aim at investigating the links between energy forms and geometry that can arise in such cases. Does one obtain the same Laplacian as for the triangle? The question appears as worth to be investigated.

FRAMEWORK OF THE STUDY
In the sequel, we place ourselves in the Euclidean plane of dimension 2, referred to a direct orthonormal frame. The usual Cartesian coordinates are (x, y). We will denote by N the set of natural integers, and set: Definition 2.1. Given a strictly positive integer k, two integers a and b will be called congruent modulo k if a´b is divisible by k. The congruence Laplacian, on the Arrowhead Curve 5 modulo k between a and b will be denoted: Notation 2.2. We will denote by: ln 3 ln 2 the box dimension (equal to the Hausdorff dimension), of the Sierpiński arrowhead curve SG C . For the sake of simplicity, we will from now on denote it by D SG . Notation 2.3. Given a point X P R 2 , we will denote by: the similarity of ratio 1 2 , the center of which is X and the angle π 3 ; ii) Sim X, 1 2 ,´π 3 the similarity of ratio 1 2 , the center of which is X and the angle´π 3 .

Definition 2.4.
Consider the following points of R 2 : We will denote by V 1 the ordered set, of the points: tA, B, C, Du.
The set of points V 1 , where A is linked to B, B is linked to C, and where C is linked to D, constitutes an oriented graph, that we will denote by SG C 1 . The set V 1 is called the set of vertices of the graph SG C 1 . Let us build by induction the sequence of points: and ℓ P Nz2 N. The set of points V m , where two consecutive points are linked, is an oriented graph, which we will denote by SG C m . The set V m is called the set of vertices of the graph SG C m . 6 Claire David Property 2.5. For any strictly positive integer m: Property 2.6. If one denotes by (SG m ) mPN the sequence of graphs that approximate the Sierpiński gasket SG, then, for any strictly positive integer m: SG C m Ĺ SG m . Definition 2.7. We will denote by SG C the limit, in the space of continuous functions from the interval to the plane: This limit will be called the Sierpiński arrowhead curve.
Property 2.8. Denote by SG the Sierpiński gasket. Then: The sequence of graphs (SG m ) mPN ‹ can also be seen as a Lindenmayer system ("L-system"), i.e. a set (V, ω, P ), where V denotes an alphabet (or, equivalently, the set of constant elements and rules, and variables), ω, the initial state (also called "axiom"), and P , the production rules, which are to be applied, iteratively, to the initial state.
In the case of the Sierpiński arrowhead curve, if one denotes by: i) F the rule: "Draw forward, on one unit length"; ii) + the rule: "Turn left, with an angle of π 3 "; iii)´the rule: "Turn right, with an angle of π 3 "; then: i) the variables can be denoted by X and Y ; ii) the constants are F , +,´; iii) the initial state is XF ; iv) the production rules are: Notation 2.10. Given a point X P R 2 , we will denote by Sim X, 1 2 ,0 the homothety of ratio 1 2 , the center of which is X. Property 2.11 (Self-similarity properties of Sierpiński arrowhead curve). Denote by E the point of R 2 such that A, D and E are the consecutive vertices of a direct equilateral triangle. One may note that A, D and E are, also, the frontier vertices of the Sierpiński gasket SG. The Sierpiński Laplacian, on the Arrowhead Curve 7 The graph SG C

1
The graph SG C 2 The graph SG C

4
The graph SG C 7 FIGURE 2.1.
arrowhead curve is self similar with the three homotheties: Proof. The result comes from the self-similarity of the Sierpiński gasket with respect to those homotethies:  This leads to: One has to consider those polygons as semi-closed ones, since, for any of those 4-gons, the starting vertex, i.e. the point number j, is not connected, on the graph SG C m , to the extreme one, i.e. the point number j´3, if j "´1 [4]. These trapezes generate a Borel set of R 2 .
In the sequel, we will denote by T 1 the initial trapeze, the vertices of which are, respectively: A, B, C, D. Remark 2.17. The trapezoids T m,j , 1 ď j ď 3 m´1 , are required in order to define a two-dimensional measure, as it will be exposed in the sequel.

Property 2.18.
In the sequel, we set, for the Lebesgue measure of the first trapeze T 1 :   By considering the prefractal graphs that approximate the Sierpiński gasket, as in [21] (see   1 3 m , then, for any natural integer m ě 2, the measure of a trapeze T m,j , 1 ď j ď 3 m´1 is given by: This leads to the following correspondence, which, for the sake of clarity, is given in Table 2.1.

