Meccanica (2005) 40: 419–436 DOI 10.1007/s11012-005-2107-0

© Springer 2005

Analysis on Fractal Objects U. R. FREIBERG∗ Mathematisches Institut, Friedrich-Schiller-Universit¨at Jena, Ernst-Abb´e-Platz 1–4, D-07740 Jena, Germany (Accepted: 27 July 2005) Abstract. Irregular objects are often modeled by fractals sets. In order to formulate partial differential equations on these nowhere differentiable sets the development of a “new analysis” is necessary. With the help of the model case of the Sierpinski gasket the definition of energy forms and Laplacians on self-similar finitely ramified fractals is explained. Moreover, some results for certain classes of non-self-similar fractals are presented. 2000 Math. Subj. Class.: Primary 28A80, 35J15; Secondary 31C25, 35P05 Key words: Fractals, Hausdorff dimension, Self-similarity, Dirichlet form, Laplacian, Lagrangian.

1.

Introduction

Many physical phenomena in our world can be described by second-order partial differential equations of the classical functional analysis as, for example, the heat-, ¨ wave-, or Schrodinger equation. These approaches deeply rely on the assumption that the underlying space is smooth. However, many things in the world around us are “wild,” “irregular,” and “rough.” They have to be modeled by non-smooth “fractal” sets, which are in addition typically of non-integer Hausdorff dimension. (The reader will be certainly convinced of this fact after reading Mandelbrot’s nice book [27].) In fact, in the last decades scholars have become increasingly interested in the study of physical phenomena such as percolation or diffusion through porous media (see e.g. [12]) or diffusion across highly conductive layers (see e.g. [23, 32]). In this set of problems the media was modeled by a fractal set. Due to this fact there were many efforts to develop some tools of analysis on fractals (see as a standard reference Kigami’s monograph [20] and the references listed there). The main problem is obvious: fractals are non-differentiable objects because the notion of a “tangent space” is not available. Hence, is it not clear what the (partial) derivative of a function could be. In the last 20–30 years several approaches have been developed in order to build a potential theory, that is to define a Laplacian, on certain classes of fractals. They split in a natural way into two main theories: the first possibility is to regard a fractal as a subset of a higher dimensional Euclidean space or a manifold. Then the “ambient analysis” somehow induces an analysis on the fractals. The main difficulty ∗

e-mail: [email protected]

420 U. R. Freiberg in doing so is the definition of the corresponding function spaces, which arise as “traces” of classical Sobolev- or Besov-spaces applying “sophisticated modifications” of Whitney’s extension theorem. Here we will not be concerned with these “functionspace”-approaches, we refer the interested reader to [16] or [34]. The present paper is devoted to more “intrinsic” approaches – where the analysis is constructed from the fractal itself. There are two main directions within this “intrinsic” approach: the first one – roughly spoken – uses probability theory and founds on the following observation: in Rn , the Laplacian is the infinitesimal generator of the standard Brownian motion, which can be obtained as the limit of random walks. On the other hand, the construction of a random walk does not require a differentiable structure. Hence, one defines the Brownian motion on suitable fractals as the limit of a sequence of “suitable” normalized random walks and calls the infinitesimal generator of the limit process “Laplacian” on this fractal. This construction was done for the class of so-called nested fractals (see [22] and [24]); a very nice survey on this probabilistic way of “thinking about analysis on fractals” can be found in [1]. The other approach – or the analytic counterpart of the probabilistic one – goes back to Kigami (see [18–20]). An energy form (and hence – via the Gauß–Green– formula – a Laplacian) is constructed on so-called post critically finite self-similar fractals as a limit of approximating energies which are defined by suitable difference schemes on a sequence of “pre-fractals”. Note that sets from both these families “nested” and “post critically finite” are in particular self-similar and finitely ramified. The construction of the energy form deeply relies on this property. “Self-similarity” means that the fractals consist of smaller similar copies of themselves (see Section 2.2), that is they are allowed to be “irregular,” but at the same time they carry a very rigid recursive structure. Unfortunately, a lot of applications require dealing with “wilder objects” which are no longer self-similar. At the end of this paper (see Section 4) we illustrate how to overcome the lack of self-similarity in some special cases. Here the energy is obtained by integrating a local energy measure, the so-called Lagrangian (see [11] for the concept of Lagrangians, see [29–33] for Lagrangians on fractals). “Finitely ramification” means that the fractal may become a disconnected set by removing a finite number of points. A standard example of a finitely ramified selfsimilar fractal is the well-known Sierpinski gasket (see Figure 3 on page 425). Finitely ramified fractals can be approximated by an increasing sequence of points (which form the “nodes” of the underlying “resistance networks”) – this is fundamental in Kigami’s approach. Analysis on unfinitely ramified (i.e. “fatter”) fractals as for example Sierpinski carpet (see Figure 4 on page 426) is much more difficult (see e.g. [2]). Note that every nested fractal is post-critically finite. On the intersection of both classes both approaches – probabilistic and analytic – determine the same object. The paper is organized as follows: in Section 2, we provide the concepts of Hausdorff dimension and self-similarity. For self-similar fractals, which satisfy the open set condition, the calculation of the Hausdorff dimension is very simple because in these cases it coincides with the similarity dimension. This fact will be illustrated with the help of some “popular” fractals as Cantor set, von Koch curve, Sierpinski gasket, and Sierpinski carpet.

