Power dissipation in fractal Feynman-Sierpinski AC circuits
Patricia Alonso Ruiz

	

This paper studies the concept of power dissipation in infinite graphs and fractals associated with passive linear networks consisting of non-dissipative elements. In particular, we analyze the so-called Feynman-Sierpinski ladder, a fractal AC circuit motivated by Feynman's infinite ladder, that exhibits power dissipation and wave propagation for some frequencies. Power dissipation in this circuit is obtained as a limit of quadratic forms, and the corresponding power dissipation measure associated with harmonic potentials is constructed. The latter measure is proved to be continuous and singular with respect to an appropriate Hausdorff measure defined on the fractal dust of nodes of the network.

Level crossings induced by a longitudinal coupling in the transverse field Ising chain
    Frédéric Mila
    December 30, 2023
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§ INTRODUCTION

Passive linear networks have a wide range of applications, and especially electrical circuits have since long been intensively studied in different research areas such as electrical engineering <cit.>, physics <cit.> and mathematics <cit.>. In particular for the latter, Dirichlet forms on finite sets and graphs can be interpreted in terms of electric linear networks by considering the current flow between nodes (vertices) connected by resistors (edges). This is the key idea behind the theory of Dirichlet and resistance forms on fractals introduced by Kigami <cit.>. In this context, one may associate these forms with “fractal networks”.



Resistors are just one type of passive components, or impedances, of an electrical network. Impedances are characterized by the fact that they produce no energy by themselves. A resistor is a dissipative element because power is lost (energy is absorbed) when an alternating current runs through it. On the contrary, no loss is caused when the current flows through a non-dissipative element such as an inductor or a capacitor. Finite linear networks consisting only of inductors and capacitors are uninteresting since no power dissipation is expected.




However, what if the network is infinite (as for instance fractal networks are)?



In the 60s, Feynman posed this “amusing question”, see <cit.>; to give an answer, he constructed an infinite ladder network as depicted in Figure <ref>. He found its behavior surprising and noticed a very interesting connection with wave propagation: 


depending on the driving frequency of the signal, power will either dissipate, allowing waves to propagate along the network, or it will not dissipate at all, preventing waves from getting through. As a consequence, voltage will either stay constant, merely changing its phase, or it will die away rapidly.
This particular infinite ladder network is what is called a low-pass filter because low frequencies “pass” while higher frequencies are “rejected”. Although such an infinite network cannot actually occur, it is often possible to realize fairly good approximations that have many technical applications, see e.g. <cit.>.




Also fractal structures present unusual, a priori unexpected, physical properties <cit.>. Feynman's example motivated in <cit.> the construction of the so-called Feynman-Sierpinski ladder (F-S ladder for short), see Figure <ref>, as a first prototype of a fractal network consisting solely of inductors and capacitors that exhibits power dissipation, and hence wave propagation, at some frequencies.



The present paper aims to set up the mathematical framework to study the concept of power dissipation in infinite graphs and fractals, working out in detail the case of the F-S ladder. One of the main novelties lies in the fact that passive linear networks are studied in the frequency domain and this requires voltage, current and impedances to be considered as complex quantities. Following the classical intrinsic approach from analysis on fractals, the power dissipation in the F-S ladder will be defined as the limit of a suitable sequence of quadratic forms over complex-valued functions on its finite graph approximations.



A crucial role in this definition is played by the harmonic functions on the fractal dust that represents the nodes of the network. These functions describe the equilibrium potentials in a circuit when a signal is applied to the boundary nodes and they guarantee the existence of the aforementioned limit. Furthermore, proving them to be continuous will allow us to fully define power dissipation for harmonic potentials, as well as to construct the power dissipation measure associated with them. The latter measure will turn out to be singular with respect to an appropriate Hausdorff measure defined on the fractal dust related to the network.



The paper is organized as follows: In section <ref>, we review the classical notion of power dissipation in electric passive linear networks and transfer it to graphs and infinite networks. Here, we recall the construction of the F-S ladder and set the first step towards the definition of power dissipation in this network. Section <ref> discusses some geometric properties of the projection of the F-S ladder onto ^2. In particular, we prove this set to be a fractal quantum graph. In order to complete the definition of power dissipation, we analyze in section <ref> the harmonic potentials and prove in Theorem <ref> that they are continuous functions on the (fractal) set of nodes of the F-S ladder. Finally, section <ref> deals with the construction of a measure associated with power dissipation for harmonic potentials, c.f. Theorem <ref>. Further, we prove in Theorem <ref> that this measure is singular with respect to a suitable Hausdorff measure on the set of nodes of the F-S ladder.




§ BACKGROUND AND PRELIMINARIES




 §.§ Complex AC currents and power dissipation

Electric linear networks are characterized by the well-known Ohm's relation V=R· I, where V denotes voltage, I current and R electric resistance. In general, passive linear networks are studied through Fourier transforms in the so-called frequency domain, requiring voltage and current to be time dependent. They are typically considered as complex quantities given by

    V(t)=V̂e^i(ω t-φ_V),     I(t)=Îe^i(ω t-φ_I),

where ω denotes the frequency and φ_I-φ_V is the phase shift or phase difference. The corresponding generalization of Ohm's relation <cit.> now becomes

    V(t)=Z· I(t),

where Z=|Z|e^i(φ_I-φ_V) is called impedance and represents a “complex resistance” whose absolute value depends on the frequency ω. For ease of the notation, we will assume φ_V=0 and write φ:=φ_I for the phase shift, so that Z=|Z|e^iφ.