Definition 2.19.
We will call trapezoidal domain delimited by the graph SG C , and denote by D(SG C ), the limit: in the sense that, for any continuous function u on the graph, and any measure with full support µ on R 2 : where: Laplacian, on the Arrowhead Curve 11 ‚ p(X) = 1 if X is not a junction point between two consecutive trapezes; ‚ p(X) = 1 2 if X is the junction point between two consecutive trapezes P m,j and P m,j+1 . Notation 2.20. In the sequel, we will denote by d R 2 the Euclidean distance on R 2 . Definition 2.21 (Edge relation, on the graph SG C ). Given a natural integer m, two points X and Y of SG C m will be called adjacent if and only if X and Y are two consecutive vertices of SG C m . We will write: Given two points X and Y of the graph SG C , we will say that X and Y are adjacent if and only if there exists a natural integer m such that:

THE LAPLACIAN, BY MEANS OF A NATURAL APPROACH
where ℓ m = 1 2 m and c m is a strictly positive constant.
Since: #V m = 3 m+1 + 1, the total length is thus: This amounts to place K m = 3 m+1 + 1 points on a circle, of perimeter: For m P N, the set of points on the circle is of the form: Given a continuous function u defined on V m , we set for k P t0, . . . , K m´1 u: Km Thus: Km and: For k P t0, . . . , K m´1 u: The spectrum of the normalized discrete Laplacian is thus: For j in t0, . . . , K m´1 u, when m Ñ +8: The existence of the limit of the spectrum, when the integer m tends towards infinity, requires: One thus retrieves, as it could have been expected, the value of the Hausdorff dimension of the Sierpiński gasket: (1) For any real-valued function u defined on D: E(u, u) ě 0.
(2) D, equipped with the inner product which, to any pair (u, v) of DˆD, associates: is a Hilbert space. (3) For any real-valued function u defined on D, if:  A Dirichlet form on V is a symmetric bilinear form E, such that: (1) For any real valued function u defined on V : E(u, u) ě 0. i.e.
4.4. Energy forms. Before introducing energy forms on our curve, we would like to recall that, a priori, Dirichlet forms solely depend on the topology of the sequence of graphs that approximate our curve. One may thus rightly think that, in this case, geometry won't be taken into account. It is not so. We shall see, further, how it plays its part.
The link between E SG C m and E SG C m´1 is obtained through the introduction of two strictly positive constants r m and r m+1 such that: Let us set so, by induction, one gets: , of harmonic extensionṽ, we will write: ) .
The constant r´1, which can be interpreted as a topological one, will be called ramification constant. For further precision on the construction and existence of harmonic extensions, we refer to [17].  One has: Since the harmonic extensionũ minimizes E SG C

2
, the values of e, f , g, h, i, j are to be found among the critical points e, f , g, h, i, j such that: This leads to: and: Thus: r´1 = 3. One may note that the ramification constant is exactly equal to one plus the number of points that arise in V m+1 , for any value of the strictly positive integer m, between two consecutive vertices of V m . Definition 4.7. By definition, the energy scaling factor is the strictly positive constant ρ such that, for any integer m ą 1, and any real-valued function u defined on V m : ) .
For the sake of simplicity, we will write it under the form: ) .

Property 4.10.
Given a natural integer m, the map, which, to any pair of real-valued, continuous functions (u, v) defined on the set V m of the N m vertices of SG C m , associates: is a Dirichlet form on V m . Moreover: . We define the Dirichlet form E which, to any pair of real-valued, continuous functions (u, v) defined on the Sierpiński arrowhead curve SG C , associates, subject to its existence: ) .

Definition 4.12 (Normalized energy, for a continuous function u, defined on the Sierpiński arrowhead curve). Taking into account that the sequence
one defines the normalized energy, for a continuous function u, defined on the curve SG C , by: Notation 4.13. We will denote by dom E the subspace of continuous functions defined on SG C , such that: E(u) ă +8.
Notation 4.14. We will denote by dom 0 E the subspace of continuous functions defined on SG C , which vanish on V 1 , and such that: E(u) ă +8.

Measures.
Definition 4.16 (Self-similar measure, on the graph of the Sierpiński arrowhead curve). A measure µ on R 2 will be said to be self-similar for the domain delimited by the Sierpiński arrowhead curve, if there exists a family of strictly positive pounds (µ 1 , µ 2 , µ 3 ) such that: For further precisions on self-similar measures, we refer to the works of J. E. Hutchinson (see [9]). Property 4.17 (Building of a self-similar measure, for the domain delimited by the Sierpiński arrowhead curve). The Dirichlet forms mentioned in the above require a positive Radon measure with full support. The choice of a self-similar measure, which is, most of the time, built with regards to a reference set, of measure 1, appears, first, as very natural. R. S. Strichartz [22,20], showed that one can simply consider auto-replicant measuresμ, i.e. measuresμ such that: where (μ 1 ,μ 2 ,μ 3 ) denotes a family of strictly positive pounds. This latter approach appears as the best suited in our study, since, in the case of the graph SG C , the initial set consists of the trapeze T 0 , the measure of which, equal to its surface, is not necessarily equal to 1.
Let us assume that there exists a measureμ satisfying (4.1). Relation (4.1) yields, for any set of trapezes T m,j , m P N, and 1 ď j and, in particular: The convenient choice, for any i of t1, 2, 3u, is: One can, from the measureμ, build the self-similar measure µ, such that: where (µ i ) 1ďiď3 is a family of strictly positive pounds, the sum of which is equal to 1.
One has simply to set, for any i of t1, 2, 3u: The measure µ is self-similar, for the domain delimited by the Sierpiński arrowhead curve.     [1]). We will denote by dom ∆ the existence domain of the Laplacian, on the graph SG C , as the set of functions u of dom E such that there exists a continuous function on SG C , denoted ∆ u, that we will call Laplacian of u, such that :