Analysis on Fractal Objects 421 In Section 3, we explain with the help of the example of the Sierpinski gasket how a “reasonable analysis” on self-similar finitely ramified fractals has been defined in the literature. Our starting point is the
construction of a fractal analog of the standard Euclidean Dirichlet form E[u] = |∇u|2 dx. This form turns out to be closed, regular, and strongly local on the underlying Hilbert space. Therefore – following the classical potential theory (see e.g. [17]) – there is a corresponding densely defined selfadjoint operator which is the Laplacian on the fractal. We will point out two main differences between this fractal construction and the Euclidean one: the first is that the energy E[u] is in general not absolutely continuous with respect to the volume measure µ, that is we do not have a Radon–Nikodym-density that allows an interpretation in terms of the square of a gradient. This will motivate us to introduce the Lagrangian, that is a measure valued energy. The second difference will occur in the spectral asymptotics of the Laplacian (see Section 3.3). Finally, in Section 4, we give some ideas on how to overcome a possible lack of self-similarity. More precisely, we will consider fractal sets G where self-similarity is destroyed by matching or deforming some given self-similar sets. In these cases the energy will be obtained by integrating the Lagrangian defined in Section 3.

2. What are Fractals? Let us start saying that there is no widely accepted definition of the term “fractal.” Most mathematicians call “fractal” a set for which the Hausdorff dimension strictly exceeds the topological dimension. In addition, in most cases, the Hausdorff dimension of such sets is a non-integer number. The best studied and most popular examples of such sets are so-called self-similar fractals. In this section, we briefly present both of these fundamental concepts. Details and proofs can be found in [4]. As we have physical application in mind we restrict ourselves to fractal subsets of Rn equipped with the standard Euclidean distance. 2.1. The Hausdorff Dimension Let us recall the connection between measures and dimension: somehow, we call a plane or a more general surface in a higher dimensional space two-dimensional if it is measurable by a two-dimensional (i.e. area) measure. Analogously, we call a (rectifiable) curve one-dimensional, if the notion of its “length” makes sense, or a solid body three-dimensional, if we may measure its volume. Interpolating these observations the definition of the Hausdorff dimension is done by introducing Hausdorff measures. Let be F ⊆ Rn and δ > 0. A countable family of sets {Ui } is a δ-cover of F if F⊆

∞ 

Ui

i=1

and 0 < |Ui |
δ.

422 U. R. Freiberg Recall that the diameter of a set U is defined by |U | := sup{|x − y| :

x, y ∈ U }.

For a fixed number s  0 we introduce  ∞  Hδs (F ) := inf |Ui |s : {Ui } is a δ-cover of F .

(2.1)

i=1

Obviously, Hδs (F ) increases as δ decreases. Therefore, the limit Hs (F ) := lim Hδs (F ) δ→0

is well-defined (as an element of [0, +∞]). Hs (F ) is called s-dimensional Hausdorff measure of F . Note that for subsets of Rn , n ∈ N, the n-dimensional Hausdorff measure coincides, up to a multiplicative constant, with the n-dimensional Lebesgue measure. More precisely, for any Borel subset B of Rn , we have Hn (B) = cn voln (B),   n where the constant cn = 2n n2 !/π 2 is the reciprocal of the n-volume of an n-dimensional ball of diameter 1. Going back to formula (2.1) we observe that for δ < 1, the quantity Hδs (F ) is a non-increasing function of s, hence, Hs (f ) is also non–increasing. In particular, if t > s and {Ui } is a δ-cover of F we have ∞ 

|Ui |
δ t

i=1

t−s

∞ 

|Ui |s ,

i=1

which implies Hδt (F )
δ t−s Hδs (F ). Taking the limit δ → 0, we find that Hδs (F ) < ∞ implies Hδt (F ) = 0 for t > s. Therefore, the graph of Hs (F ) as a function of s has a critical value where Hs (F ) jumps from +∞ to 0. This jumping point is referred to as Hausdorff-dimension of the set F and denoted by dimH F : dimH F = inf {s : Hs (F ) = 0} = sup{s : Hs (F ) = +∞}. If s = dimH F , then Hs (F ) can be zero, infinite or take a value from (0, +∞). In the latter case F is called s-set. Note that the mappings on a set preserving the Hausdorff dimension are just given by the bi-Lipschitz transforms of this set. A biLipschitz transform is a bijective mapping which is Lipschitz continuous in both directions. Typical examples are C1 -diffeomorphisms which are considered in Section 4.2.