Due to Kirchhoff's rules, see e.g. <cit.>, the electromotive force of a generator connected to a linear circuit of several impedances satisfies that 

    (t)=Z^eff· I(t)=|Z^eff|e^iφI(t)=|Z^eff|Îe^iω t,

where Z^eff is the so-called effective/characteristic impedance, which represents an im­pe­dance equivalent to the initial set. 

The power dissipation
, also called energy dissipation or average rate of energy loss, is given by


    P   =1/T∫_0^T((t))(I(t)) dt
    
       =1/T∫_0^T|Z^eff|Î^2cos^2(ω t)cosφ dt+1/T∫_0^T|Z^eff|Î^2cos(ω t)sin(ω t)sinφ dt
       =1/2|Z^eff|Î^2cosφ=1/2|I(t)|^2(Z^eff).

Notice that this quantity only depends on the real part of the effective impedance. Consequently, a purely complex impedance is called a non-dissipative element because, on average, there is no loss of electrical power (energy) when an alternating current runs through it.  
In order to have non-trivial power dissipation, 
it is thus necessary that the effective impedance of the circuit has positive real part. Motivated by the physical concept of power dissipation, we introduce next a quadratic form on graphs resembling this phenomenon.



 §.§ Power dissipation in finite graphs


Let us start by considering a simple graph with two vertices x,y joined by an edge {x,y} and a network  consisting in a single impedance Z_xy with nonzero real part. In this case, Z_xy coincides with the effective impedance of the circuit.



The voltage across the edge {x,y} is defined as the difference between the potential at each node. Following Ohm's law (<ref>), for any potential function (v(x),v(y))∈^2 the current flowing from x to y is given by





    I_xy=v(y)-v(x)/Z_xy.

Although these quantities actually depend on time, we will consider this parameter fixed and omit it hereafter in the whole discussion.

The power dissipation associated with the network ={Z_xy} is defined as the quadratic form ω,^2→ given by

    ω,[v]_xy=1/2(Z_xy)/|Z_xy|^2|v(x)-v(y)|^2,

equivalently

    ω,[v]_xy=cosφ_xy/2|Z_xy||v(x)-v(y)|^2,

where φ_xy is the phase shift associated with Z_xy.

The subindex ω refers to the dependence of this expression on the frequency. For ease of the reading we will drop it off in the sequel and refer to this dependence explicitly only when confusion may occur.







Once power dissipation is defined for functions on a simple (2-node) network, this concept naturally extends to networks in graphs with several vertices and edges. In order for (<ref>) to provide a one-to-one correspondence between potentials and currents and thus identify functions on edges with functions on vertices, we will restrict our discussion to graphs/networks without multiple edges.


Let us now consider a finite graph =(V,E) and a network ={Z_xy | {x,y}∈ E} consisting of impedances Z_xy attached to each edge {x,y}. Further, we denote by ℓ(V) the space of complex-valued functions on V. 

The quadratic form ℓ(V)→ given by

    [v]:=∑_{x,y}∈ E[v]_xy

is called the power dissipation in  associated with the network . 



In the case of time-independent circuits with purely real impedances (resistors), power dissipation coincides with the classical definition of a resistance/energy form. Indeed, if the quantities Z_x,y, I_xy and the function v are real, (<ref>) becomes

    [v]=1/2∑_{x,y}∈ E1/Z_xy(v(x)-v(y))^2,

where Z_xy is the resistance between x and y. 










In the next section we extend the notion of power dissipation to infinite graphs and networks, focusing on the particular case of the Feynman-Sierpinski ladder (F-S ladder), for which some useful computations and results have been obtained in <cit.>.



 §.§ Power dissipation in infinite networks. The F-S ladder

The F-S ladder circuit that we denote by _ was introduced in <cit.> as a fractal network whose underlying graph structure is described in Figure <ref>. The infinite graph that arises in the limit can be formally embedded in ^2; existence and geometric properties of this set are discussed in Section <ref>.




Let _n=(V_n,E_n), n≥ 0, be the graphs displayed in Figure <ref> and let _∞ denote the limit (in the Gromov-Hausdorff metric) of the sequence {_n}_n≥ 0. Notice that this limit exists in view of Proposition <ref> and Remark <ref>.










The F-S ladder _ is the infinite network on _∞ whose edges have impedances that are capacitors Z_C=1/iω C or inductors Z_L=iω L with C,L>0 as shown in Figure <ref>. By convention, the symbol ⊣ ⊢ is employed for capacitors, and  for inductors. 


As for any passive linear network, power dissipation in _ is only meaningful if its effective impedance , that depends on the frequency ω, has positive real part. In the case of the F-S ladder, this is satisfied under the filter condition

    9(4-√(15))<2ω^2LC<9(4+√(15)),

c.f. <cit.>.



Following the intrinsic approach from analysis on fractals, we introduce next a sequence of networks on the approximating graphs _n that will eventually lead to the definition of the power dissipation in _∞ associated with _.



 §.§ Networks _ε,n={Z_ε,n,xy | {x,y}∈ E_n}

At each level n≥ 1, the network _ε,n is constructed by adding a small positive resistance ε in series with each of the impedances of _, see Figure <ref>.  Thus, for each {x,y}∈ E_n,

    Z_ε,n,xy=Z_,xy+ε,

where Z_,xy∈{Z_C, Z_L} according to the previous construction in Figure <ref>. Under the filter condition (<ref>), we know from <cit.> that the effective impedance of the F-S ladder, , is the regularized limit of the effective impedances of the networks {_ε,n}_n≥ 1, i.e.