Definition 4.23 (Harmonic function).
A function u belonging to dom ∆ will be said to be harmonic if its Laplacian is equal to zero.

Notation 4.24.
In the following, we will denote by H 0 Ă dom ∆ the space of harmonic functions, i.e. the space of functions u P dom ∆ such that: Given a natural integer m, we will denote by S(H 0 , V m ) the space of spline functions "of level m", u, defined on SG C , continuous, such that u |Vm is harmonic, i.e.: ∆ τ m (u |Vm ) = 0. Property 4.25. For any natural integer m: Property 4.26. Let m be a strictly positive integer, X R V 0 a vertex of the graph SG C , and ψ k X P S (H 0 , V k ) a spline function such that: Given a self-similar measure µ on the domain delimited by the Sierpiński arrowhead curve, one has: Thus

THE MOSCO APPROACH
Let us consider again the problem of energy forms on our curve, which cannot be obtained by means of a classical iterated function system, as it is the case with the Sierpiński gasket.
Such a problem was studied by U. Mosco [15], who specifically considered the case of what he called "the Sierpiński curve", or "Sierpiński string". Yet, he did not dealt with the curve itself, but with the Sierpiński gasket: "2D branches (...) meet together". Contrary to the arrowhead curve, the Sierpiński gasket exhibits self-similarity properties which turn it into a postcritically finite fractal (pcf fractal).
Yet, one can find interesting ideas in the work of U. Mosco. For instance, he suggests to generalize Riemannian models to fractals and relate the fractal analogous of gradient forms, i.e. the Dirichlet forms, to a metric that could reflect the fractal properties of the considered structure. The link is to be made by means of specific energy forms.
There are two major features that enable one to characterize fractal structures: (i) their topology, i.e. their ramification; (ii) their geometry. The topology can be taken into account by means of classical energy forms (we refer to [11,12,20,21]). As for the geometry, again, things are not that simple to handle. U. Mosco introduces a strictly positive parameter, δ, which is supposed to reflect the way ramification -or the iterative process that gives birth to the sequence of graphs that approximate the structureaffects the initial geometry of the structure. For instance, if m is a natural integer, X and Y two points of the initial graph V 1 , and M a word of length m, the Euclidean distance d R 2 (X, Y ) between X and Y is changed into the effective distance: ( This parameter δ appears to be the one that can be obtained when building the effective resistance metric of a fractal structure (see [21]), which is obtained by means of energy forms. To avoid turning into circles, this means: (i) either working, in a first time, with a value δ 0 equal to one, and, then, adjusting it when building the effective resistance metric; (ii) using existing results, as done in [3]. In the case of the Sierpiński gasket, at a step m P N ‹ of the iteration process, the effective distance between two adjacent points, as given in the work of U. Mosco, is