Analysis on Fractal Objects 423 2.2. Self-Similar Fractals In this section, we briefly recall some fundamental properties of self-similar sets and self-similar measures. These – maybe most popular – examples of fractal sets have the property that they consist of a finite number of smaller copies of themselves. We will see later that the development of an analysis on such sets deeply relies on this property. Let S = {S1 , . . . , SM }, M  2, be a finite family of contractive similitudes acting on Rn , n  1, that is |Si (x) − Si (y)| = ri |x − y|,

x, y ∈ Rn

for some numbers ri ∈ (0, 1), i = 1, . . . , M. Further, we are given a M-dimensional vector of weights  ρ = (ρ1 , . . . , ρM ), that is ρ1 , . . . , ρnM are real numbers from the interval (0, 1) and M i=1 ρi = 1. We call a subset F of R self-similar with respect to S if F=

M 

Si (F )

i=1

and a Borel probability measure µ self-similar with respect to S and ρ if µ(A) =

M 

ρi µ(Si−1 (A))

i=1

for any Borel set A in Rn . Results of Hutchinson (see [14, 28]) imply the following properties: (i) For any finite family S = {S1 , . . . , SM } as above there exists a unique set F = F (S) ⊆ Rn , which is self-similar with respect to S. Furthermore, F is compact. (ii) For any pair S and ρ = (ρ1 , . . . , ρM ) as above there exists a unique Borel probability measure µ = µ(S, ρ),which is self-similar with respect to S and ρ. Further, it holds that supp µ(S, ρ) = F (S). Now let d be the unique positive solution of N 

rid = 1,

i=1

where the numbers ri are the contraction ratios of the mappings Si , i = 1, . . . , M. The number d is the so-called similarity dimension of the family S. Further we assume that the family S satisfies the so-called open set condition, that is we assume that there exists a non-empty bounded open set O such that Si (O) ⊆ O, i = 1, . . . , M, and Si (O) ∩ Sj (O) = ∅, i = j . Then we have: (iii) d = dimH (F ) and 0 < Hd (F ) < ∞. (iv) Hd (Si (F ) ∩ Sj (F )) = 0 for i = j . (v) If ρi = sid (which is the “natural choice” of the weights) then we have µ(S, ρ)(A) =

Hd (A ∩ F ) , Hd (F )

that is the unique self-similar measure is just given by the normalized d-dimensional Hausdorff measure on F .

424 U. R. Freiberg For example, the well known classical (middle third) Cantor set C ⊆ [0, 1] is the unique self similar set with respect to the family S = {S1 , S2 } where S1 (x) = x/3 and S2 (x) = (x + 2)/3, x ∈ [0, 1]. Its Hausdorff dimension equals its similarity dimension (the open set satisfying the open set condition can be chosen to be the open interval (0, 1)), hence, it is given by dimH C = (ln 2/ln 3). For any vector  = (1 , 1 − 1 ), 1 ∈ (0, 1), there exists a unique self–similar measure µ with respect to S and . If we choose  = (1/2, 1/2) we obtain the normalized (ln 2/ln 3)-dimensional Hausdorff measure on C. Of course, from a physical point of view, the Cantor set may be an uninteresting fractal because it is totally disconnected. Note however, that there is a well-developed theory of generalized (measure geometric) analysis of the real line, which forms the analytic counterpart of the so called quasi- or gap-diffusions on one hand and generalizes the notion of Sturm–Liouville-operator on the other hand (see [5–7, 25, 26]).

Figure 1. The middle third Cantor set.

Another well-known self-similar fractal is the von Koch curve, which is constructed as follows. Pose P1 := (0, 0) and P2 := (1, 0). Remove from the segment P1 P2 the middle (open) third and construct above this hole the two other sides of a regular triangle. Do the same with the four segments of length 1/3 of the arising set and continue to do so.

Figure 2. The von Koch curve.