    =lim_ε→ 0_+lim_n→∞ε,n.


Furthermore, we set Z_ε,0,xy=ε:=lim_n→∞ε,n for all {x,y}∈ E_0, and from now on assume that the F-S ladder satisfies the filter condition (<ref>).



In view of (<ref>), the sequence of networks {_ε,n}_n≥ 0 provides the base towards the desired definition of power dissipation.



Let V_*=⋃_n≥ 0V_n. The power dissipation in _∞ associated with the F-S ladder is the quadratic form → given by

    [v]:=lim_ε→ 0_+lim_n→∞[v_|_V_n]

and ={v∈ℓ(V_*) | [v]<∞}.


The embedding of the infinite graph _∞ in ^2 presented in Section <ref> will reveal that the set of vertices of _∞ is actually larger than V_*, so that the latter form is actually incomplete. Its definition for potentials defined on the whole network will appear in Section <ref>, c.f. Definition <ref>.

At each approximating level, the network _ε,n is equivalent to a triangular network with impedances ε,n. Thus, for any n≥ 0 and u∈ℓ(V_0),

    min{[v] | v∈ℓ(V_n),v_|_V_0=u}=(ε,n)/2|ε,n|^2∑_{x,y}∈ E_0|u(x)-u(y)|^2.




The latter remark is directly related to harmonic functions. These describe the equilibrium states of the F-S ladder when a potential is connected to the boundary vertices of the circuit, which in this case consists of the three vertices in V_0. More precisely, a function h∈ℓ(V_*) is said to be harmonic if for any ε>0 and n≥ 1

    _ε,0[h_|_V_0]=[h_|_V_n].

In particular, _[h]=lim_ε→ 0_+_ε,n[h_|_V_n] for any n≥ 0. Since V_0 has three elements, the space of harmonic functions on V_*, that we denote by _(V_*), is a 3-dimensional subspace of . 



In connection with harmonic functions we introduce the following auxiliary networks on the approximating graphs _n. 


 §.§ Networks _n={Z_n,xy | {x,y}∈ E_n}

At each n≥ 1, this network is constructed by changing the impedance of edges building triangles in the “deepest approximation level”, i.e. {x,y}∈ E_n∖ E_n-1 with x,y∈ V_n∖ V_n-1, to equal the effective impedance of the whole network _. The elements of _n are thus given by

    Z_n,xy={[                                       if {x,y}∈ E_n∖ E_n-1,x,y∈ V_n∖ V_n-1,;                                      ;                                 Z_,xy                            otherwise. ].

For completeness, we set Z_0,xy= for all {x,y}∈ E_0. One of the most relevant differences between _n and _ε,n is that the impedance of the edges changes with the approximation level. Moreover, the impedance of edges building triangles have non-zero real part, whereas the impedance of the remaining edges is purely imaginary.



In view of <cit.>, the networks _n are all electrically equivalent. This fact relates them directly to harmonic functions and power dissipation, as the next proposition shows.

For any h∈_(V_*) it holds that

    _[h]=_n[h_|_V_n]     ∀ n≥ 0.




Here, [h] acts in place of what one would in general define as “trace” of the “limit” power dissipation.

We prove the equivalent statement that for each n≥ 0, measuring potential across vertices in 
the F-S ladder network is equivalent to measuring them in the network _n.

By definition of _n, the impedance between edges {x,y}∈ E_n-1 with at least one vertex in V_n-1 is the same in _ and _n. Thus, it suffices to show that triangular cells of level n (n-cells) are electrically equivalent in both networks. 

On the one hand, notice that an n-cell of the network _n is a triangular network with impedances . On the other hand, an n-cell of _ is itself a F-S ladder, which is  electrically equivalent to a triangular network with impedances  as well.


From the definition of _n it follows that for any v∈ℓ(V_*) and n≥ 1,

    _n[v_|_V_n]=1/2∑_{x,y}∈ E_n∖ E_n-1
    x,y∈ V_n∖ V_n-1()/||^2|v(x)-v(y)|^2,

which is a multiple of the energy of the n-th graph approximation of the Sierpinski gasket.




§ GEOMETRIC PROJECTION OF THE INFINITE GRAPH _∞


The power dissipation associated with the F-S ladder has so far been defined for potentials on V_*. The present section investigates some geometric properties of the subset of ^2 that corresponds to the graphical representation of _∞. Among them, this set turns out to be a fractal quantum graph whose set of vertices is a fractal dust larger than V_*. 



Let S={1,2,3} and let {p_1,p_2, p_3}∈^2 denote the set of vertices of an equilateral triangle of side length 1 with baricentre p_0.


For each i∈ S, define the map G_i^2→^2 as

    G_i(x):=F_i∘ G_0(x),

where F_i,G_0^2→^2 are given by

    F_i(x)=1/2(x-G_0(p_i))+G_0(p_i),





respectively

    G_0(x)=α(x-p_0)+p_0,





with α∈(0,1). Moreover, set p_ij=G_i(p_j) for each i,j∈ S.






Notice that p_i is not the fixed point of G_i for any i∈ S, and therefore p_i≠ p_ii for any i∈ S. On the other hand, since p_ij=p_ji for all i≠ j, we will restrict ourselves to writing p_ij only for i≤ j. Although the mappings G_i actually depend on α, we will see in Proposition <ref> that all lead to topologically equivalent sets, which eventually makes the parameter α irrelevant.


Let W_0 = {∅} and define for n ≥ 1

    W_n = {w | w = w_1…w_n, w_i ∈ S, i = 1, …, n}.