THREE APPROACHES, ONE LAPLACIAN
Let us note that: (i) The natural approach enables one to retrieve the usual Laplacian: (ii) In the Kigami and Strichartz approach, in the case of the Sierpiński gasket SG, the discrete Laplacians are graph ones (thus solely topological ones), which do not take account any metric. But, in fact, the geometry of the structure is indeed taken into account since, for a function u in the domain of the Laplacian: 3 m is exactly the Lebesgue measure of the m th -order triangular cell to which the vertex X belongs.
If one denotes by l m ą 0 a characteristic length of the m th -order cell (the length of a side, for instance): and thus retrieve a Riemannian expression, of the form: (iii) In the case of the Mosco approach, topology and geometry are taken into account by means of a quasi-distance, built from the euclidean one d eucl between adjacent points X and Y such that X " m Y : The related energy writes: This yields Satisfactorily, those three approaches generate the same Laplacian.
6.1. Spectrum of the Laplacian. In the following, u denotes a continuous function on SG C which belongs to dom ∆. We will apply the spectral decimation method developed by R. S. Strichartz [21], in the spirit of the works of M. Fukushima et T. Shima [4]. In order to determine the eigenvalues of the Laplacian ∆ u built in the above, we concentrate first on the eigenvalues (´Λ m ) mPN of the sequence of graph Laplacians (∆ m u) mPN , built on the discrete sequence of graphs (Γ Wm ) mPN . For any natural integer m, the restrictions of the eigenfunctions of the continuous Laplacian ∆ u to the graph Γ Wm are, also, eigenfunctions of the Laplacian ∆ m , which leads to recurrence relations between the eigenvalues of order m and m + 1.
We thus aim at determining the solutions of the eigenvalue equation: as limits, when the integer m tends towards infinity, of the solutions of Let m ě 2. We consider an eigenfunction u m´1 on V m´1 zV 1 , for the eigenvalue Λ m´1 . The aim is to extend u m´1 on V m zV 1 in a function u m , which will itself be an eigenfunction of ∆ m , for the eigenvalue Λ m , and, thus, to obtain a recurrence relation between the eigenvalues Λ m and Λ m´1 . Given three consecutive vertices of SG C m´1 , X k , X k+1 , X k+2 , where k denotes a generic natural integer, we will denote by Y k+1 , Y k+2 the points of V m zV m´1 such that: Y k+1 , Y k+2 are between X k and X k+1 , by Y k+4 , Y k+5 , the points of V m zV m´1 such that: Y k+4 , Y k+5 are between X k+1 and X k+2 , and by Y k+7 , Y k+8 , the points of V m zV m´1 such that: Y k+7 , Y k+8 are between X k+2 and X k+3 . For the sake of consistency, let us set: The eigenvalue equation in Λ m leads to the following system: where 0 ď i ď 2.
The sequence (u m (Y k+i )) 0ďiď9 satisfies a second order recurrence relation whose characteristic equation is r 2 + tΛ m´2 u r + 1 = 0. Its discriminant is δ m = tΛ m´2 u 2´4 = ω 2 m , ω m P C, and the roots r 1,m and r 2,m of the characteristic equation are the scalar given by One has then, for any natural integer i of t0, . . . , 9u where α m and β m denote scalar constants.
The extension u m of u m´1 to V m zV 1 has to be an eigenfunction of ∆ m , for the eigenvalue Λ m .
Since u m´1 is an eigenfunction of ∆ m´1 , for the eigenvalue Λ m´1 , the sequence (u m´1 (X k+i )) 0ďiď9 must itself satisfy a second order linear recurrence relation which be the pendant, at order m, of the one satisfied by the sequence (u m (Y k+i )) 0ďiď9 whose characteristic equation is: tΛ m´1´2 u r =´1´r 2 and the discriminant δ m´1 = tΛ m´1´2 u 2´4 = ω 2 m´1 , ω m´1 P C. The roots r 1,m´1 and r 2,m´1 of this characteristic equation are given by For any integer i of t0, . . . , 9u u m´1 (Y k+i ) = α m´1 r i 1,m´1 + β m´1 r i 2,m´1 , where α m´1 and β m´1 denote scalar constants.
One has then: This yields: which leads to: (ii) Second case. Notice that for any natural integer m r 1,m P CzR, r 2,m = r 1,m P CzR
The system is satisfied if which leads to the same relation as in the previous case: where ε P t´1, 1u.
, (we refer to [20,21] for further details). Thus, the respective spectrums of the Sierpiński gasket and arrowhead curve are distinct.

DETAILED STUDY OF THE SPECTRUM OF THE LAPLACIAN
As exposed by R. S. Strichartz in [21], one may bear in mind that the eigenvalues can be grouped into two categories: i) initial eigenvalues, which a priori belong to the set of forbidden values (as for instance Λ = 2); ii) continued eigenvalues, obtained by means of spectral decimation.
We present, in the sequel, a detailed study of the spectrum of ∆.
In the same way, the eigenvalue Λ = 3 yields a three-dimensional eigenspace and its multiplicity is 3.
Since the cardinal of V 2 zV 1 is: N S 2´4 = 6 one may note that we have the complete spectrum. 7.2. Eigenvalues of ∆ m , m P N, m ě 3. As before, one can easily check that the forbidden eigenvalue Λ = 2 is not a Dirichlet one.
One can also check that Λ m = 1 and Λ m = 3 are eigenvalues of ∆ m . By induction, one may note that, due to the spectral decimation, the initial eigenvalue Λ 2 = 1 gives birth, at this m th step, to eigenvalues Λ ãÑ1,m , and, in the same way, the initial eigenvalue Λ 2 = 3 gives birth, at this m th step, to eigenvalues Λ ãÑ3,m .
The dimension of the Dirichlet eigenspace is equal to the cardinal of V m zV 1 , i.e. If one looks for an asymptotic growth rate of the form then α SG C = 1 2 which is not the same value as in the case of the Sierpiński gasket (we refer to [21]):