On the other hand, this fractal is given as the unique nonempty set, which is selfsimilar with respect to the family of affine contractions  := {ψ1 , . . . , ψ4 }, where the mappings are given by

Analysis on Fractal Objects 425 ψ1 (z) := 3z , √ π ψ3 (z) := 3z e−i 3 + 21 + i 63 and

π ψ2 (z) := 3z ei 3 + 13 , ψ4 (z) := z+2 . 3

Here z denotes an element of C (for the moment we identify R2 with C). The family  satisfies the open set condition, hence the Hausdorff dimension of the von Koch curve equals (ln 4/ln 3). The von Koch curve belongs to the class of nested fractals, but as it is homeomorphic to the unit interval it is not a typical one (see also Remark 3.5). Fractals of this type often serve as models for highly conductive layers (see [23, 32]). The next set of interest is the famous Sierpinski gasket, which in Section 3 will serve as our standard example in defining an energy form on a nested fractal. Pose √ P1 := (0, 0), P2 := (1, 0), and P3 := ( 21 , 23 ). The Sierpinski gasket K is defined to be the unique nonempty compact set which is self-similar with respect to the family of affine contractions  := {ψ1 , ψ2 , ψ3 } (i.e. K = 3i=1 ψi (K)) where the mappings ψi : R2 −→ R2 are just given by the unique contractive similitude with contraction ratio 1/2 and fixed point Pi , i = 1, 2, 3; that is ψ1 (x1 , x2 ) := 21 (x1 , x2 ), ψ2 (x1 , x2 ) := 21 (x1 + 1, x2 ), ψ3 (x1 , x2 ) := 21 (x1 + 21 , x2 +

√

3 ). 2

It is easy to check that the open set condition is satisfied by choosing the open set O to be the interior of the triangle P1 P2 P3 . Hence the Hausdorff dimension of the Sierpinski gasket is (ln 3/ln 2). Obviously, the Sierpinski gasket is finitely ramified: Removing the middle points from the line segments, P1 P2 , P1 P3 and P2 P3 respectively, makes it a disconnected set. One can imagine lots of applications as for example heat or wave propagation on “spider–web like sets.”

Figure 3. The Sierpinski gasket.

426 U. R. Freiberg

Figure 4. The Sierpinski carpet.

On the contrary, the Sierpinski carpet is obviously not finitely ramified. In Figure 4, one can easily see what the eight similitudes (all of ratio 1/3) realizing its self-similarity are. The open set condition is satisfied (by choosing the interior of the big square). Hence the Hausdorff dimension of the Sierpinski carpet is (ln 8/ln 3). 3. Analysis on the Sierpinski Gasket In the classical analysis, continuous quantities are often approximated by discrete structures. For example, an integral is obtained as the limit of Riemann sums or the derivative of a function is given as the limit of difference quotients. The same procedure serves for defining the energy of a function on
fractals where the “fractal analog” of the Euclidean standard energy form E[u] = |∇u|2 dx is obtained as the limit of certain discrete “pre-energies” defined on finite sets approximating the fractal. The approach deeply relies on the self-similarity and the finite ramification of the underlying set. In this section, we explain this construction for the model case that the fractal F is the Sierpinski gasket. For a general outline of the theory we refer the reader to [20] and [22]. The Sierpinski gasket is a fractal subset of R2 . For any B ⊆ R2 we denote by C(B) the space of real valued continuous functions on B, and by C(B) its dual. C0,β (B) ¨ ¨ denotes the space of all Holder continuous functions on B with Holder exponent β. 3.1. Approximation of the Sierpinski Gasket As we have seen in Section 2.2, the Sierpinski gasket K is the unique self-similar set with respect to the family of the three contractions  := {ψ1 , ψ2 , ψ3 }, ψi : R2 −→ R2 , defined by

Analysis on Fractal Objects 427 ψi (x) = 21 (x − Pi ) + Pi ,

i = 1, 2, 3,

where {P1 , P2 , P3 } are the vertices of a planar triangle with unit side length. The Hausdorff dimension of the Sierpinski gasket equals d = (ln 3/ln 2). One can approximate K by an increasing sequence of finite sets (Vn )n
0 as follows. Setting V0 := {P0 , P1 , P2 }, we define for arbitrary n–tuples of indices (j1 , ..., jn ) ∈ {1, 2, 3}n ψj1 ...jn := ψj1 ◦ ... ◦ ψjn , Vj1 ...jn := ψj1 ...jn (V0 ) and Vn =

3 

Vj1 ...jn .

j1 ...jn =1

Figure 5. The approximating sets V0 , V1 , V2 and V3 .