Moreover, let W_* = ∪_n ≥ 0 W_n and for any w = w_1…w_n∈ W_* define G_w^2→^2 by

    G_w = G_w_1∘ G_w_2∘⋯∘ G_w_n,

setting G_∅ to be the identity on ^2. Finally, define V_0 = {p_1, p_2, p_3} and

    V_n = ⋃_w ∈ W_n G_w(Ṽ_0)

for n ≥ 1, as well as V_*=∪_n≥ 0V_n.

For each i∈ S we will denote by e_ii the line segment joining p_i and p_ii, and by e_ij the line segment joining p_i and p_j, with i<j, see Figure <ref>. Moreover, we define B:= {(i,j) | i≤ j} and write e_ij^w = G_w(e_ij) for any (w,(i,j))∈ W_*× B.



For any α∈ (0,1) there exists a unique compact set Q_α⊆^2 such that

    Q_α = ⋃_i∈ SG_i(Q_α) ∪⋃_(i,j)∈ B e_ij.

Furthermore,

    Q_α = C_α∪⋃_(w,(i,j)) ∈ W_*× Be_ij^w,

where C_α is the self-similar set associated with {G_1, G_2, G_3}, i.e. C_α is the unique nonempty compact set satisfying

    C_α = ⋃_i∈ SG_i(C_α).




The mapping H(K):=⋃_i=1^3 G_i(K) is a α/2-contraction. By <cit.>, the inhomogeneous equation 
x=H(x)∪∪_(i,j)∈ B e_ij has a unique solution Q_α in the space of compact subsets of ^2 that equals the closure of 

⋃_(w,(i,j)) ∈ W_*× Be_ij^w and in particular (<ref>) holds.

Notice that 
C_α is a fractal (Cantor) dust for any α∈(0,1). In view of the next proposition, we will refer to any of Q_α and C_α with α∈(0,1) simply by Q_∞ and C_∞. Although the notation might seem misleading at first sight, we write Q_∞ to underline its relation with _∞, not meaning α=∞.

The sets Q_α are pairwise homeomorphic for any α∈ (0,1).


Let G_w^α and e^α,w_ij denote G_w and e^w_ij respectively. Moreover, let ι^α denote the canonical coding mapping that identifies S^ with C_α, which is given by ι^α(w_1w_2…)=⋂_k≥ 1G_w_1… w_k(C_α). For any α_1,α_2∈ (0,1), define φ_α_1,α_2=ι^α_2∘(ι^α_1)^-1 C_α_1→ C_α_2. Extending this mapping onto each e^α_1,w_ij by φ_α_1,α_2|_e^α_1,w_ij=G_w^α_2∘(G_w^α_1)^-1|_e^α_1,w_ij for any (w,(i,j))∈ W_*× B yields the desired homeomorphism φ_α_1,α_2 Q_α_1→ Q_α_2.


The fractals Q_∞, C_∞, and the infinite graph _∞ are related through a projection mapping π_∞→^2 that in particular maps each approximating graph _n to its graphical representation in ^2 displayed in Figure <ref>. Based on this construction, each vertex x∈ V_n∖ V_n-1 will be associated with a word of length n≥ 1, w(x)∈ W_n, so that π(x)=G_w(x)_1… w(x)_n-1(p_w(x)_n). In view of (<ref>), any accumulation point, which corresponds to a vertex not captured by V_*, will be associated with an infinite word w(x)∈ S^ provided by the canonical coding mapping associated with C_∞, so that π(x)=∩_n≥ 1G_w(x)_1… w(x)_n(C_∞).




Let V_∞ and E_∞ the set of vertices, respectively edges, of _∞. For any fixed choice of the values of |_V_0 so that (V_0)=V_0, the projection mapping π_∞→^2 is defined as

    (x)={[            G_w(x)(p_w(x)_n)           if x∈ V_n∖ V_n-1,; ⋂_k≥ 1G_w_1(x)… w_k(x)(C_∞)             if x∈ V_∞∖ V_*, ].

and

    π({x,y})   ={(x)(1-t)+t(y), t∈ (0,1)},    {x,y}∈ E_∞.
 

An each level n≥ 1, (_n) is isomorphic to the so-called cable system associated with the graph _n, see <cit.>.






The sequence {π(_n)}_n≥ 0 is monotonically increasing and

    Q_∞=⋃_n≥ 0⋃_{x,y}∈ E_n({x,y})^Eucl.












This last observation leads to the fact that Q_∞ is a fractal quantum graph, a concept introduced in <cit.>, whose definition we recall below.

A fractal quantum graph with length system {(ϕ_k,ℓ_k)}_k≥ 1 is a se­parable compact connected and locally connected metric space (X,d) that satisfies the following two conditions:


  (i) For each k≥ 1, ℓ_k>0 and ϕ_k [0,ℓ_k]→ X is an isometry such that ϕ_k([0,ℓ_k])≅[0,ℓ_k] and

    ϕ_k((0,ℓ_k)) ∩ ϕ_j((0,ℓ_j))=∅    ∀ k≠ j.


  (ii) The set

    K:=X∖⋃_k≥ 1ϕ_k((0,ℓ_k))

is totally disconnected.


In view of the definition of , the totally disconnected set K_∞ associated with Q_∞ corresponds to the closure of the set of nodes V_*. Potentials in the F-S ladder are thus defined on K_∞ and we will see in the next section how to extend our previous definition of power dissipation to a special class of them.

Q_∞ is a fractal quantum graph.