It is easy to see that the cardinality of Vn is given by Vn = 23 (3n + 1). This observation is crucial in approximating the Hausdorff measure on the Sierpinski gasket by discrete measures supported on the sets (Vn )n
0 (see formula (3.2)). We say that p, q ∈ Vn are n-neighbors if there exists a n−tuple of indices (j1 , . . . , jn ) ∈ {1, 2, 3}n such that p, q ∈ Vj1 ...jn . Every point p in Vn \V0 has four n-neighbors q ∈ Vn denoted in the following by q ∼n p. Every n-neighbor q of p has distance 2-n from p. We say that each pair of points taken from V0 forms a pair of zero-neighbors. Further, we set V∗ := n
0 Vn = limn→∞ Vn . It holds that K = V∗ . We denote by µ the normalized d-dimensional Hausdorff measure Hd , restricted to K, which is in addition self-similar with respect to the family  (and the vector of “natural” weights  = ( 13 , 13 , 31 )), that is 1  −1  µ ψi A , 3 3

µ(A) =

i=1

428 U. R. Freiberg for any Borel set A ⊆ R2 . Note that µ is a so-called d-measure, that is there exist positive constants C1 , C2 , and r0 , such that C1 r d
µ(B(x, r))
C2 r d ,

x ∈ K, r ∈ (0, r0 )

(3.1)

(see [16] for details). The discrete approximation of the measure µ is done as follows. For any n  0, we define a discrete measure on Vn by: µˆ n :=

2



3n+1 p∈V

(3.2)

δ{p} ,

n

where δ{p} denotes the Dirac measure at the point p. Note that µˆ n (Vn ) = 1 + 31n . In [15], the following result is proved: PROPOSITION 3.1. The sequence (µˆ n )n
1 is weakly convergent (i.e. in C(K) ) to the measure µ. 3.2. Energy Form and Lagrangian on the Sierpinski Gasket In this section, we present the construction of the energy form on the Sierpinski gasket K. It is based on finite difference schemes and has been published first in [19] (the special case that K is the Sierpinski gasket, was in fact already treated in [18]). In the present paper, we follow the general lines described in [22] for nested fractals. For any function u: V∗ −→ R, we define 1 5 n  En [u] := (u(p) − u(q))2 , 2 3 p∈V q∼ p n

n  0,

(3.3)

n

where the second summation has to be intended over all n-neighbors q of p. The number (5/3) is the energy scaling factor determined by the Gaussian principle (see [22] and [31]). It can be shown (see [22]) that the sequence (En [u])n
0 is non-decreasing, the limit of the right-hand side of (3.3) exists and the limit form EK [u] := lim En [u] n→∞

is non-trivial (EK ≡ ∞) with domain D∗ (EK ) := {u : V∗ −→ R : EK [u] < ∞}. Every function u ∈ D∗ (EK ) can be uniquely extended to be an element of C(K). We denote this extension still by u and we set D := {u ∈ C(K): EK [u] < ∞}, where EK [u] := EK [u|V∗ ]. Hence D ⊆ C(K) ⊆ L2 (K, µ), where L2 (K, µ) is the Hilbert space of square summable functions on K with respect to the self-similar measure

Analysis on Fractal Objects 429 µ. We now define the space D(EK ) to be the completion of D with respect to the norm ||u||EK := (||u||2L2 (K,µ) + EK [u])1/2 .

(3.4)

D(EK ) is injected in L2 (K, µ) and is a Hilbert space with the scalar product associated to the norm (3.4). Then we extend EK as usual on the completed space D(EK ). By EK (·, ·) we denote the bilinear form defined on D (EK ) × D (EK ) by polarization, that is EK (u, v) := 21 (EK [u + v] − EK [u] − EK [v]) ,

u, v ∈ D (EK ) .

It is easy to see that for any pair u, v ∈ D (EK ) the form EK (·, ·) is the limit of the sequence En (·, ·) given by 1 5 n  [u(p) − u(q)] [v(p) − v(q)] . (3.5) En (u, v) := 2 3 p∈V q∼ p n

n

EK (·, ·) with domain D(EK ) is a Dirichlet form in the Hilbert space L2 (K, µ). The form EK is regular and strongly local. Regularity means that D(EK ) ∩ C(K) is dense both in C(K) with respect to the uniform norm and in D(EK ) with respect to the intrinsic norm (3.4). This property implies that D(EK ) is not trivial (i.e. not made by only the constant functions). Moreover, the functions in D(EK ) posses a continuous ¨ representative, which is actually Holder continuous on K (see Corollary 2 in [15]): PROPOSITION 3.2. The space D(EK ) is continuously embedded in C0,β (K), the space of H¨older continuous functions with exponent β = ln(5/3) . 2 ln 2 In the following, we identify u ∈ D(EK ) with its continuous representative, still denoted by u. The Dirichlet form (EK , D(EK )) is the fractal analog of the standard Dirichlet
form on a smooth domain ⊆ Rn . Recall that EK corresponds to |∇u|2 dx while the domain D(EK ) plays the role of the usual Sobolev space W 1,2 ( ). A main difference between a smooth domain and the Sierpinski gasket K is that in general EK [u] is not absolutely continuous with respect to the volume measure µ. Hence, we do not have the notion of a gradient. However, in Section 3.3, we will see that a Laplacian on K is defined in a natural way. This is a typical feature of fractals: first-order derivatives are harder to define than second-order derivatives. Moreover, the energy of a function is a “global” quantity. On the other hand, the study of the interplay between the geometry of a set K and the resulting analysis (what will be done in Section 4) requires a “local” analytic notion. Fortunately, the lack of the gradient can be overcome by a Lagrangian. For the concept of Lagrangians on fractals, that is the notion of a measure valued local energy, we refer to [11, 29,30] (see also [3,31,33]). We observe that the approximating energy forms En on Vn , n  0, defined in (3.5), can be written as  En (u, v) = ∇n u · ∇n v dµn , (3.6) Vn

430 U. R. Freiberg where µn is the discrete measure on K supported on Vn , given by µn = 23 µˆ n (see (3.2)); and for any p ∈ Vn the “discrete gradient” is given by 1  u(p) − u(q) v(p) − v(q) , u, v ∈ D (E) , (3.7) ∇n u · ∇n v(p) = |p − q|δ 2 q∼ p |p − q|δ n

where δ = (ln 5/ln 2). Remark 3.3. As explained in [31], δ is the unique positive number which yields (in view of formulae (3.6) and (3.7)) a non-trivial limit of the sequence (En )n
0 . Note that δ is not only determined by the Hausdorff dimension of the fractal K but also by the ramification properties of the underlying “pre-fractal networks.” This means that from the viewpoint of the energy the “effective distance” on the fractals is no longer given by the Euclidean metric but by a certain power δ of it, that is by a quasi-metric. Then it holds (see [8,31,33]): PROPOSITION 3.4. Let A be any subset of K. For every u, v ∈ D (EK ), the sequence of measures given by  (n) ∇n u · ∇n v dµn , n  0, LK (u, v)(A) := A∩Vn

weakly converges in C(K) to a signed finite Radon measure LK (u, v) on K as n → ∞, the so-called Lagrangian measure on K. Moreover, it holds that  EK (u, v) = dLK (u, v), u, v ∈ D (EK ) . K

Remark 3.5. The measure valued map LK on D (EK ) × D (EK ) is bilinear, symmetric, and positive (i.e. LK [u] := LK (u, u)  0 is a positive measure). This measure valued Lagrangian takes on the fractal K the role of the Euclidean Lagrangian dL(u, v) = ∇u · ∇vdx. Note that the Lagrangian LK is not absolutely continuous with respect to the volume measure µ for most nested fractals (see [13,22]). Therefore, the Lagrangian approach turns out to be a very powerful tool to define energy forms on these fractals and their deformations (see Section 4). 3.3. The Laplacian and Spectral Asymptotics on the Sierpinski Gasket As (EK , D(EK )) is a strongly local, closed, regular Dirichlet form on L2 (K, µ) we obtain applying the classical potential theory (see e.g. Chap. 6, Theorem 2.1 in [17]) the existence of a unique self-adjoint, non-positive operator K on L2 (K, µ) – with domain D( K ) ⊆ D (EK ), dense in L2 (K, µ) – such that  u ∈ D ( K ), v ∈ D(EK ). EK (u, v) = − ( K u) v dµ, K

The operator K is called the Laplacian on K. Let us give a look to its spectral asymptotics. First, let us recall the classical result in the Euclidean case. Let be a

Analysis on Fractal Objects 431 bounded domain in Rn with smooth boundary ∂ . We regard the Dirichlet eigenvalue problem − n u = λu on , u|∂ ≡ 0,  ∂2 n where n = ni=1 ∂x 2 is the classical Laplacian in R . It is well-known (see [35]) that i the eigenvalue counting function Nn (x) := # {λk
x :

− n u = λk u for some u = 0}

(counting the eigenvalues according multiplicities) is well–defined. Moreover, for any n ∈ N it holds that Nn (x) = (2π)−n cn voln ( )x n/2 + o(x n/2 )

as x → ∞,

(3.8)

where voln ( ) denotes the n-dimensional volume of and cn is the n-dimensional volume of the unit ball in Rn . Which kind of spectral asymptotics we can expect for the Laplacian on a fractal? The natural analog of Weyl’s result (3.8) would be Nd (x) = cd Hd ( )x d/2 + o(x d/2 )

as x → ∞,

(3.9)