Equipped with the Euclidean metric, Q_∞ is a compact, and thus locally compact metric space. The length system is given by {(ϕ_ij^w,ℓ_ij^w)}_(w,(i,j))∈ W_*× B, where ℓ_ij^w= e_ij^w and ϕ_ij^w[0,ℓ_ij^w]→ e_ij^w is the curve parametrization of e_ij^w. In particular, ϕ_ij^w((0,ℓ_ij^w))=π({x,y}):=π({x,y})∖{(x),(y)}, where (x)=G_w(p_i) and (y)=G_w(p_j). In view of (<ref>) we have that 

    K_∞:=Q_∞∖⋃_(w,(i,j))∈ W_*× Bϕ_ij^w((0,ℓ_ij^w))=Q_∞∖⋃_n≥ 0⋃_{x,y}∈ E_n({x,y})=C_∞∪V_*

is a totally disconnected set.


Notice that Q_∞ is also a finitely ramified fractal <cit.> and it can be expressed as a graph directed fractal <cit.> as well. 


In our particular case we have focused on the fact that it is a fractal quantum graph because of the importance of the totally disconnected set K_∞ in the next sections.


§ CONTINUITY OF POTENTIALS

The projection mapping  allows us to identify the nodes of the F-S ladder network with a subset of ^2. In general, this kind of identification naturally transfers the notion of power dissipation in graphs to discrete subsets of ^2. The harmonic functions associated with power dissipation and in particular their continuity, proved in Theorem <ref>, are essential to define power dissipation in Q_∞ through the fractal dust K_∞.



From now on, we identify the approximating sets V_n in (<ref>) with the set of vertices V_n via V_n=(V_n) and hence use the notation V_n for any of both. The power dissipation in V_n associated with a network  (formally given by [v∘π]) will be denoted by [v].












In this manner, the power dissipation associated with the F-S ladder is given by 

    [v]:=lim_ε→ 0_+lim_n→∞_ε,n[v_|_V_n],

where :={v∈ℓ(V_*) | [v]<∞} and now V_* is a subset of ^2.




As already mentioned, Lemma <ref> and the definition of  allow us to identify the set of vertices V_∞ with the totally disconnected set K_∞ for which V_* is a dense subset. By Proposition <ref>, K_∞ is compact with respect to the topology induced by the Euclidean metric. The aim of this section is to prove the continuity of the harmonic functions on V_*, so that they can be uniquely extended to continuous (harmonic) functions on K_∞. 



Recall that a function h∈ℓ(V_*) is said to be harmonic if for any ε>0 and n≥ 0

    _ε,0[h_|_V_0]=_ε,n[h_|_V_n].

Moreover, the space of harmonic functions on V_*, denoted by _(V_*), is 3-dimensional, and for any h∈_(V_*) and n≥ 1,

    lim_ε→ 0_+_ε,n[h_|_V_n]=[h]=_n[h_|_V_n]

c.f.  Proposition <ref>.

















Starting with a function h_0∈ℓ(V_0), harmonic functions are constructed by applying recursively the harmonic extension algorithm provided in <cit.>. This result conveys an explicit expression of the 3× 3-matrices A_1,A_2,A_3, that describe the algorithm. Therefore,

    h_|_G_j(V_0)=A_j h_|_V_0.

for any h∈_(V_*) and j=1,2,3.




  (i)The eigenvalues of A_j, j=1,2,3, can be explicitly computed with any mathematical software and equal

    λ_1=1,  λ_2=3/9Z_C+5,  λ_3=1/3λ_2.


  (ii) While the eigenvector associated with λ_1 is h_1=(1,1,1) in all matrices, the eigenvectors associated with λ_2 and λ_3 vary with the choice of j. The space of constant functions on V_* is thus the 1-dimensional subspace of _(V_*) spanned by h_1.


  (iii) Under the filter condition (<ref>), substituting Z_C and  by their actual values from <cit.>, one obtains 

    |λ_2|^2=9σ^2+(27+6CLω^2)^2/2106+25σ^2+90σ+100CLω^2(9+2CLω^2),

where σ=√(144 CLω^2-(2CLω^2)^2-81)∈. Although not directly readable from this expression, it holds that |λ_2|<1. 




The eigenvalues of the matrices A_1,A_2,A_3 from the harmonic extension algorithm satisfy |λ_3|<|λ_2|<|λ_1|=1.


In view of <ref>(i) it only remains to prove that |λ_2|<1. Let us consider for instance the matrix A_1 and let h_2 be the eigenvector associated with λ_2. Further, let h∈_(V_*) be the harmonic function with h_|_V_0=h_2 and denote by D^2_0 the matrix representation of the power dissipation _0, i.e. 

    D^2_0=()/2||^2[  2 -1 -1; -1  2 -1; -1 -1  2 ].

Since h is harmonic, we have that

    _0[A_1 h_|_V_0]   =⟨ D^2_0A_1 h_2,A_1h_2⟩=|λ_2|⟨ D^2_0h_2,h_2⟩
       =|λ_2|_0[h_|_V_0]=|λ_2|[h].

On the other hand, it follows from (<ref>) and the definition of _n that 

    [h]=_1[h_|_V_1]=∑_j=1^3_0[A_j h_|_V_0].

Thus, if |λ_2|=1, then _0[A_2 h_|_V_0]=_0[A_3 h_|_V_0]=0 and hence A_2h_2 and A_3h_2 are constant, a contradiction. 



In fact, it is possible to check directly that for instance A_3h_2 is non-constant because <cit.> provides the explicit expression of A_3 and h_2, which leads to

    A_3h_2=3/9Z_c+5(3,27Z_C+10/3Z_C+2,18Z_C+8/3Z_C+2).