where is a fractal set of Hausdorff dimension d = dimH ( ), Hd denotes the d-dimensional Hausdorff measure, and cd is a constant independent of the set . In fact, (3.9) has been conjectured in the early 80’s. Later it turned out that (3.9) fails for the majority of self-similar finitely ramified fractals (see [21]). For example, the eigenvalue counting function of the Laplacian K on the Sierpinski gasket K behaves asymptotically like x ds /2 , where the so-called spectral dimension ds of the Sierpinski 9 (see [21]). Obviously, ds differs from the Hausdorff dimengasket is given by ds = log log 5 sion. This is the second “fractal feature” which we want to emphasize: Spectral asymptotics on fractals do not only depend on the Hausdorff dimension, but also on the ramification properties. In fact, one could construct two fractals with different Hausdorff dimension having the same “eigenfrequencies” of different order. So the famous question “Can one hear the shape of a fractal?” has to be responded with “NO.” 4. Remarks on Non Self-Similar Fractals Aim of this section is to illustrate how to overcome the lack of self-similarity in some special cases. To this end, we use the Lagrangian approach introduced in Section 3.2. More precisely, we will consider fractal sets G where the self-similarity is destroyed by matching or deforming some given self-similar sets. The energy form EG on G is obtained in both cases by integrating a local energy measure LG on G. 4.1. Matching Self-Similar Sets In [8] a simple example of a non self-similar fractal has been considered, namely, the closed fractal curve F obtained as F = 3i=1 Ki = 6i=4 Ki , where the sets K1 , . . . , K6 are von Koch curves (see Section 2.2).

432 U. R. Freiberg

Figure 6. (a) First decomposition; (b) second decomposition.

Due to the special feature of F (see Figure 6a and b), and combining tools from differential and fractal geometry it is possible to regard F as a “fractal manifold,” which can be described by an atlas A = {(Ui , ϕi )}6i=1 , where the charts are just given ◦

◦

by Ui =Ki , where Ki denotes the set Ki without its endpoints, and where ϕi is the unique orthogonal mapping from Ki to a fixed reference Koch curve K, i = 1, . . . , 6. Set µi := ϕi−1 µ, i = 1, . . . , 6, where µ is the normalized d-dimensional Hausdorff measure restricted to K with d = (ln 4/ln 3) (see Section 2.2). Equip F with the finite Borel measure µF := µ1 + µ2 + µ3 = µ4 + µ5 + µ6 . In this case, the Lagrangian LF is locally defined on F as the image measure of the Lagrangian LK on K with respect to the corresponding map ϕi−1 , that is LF (w, z)(A) := LK (w ◦ ϕi−1 , z ◦ ϕi−1 ) (ϕi (A)) ,

A ⊆ Ki , w, z ∈ DF

(4.1)

with domain DF := {w : F −→ R : w ◦ ϕi−1 ∈ D(EK ), i = 1, . . . , 6}, where D(EK ) is the space of all functions of finite energy on the reference Koch curve K. In [8], Section 4.1, it is shown, that the definition in (4.1) is independent of the choice of the chart. Moreover, LF (w, z) is uniquely extendible to any Borel subset of F , hence to a finite Borel measure supported on F , by using the additivity property of measures. We now define the energy form on the fractal F by integrating its local energy measure, that is  EF (u, v) := dLF (u, v), u, v ∈ DF . F

It turns out that (EF , DF ) is a strongly local, closed, regular Dirichlet form on L2 (F, µF ), that is there exists (see e.g. Chap. 6, Theorem 2.1 in [17]) a unique selfadjoint, non-positive operator F on L2 (F, µF ) – with domain D( F ) ⊆ DF , dense in L2 (F, µF ) – such that  u ∈ D ( F ) , v ∈ DF . EF (u, v) = − ( F u) v dµF , F

Analysis on Fractal Objects 433 This Laplacian on F is locally given by the localized Laplacians on the Koch curves building F (see Section 5 in [8]). The latter fact has a nice stochastic interpretation in terms of a strong reflection principle (see Section 6 in [8]). The analogue of this in the language of the Dirichlet forms is given by the “natural” fact that the energy of a function u on F can be obtained as the sum of the energies of the restrictions of u to the Koch curves K1 , K2 , and K3 , or K4 , K5 , and K6 respectively (see Theorem 4.6. in [8]). Note that no matching condition at the junction points (except continuity) is needed. 4.2. Deforming Self-Similar Sets Other examples of non-self-similar fractals obtained by suitably deforming a nested fractal have been considered in [9]. Let K be the Sierpinski gasket as in Section 3, let g: U ⊂ R2 −→ R2 be a conformal C1 -diffeomorphism where U is an open set in R2 containing the set K. This yields that the differential Dg is given by (Dg)(x) = f (x)O(x), x ∈ U, where f (·) is a real valued, positive, continuous function on U , and O(x) is an orthogonal 2 × 2-matrix for any x ∈ U . Let G := g(K) denote the deformed fractal. The Hausdorff dimension dimH of G remains unchanged because g is in particular a bi-Lipschitz mapping. From [4], Proposition 2.2, it follows that 0 < Hd (G) < ∞, hence dimH G = dimH K = ln 3/ln 2. As it was done in Section 3.1, we approximate G by an increasing sequence of finite sets. Set Wn := g(Vn ) and W∗ := n
0 Wn = g(V∗ ). It holds that G = W∗ . For any n  0, two points p and q in Wn are n-neighbors – denoted in the following also by p ∼n q – if and only if g −1 (p) and g −1 (q) are n-neighbors in Vn . We equip G with the image measure µ˜ := gµ of µ under g, that is µ(A) ˜ := µ(g −1 A) for any Borel subset A of g(U ). Of course, supp µ˜ = G and µ(G) ˜ = 1. On the other hand, µ˜ can be described as the weak limit of a sequence of discrete measures which are supported on the approximating sets (Wn )n
0 . Define µ˜ n := g µˆ n , then it holds that 2  supp µ˜ n = Wn , µ˜ n = 3n+1 δ (see Section 3.1). From the weak convergence of p∈Wn {p} the sequence µˆ n it follows that µ˜ n  µ. ˜ Moreover, µ˜ is a d-measure on G (see (3.1)), and therefore it is equivalent to Hd (see [16], Chap. III). We introduce the Lagrangian LG on the deformed fractal G which is obtained as the weak limit of a sequence of suitable defined discrete Lagrangians L(n) G supported on Wn , n  0. Let LK and (EK , D(EK )) be as in Section 3.2. We introduce the linear space DG := {u : G −→ R :