If this were to be constant, then 27Z_C+10=18Z_C+8, equivalently =-9/2Z_C. But Z_C is purely imaginary, whereas  has positive real part, a contradiction.


The next theorem is the main result of this section. It justifies the extension of power dissipation in the F-S ladder to potentials defined on the whole K_∞.


Harmonic functions are continuous on V_*, i.e. _(V_*)⊆ C(V_*).


Before proving this result we show the following key lemma.


There exists r∈(0,1) such that for any non-constant h_0∈ℓ(V_0) 

    _0[A_jh_0]≤ r^2_0[h_0],       j=1,2,3.



Let h_0∈ℓ(V_0) be non-constant and let D^2_0 be the matrix representation (<ref>) of _0. Since D^2_0 is non-negative definite and symmetric, ⟨ D^2_0A_jh_0,h_0⟩≥ 0 for any j=1,2,3. Consider j arbitrary but fixed.

Let h_2 and h_3 denote the eigenvectors of A_j associated with the eigenvalues λ_2, resp. λ_3 given in Remark <ref> (i). Non-constant harmonic functions are thus the 2-dimensional subspace of _(V_*) spanned by {h_2,h_3} and hence h_0=∑_k=2^3a_kh_k, a_k∈. Then,

    _0[A_jh_0]   =⟨ D^2_0A_jh_0,A_j h_0⟩=|∑_k,l=2^3a_ka_l⟨ D^2_0λ_kh_k,λ_lh_l⟩|
       ≤∑_k,l=2^3|λ_kλ_l|⟨ D^2_0a_kh_k,a_lh_l⟩≤ r^2⟨ D^2_0h_0,h_0⟩

with r=|λ_2|<1 in view of Lemma <ref>.



Without loss of generality, let h∈_(V_*) be non-constant and such that [h]=1. For each ε >0 and m≥ 0 large enough, points inside a m-cell G_w(V_*), w∈ W_m, satisfy |x-y|<δ for some δ>0. Since h is harmonic, the maximum principle guarantees that h takes its maximum and minimum value within G_w(V_*) on the boundary G_w(V_0). Thus, for any x,y∈ G_w(V_0), by definition of _m we have that

    |h(x)-h(y)|^2≤2||^2/()∑_x,y∈ G_w(V_0)
    {x,y}∈ E_m_0[h]_xy=2||^2/()_0[h_|_G_w(V_0)]

and since h is harmonic, h_|_G_w(V_0)=A_w_1⋯ A_w_mh_|_V_0, with w=w_1… w_m. Applying repeatedly Lemma <ref> yields

    _0[h_|_G_w(V_0)]≤ r^2m_0[h_|_V_0]

and hence

    |h(x)-h(y)|≤ ||√(2/()) r^m<ε

for m large.


As an immediate consequence of this result, the space of harmonic functions on K_∞, denoted by _(K_∞), is well-defined. For any h∈_(K_∞) we will identify [h] with the former [h_|_V_*] to obtain the power dissipation associated with the F-S ladder for harmonic potentials on the fractal dust K_∞.


The power dissipation in K_∞ associated with _ of a function h∈_(K_∞) is given by

    [h]=lim_ε→ 0_+lim_n→∞_ε,n[h_|_V_n].





§ CONTINUITY OF THE POWER DISSIPATION MEASURE

In analogy to energy measures, this section aims to construct a measure on K_∞ that can be understood as the “power dissipation measure” associated with harmonic potentials. The main theorem states the existence of this continuous (atomless) measure.


For each non-constant harmonic function h∈_(K_∞), the power dissipation  induces a continuous measure ν_h on K_∞ with ν_h=C_∞.


Before proving this result, we provide some useful observations. In this and the next section, n-cells will be denoted by T_w=G_w(K_∞) for any w∈ W_n, n≥ 0. 


The following hold.








  (i) For any h∈_(K_∞) and {x,y}∈ E_m, m≥ 0,

    lim_ε→ 0_+lim_n→∞_ε,n[h]_xy=lim_ε→ 0_+_ε,m[h]_xy=0.





  (ii) For any h∈_(K_∞) and w∈ W_m, m≥ 1,

    lim_ε→ 0_+lim_n→∞∑_x,y∈ T_w∩ V_n 
    {x,y}∈ E_n_ε,n[h]_xy=()/2||^2∑_x,y∈∂ T_w|h(x)-h(y)|^2.
 













Let h∈_(K_∞) be non-constant. For each m-cell T_w define

    ν_h(T_w):=lim_ε→ 0_+lim_n→∞∑_x,y∈ T_w∩ V_n 
    {x,y}∈ E_n_ε,n[h]_xy.

Notice that since h is harmonic,

    0≤ν_h(T_w)≤ν_h(K_∞)=[h]<∞.


Applying the same definition of ν_h to isolated points, we have that ν_h({x})=0 whenever x∈ V_m for some m≥ 0 
because no edges are involved. If x∈ K_∞ is an accumulation point, it satisfies x=⋂_k≥ 1T_w_1… w_k

for some infinite word w_1w_2…∈ S^. In view of Remark <ref>(ii) we have that

    ν_h({x})   =lim_ε→ 0_+lim_n→∞lim_m→∞∑_x,y∈ T_w_1… w_m∩ V_n 
    {x,y}∈ E_n_ε,n[h]_xy
    
       =lim_ε→ 0_+lim_n→∞lim_m→∞(ε)/2|ε|^2∑_x,y∈∂ T_w_1… w_m
    {x,y}∈ E_n|h(x)-h(y)|^2.

By Theorem <ref>, h is continuous and therefore for any δ>0 we find m_0≥ 0 large enough such that 




ν_h({x})<3()/2||^2δ^2 for all m≥ m_0. Thus, ν_h({x})=0.

Let us now prove finite-additivity of ν_h: Let T_w,T_v be two disjoint cells of levels n_1, n_2. If these cells can be connected by edges without additional vertices, there is at most one such connecting edge {p_1,p_2}∈ E_m with m=max{n_1,n_2}. Let us suppose this is the case. In view of Remark <ref>(i) we have 

    ν_h(T_w∪ T_v)   =lim_ε→ 0_+lim_n→∞∑_x,y∈ T_w∩ V_n 
    {x,y}∈ E_n_ε,n[h]_xy+lim_ε→ 0_+lim_n→∞_ε,n[h]_p_1p_2
       +lim_ε→ 0_+lim_n→∞∑_x,y∈ T_v∩ V_n 
    {x,y}∈ E_n_ε,n[h]_xy=ν_h(T_w_1)+ν_h(T_w_2).

If there is no possible connecting edge, the above equality follows directly. The same argument applies for any finite union (both of cells or isolated points), since they can be connected by at most finitely many single edges.
 
In order to prove σ-additivity, consider a sequence of pairwise disjoint cells {T_v(k)}_k≥ 1, where T_v(k) is a n_k-cell. Without loss of generality, assume that n_k≤ n_k+1. Notice that, since the cells are pairwise disjoint, it is only possible to connect T_v(k) and T_v(k+1) by an edge {x,y}∈ E_n_k+1 if n_k+1=n_k+1. Hence, we can assume that there will be at most and edge {x_k+1,y_k+1}∈ E_n_k+1 joining T_v(k) and T_v(k+1). Then,

    ν_h(⋃_k≥ 1T_v(k))   =lim_ε→ 0_+lim_n→∞lim_m→∞∑_k=1^m∑_x,y∈ T_v(k)∩ V_n 
    {x,y}∈ E_n_ε,n[h]_xy
       +lim_ε→ 0_+lim_n→∞lim_m→∞∑_k=2^m+1_ε,n_k[h]_x_ky_k=∑_k=1^∞ν_h(T_w(k))+0,

where last equality follows by Remark <ref>(i) after interchanging the order of the limits, which is possible because both summands are uniformly bounded by [h].





The same argument applies to countable unions of isolated points, which in particular implies that ν_h =C_∞.





Finally, by Carathéodory's extension theorem, ν_h admits a unique extension to a (finite) measure on K_∞ and this measure is continuous because isolated points have no mass.










For any m≥ 0, w∈ W_m, and any m-cell T_w it holds that

    ν_h(T_w)≍(h_|_T_w)^2,

where (h_|_T_w)=max_x∈ T_wh(x)-min_y∈ T_wh(y).


First of all, recall from Remark <ref>(ii) that

    ν_h(T_w)=()/2||^2∑_x,y∈∂ T_w|h(x)-h(y)|^2.

By the maximum principle and since h is harmonic, h_|_T_w takes its maximum and minimum on the boundary ∂ T_w. Hence, (h_|_T_w)=|h(x)-h(y)| for some x,y∈∂ T_w and the definition of ν_h yields

    ()/2||^2(h_|_T_w)^2≤ν_h(T_w)≤3()/2||^2(h_|_T_w)^2.




§ SINGULARITY OF THE POWER DISSIPATION MEASURE

This section is devoted to proving that the power dissipation measure ν_h discussed in the preceding section is singular with respect to the Bernouilli measure μ on K_∞ that satisfies

    μ(T_w_1… w_n)=μ_w_1⋯μ_w_n

for any n-cell T_w_1… w_n, w_1… w_n∈ W_n, where ∑_i∈ Sμ_i=1. In particular in this case, we will consider μ_1=μ_2=μ_3=1/3. Together with the measure μ, K_∞ can be seen as a probability space. Notice that, as it happened with ν_h, μ =C_∞, where C_∞ is the Cantor dust defined in (<ref>).






Recall that any element in the support of μ is a non-isolated point of K_∞ such that x=⋂_n≥ 1T_w_1… w_n for some w_1w_2…∈ S^. For these points, we define the (random) matrices M_n(x)=A_w_n, where A_j, j∈ S, are the matrices of the harmonic extension algorithm (<ref>). The matrices M_n(x) are statistically independent with respect to μ.



The next result is based on a special case of <cit.> and we will mainly follow the proof given in <cit.>, including details for completeness. Since we are only dealing with non-constant harmonic functions, we will restrict to the 2-dimensional subspace of _(K_∞) spanned by the two harmonic functions h_2,h_3 associated to the eigenvalues λ_2,λ_3 from Remark <ref>(i). 


Assume that for a non-constant h∈_(K_∞) there exists m≥ 1 such that the mapping x↦D_0M_m(x)⋯ M_1(x)h_|_V_0 is non-constant. Then, ν_h is singular with respect to μ.


The proof of this theorem essentially consists in proving the condition stated in the following lemma, which is a consequence of the generalized Lebesgue differentiation theorem for metric measure spaces.

The measure ν_h is singular with respect to μ if for μ-a.e. x∈ C_∞

    lim_n→∞ν_h(T_w_1… w_n)/μ(T_w_1… w_n)=0,

where x=⋂_n≥ 1T_w_1… w_n.