u ◦ g ∈ D(EK )} = g −1 [D(EK )].

Note that these are the functions of finite energy on G because g is a ¨ ¨ C1 -diffeomorphism; and they are still Holder continuous with the same Holder exponent.

 For any u ∈ DG , we define a sequence of measures L(n) by G [u]  L(n) G [u](A) :=

n
0

n u|2 dµ˜ n , |∇

A∩Wn

where A is a Borel subset of G.

434 U. R. Freiberg n u the “effecThe crucial point is that in the definition of the discrete gradients ∇ tive distance” is now given by a suitable power of the arc length, instead of a suitable power of the Euclidean distance. This exponent turns out to be the same as for the undeformed fractal (see Section 3 in [9]). Fix n  0, p ∈ Wn and u ∈ DG . Proceeding as in Section 3.2, we define the square of the discrete gradient of u in p ∈ Wn by 2   u(p) − u(q) 1 2 n u| (p) = , |∇ δ 2 q∼ p lpq n

where lpq denotes the arc length of the curve  defined as the image (under g) of the line segment joining g −1 (p) and g −1 (q). Then it holds (see [9]): PROPOSITION 4.1. For any u ∈ DG there exists a unique finite Radon measure LG [u] supported on G, which we call Lagrangian measure on G, such that L(n) G [u]  LG [u] as n → ∞. Moreover, this limit measure is given by dLG [u](x) = [f (g −1 (x))]−2δ dLK [u ◦ g](g −1 (x)), that is



 dLG [u](x) =

LG [u](A) = A

g −1 (A)

(4.2)

[f (y)]−2δ dLK [u ◦ g](y)

for any Borel set A in G. The energy form on G is defined by
integrating the Lagrangian given in (4.2), that is for any u ∈ DG we set EG [u] := G dLG [u], and the energy norm is given by 1/2

. The corresponding bilinear form (EG , DG ) is a reg|| · ||EG := EG [·] + || · ||2L2 (G,µ) ˜

˜ – as well as on L2 (G, µ), ˆ where µˆ := ular, strongly local Dirichlet form on L2 (G, µ) Df 1 H – (see Theorem 4.3 and Corollary 4.4 in [9]). G HDf (G) The above result has a probabilistic counterpart, that is, there exists a strong Markovian process (Xt )t
0 with continuous paths on G, which can be regarded as the “natural Brownian motion” on G. Proceeding analogously as in Section 4.1, it  G on L2 (G, µ) follows that there exists a unique self-adjoint, non-positive operator

˜ 2 2  (or, on L (G, µ)) ˆ – with domain D( G ) ⊆ DG , dense in L (G, µ) ˜ (or, in L2 (G, µ) ˆ resp.) – which is the “natural” Laplacian on the “curved” fractal G, hence a “frac G will be determined in the tal Laplace–Beltrami-operator.” Spectral asymptotics of

forthcoming paper [10]. As a concluding remark we admit that the fractals considered in this section are somehow “very special non self-similar” fractals. They are obtained by destroying a pre-assumed self-similarity which ensures to treat them with the help of “modified self-similarity techniques”. Up to now there exists no “intrinsic approach” which allows to define a reasonable analysis on “a priori non self-similar” fractals (at the contrary, the “extrinsic approach” is available for general d-sets). Advances in this direction would contribute answering the fundamental question how the geometry of a set affects the analysis on this set.

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