For each n≥ 1, T_w_1… w_n is a neighborhood of x and lim_n→∞μ(T_w_1,… w_n)=0. Let us suppose that ν_h is absolutely continuous with respect to μ. Then, there exists a measurable function f (the Radon-Nikodym derivative of ν_h with respect to μ) such that

    ν(T_w_1… w_n)/μ(T_w_1… w_n)
    =1/μ(T_w_1… w_n)∫_T_w_1… w_nf(x) μ(dx).

Due to the definition of μ and since C_∞ is self-similar, it is Ahlfors regular (i.e. μ(B_d_E(x,r)∩ C_∞)≍ r^γ, in this case with γ being the Hausdorff dimension of C_∞). Thus, (C_∞,μ) equipped with the Euclidean metric is volume doubling 
and the generalized Lebesgue differentiation theorem, see e.g. <cit.> holds, so that (<ref>) equals f(x) for μ-a.e. x∈ C_∞. By assumption, this implies that f is zero μ-a.e., a contradiction.




















































For any h∈_(K_∞), w∈ W_m and n large it holds that

    ∑_x,y∈ T_w∩ V_n
    {x,y}∈ E_n_ε,n[h]_xy   = ∑_x,y∈ T_w∩ V_n-m
    {x,y}∈ E_n_ε,n[h∘ G_w]_xy
       =∑_x,y∈ T_w∩ V_n-m
    {x,y}∈ E_n-m_ε,n-m[h∘ G_w]_xy=_ε,n-m[h∘ G_w_|_V_n-m]
       =(ε)/2|ε|^2∑_{x,y}∈ E_0|h∘ G_w(x)-h∘ G_w(y)|^2.

Letting ε→ 0_+ and n→∞ in both sides of the equality yields

    ν_h(T_w)   =()/2||^2∑_{x,y}∈ E_0|h∘ G_w(x)-h∘ G_w(y)|^2
    
    
       =D_0A_w_m⋯ A_w_1h_|_V_0^2.




On the other hand, it follows from (<ref>) and the definition of μ that 


    D_0h_|_V_0^2   =⟨ D^2_0h_|_V_0,h_|_V_0⟩=_0[h_|_V_0]=∑_i=1^3_0[h∘ G_i_|_V_0]
       =∑_i=1^3⟨ D^2_0A_ih_|_V_0,A_ih_|_V_0⟩=∑_i=1^3D_0A_ih_|_V_0^2
       =3∑_i=1^3μ_iD_0A_ih_|_V_0^2
    
    
    =3∫_K_∞D_0M_1(x)h_|_V_0^2μ(dx).

and induction leads to

    D_0h_|_V_0^2=3^n∫_K_∞D_P_0M_n(x)⋯ M_1(x)h_|_V_0^2μ(dx)

for any n≥ 1. Notice that all computations are in fact basis independent.



Furthermore, by assumption, there exists m≥ 1 such that Jensen's and the Cauchy-Schwartz inequality yield

    ∫_K_∞logD_0M_m(x)⋯ M_1(x)h_|_V_0 μ(dx)<log∫_K_∞D_0M_m(x)⋯ M_1(x)h_|_V_0 μ(dx)
    
       ≤1/2log∫_K_∞D_0M_m(x)⋯ M_1(x)h_|_V_0^2μ(dx)=1/2log1/3^mD_0h_|_V_0^2,

where last equality follows from (<ref>). Hence, 

    β:=sup_h∈_1∫_K_∞logD_0M_m(x)⋯ M_1(x)h_|_V_0 μ(dx)<-m/2log 3,

where _1:={h∈_(K_∞) non-constant, D_0h_|_V_0=1}.



Let now h∈_1 and n≥ 1. Multiplying and dividing by D_0M_m(n-1)(x)⋯ M_1(x)h_|_V_0 and since the matrices M_i(x) are statistically independent, Jensen's inequality yields

    ∫_K_∞logD_0M_mn(x)⋯ M_mn-m(x)⋯ M_1(x)h_|_V_0 μ(dx)
    
    
    
    
       ≤β+∫_K_∞logD_0M_m(n-1)(x)⋯ M_1(x)h_|_V_0 μ(dx).

By induction we obtain

    ∫_K_∞logD_0M_mn(x)⋯  M_1(x)h_|_V_0 μ(dx)≤ nβ

and thus

    ∫_K_∞logD_0M_m_n(x)⋯  M_1(x)h_|_V_0 μ(dx)≤m_n/mβ

for the subsequence m_n=mn. Consequently,

    lim sup_m_n→∞1/m_nlogD_0M_m_n(x)⋯  M_1(x)h_|_V_0≤1/mβ<-1/2log 3,

where the existence of the limit is guaranteed by Furstenberg's Theorem <cit.> because the matrices M_m_n(x) are i.i.d. By definition of μ and (<ref>) we thus have that

    ν_h(T_w_1… w_n)/μ(T_w_1… w_n)=3^nD_0M_w_n(x)⋯ M_w_1(x)h_|_V_0^2

for μ-a.e. x∈ C_∞, hence μ-a.e. in K_∞, and (<ref>) yields

    1/n(logD_0M_w_n(x)⋯ M(x)_w_1h_|_V_0^2+nlog 3)<-nlog 3-nlog 3=0.

Finally, this implies that

    lim_n→∞ν_h(T_w_1… w_n)/μ(T_w_1… w_n)=0

for μ-a.e. x∈ K_∞. By Proposition <ref>, ν_h is singular with respect to μ.



 §.§ Acknowledgments

The author would like to thank A. Teplyaev and L. Rogers for very fruitful discussions.
amsplain

