Boltzmann and the Statistical Multifractals

The multifractal formalism introduced by Halsey et al [1] can be understood in a simple way by applying a similar reasoning that was used by Boltzmann for obtaining the thermodynamics of an ideal gas using statistical arguments instead of the microscopic description of a system conformed by 10²³ particles. The central idea was to introduce the relation between the entropy and the probability associated with a macrostate [2, 3]. In the second section recalls briefly the Boltzmann fundamental ideas for obtaining the statistical description of an ideal gas in thermodynamic equilibrium. In the third section, we obtain the Eggleston’s theorem, which relates the Hausdorff dimension with the Shannon entropy; this theorem plays a similar role in fractals like that the relation between entropy and probability in the Boltzmann treatment. The theorem is showed using a multiplicative process to decompose the unitary interval in fractals M(φ⃗), conformed by points with the same frequency of digits φ⃗; evaluate the Hausdorff dimension of M(φ⃗) and obtain that it is related to the Shannon entropy. In the fourth section, we introduce a Bernoulli measure with a probability vector p⃗; make the multifractal decomposition in terms of sets J_α, conformed by points with the same pointwise dimension α, and show that they are conformed by an infinite number of sets M(φ⃗). Therefore, each J_α has a multifractal structure, to determine the Hausdorff dimension D(α), we use a basic property of the Hausdorff Dimension which plays the rule of the Boltzmann principle of maximum entropy.

In the fifth section, the Boltzmann procedure is extended to determine the distribution P⃗(q), which maximize the Eggleston’s relation, given a value of α, D(α) is determined using P⃗(q), and D(α) singularity spectrum is obtained.

In the sixth section, we introduce a family of Bernoulli q-measures with a probability vector P⃗(q). We evaluate the q-measure of the sets J_α, finding that for a particular value of q the measure is concentrated in the set J_α^*, as a consequence of the singular behaviour of the q-measure, we obtain that τ(q) and -D(α) are Legendre transform of each other. The explicit values α^* = α(q) and the Hausdorff dimension D(α(q)) are the functions obtained in the Boltzmann procedure to determine the D(α) singularity spectrum.

Boltzmann analyzed a system conformed by N particles which interact only through elastic collision. For establishing a relation between the mechanics and thermodynamics, Boltzmann introduced a probabilistic description of this system dividing in K ⋘ N cells the phase space of a particle, and described the state of the system by the occupation number of particles of these cells:

n1(t),n2(t),…,nK(t)=n→(t) $$\begin{array}{} \displaystyle \left\lbrace n_1(t),n_2(t ),\ldots ,n_K (t) \right\rbrace = \left\lbrace \vec{\bf{n}}(t) \right\rbrace \end{array} $$(1)

The number of microstates corresponding with this distribution is given by:

Wn→(t)=N!∏i=1Kni(t)!=∏i=1Kni(t)Nni(t)−1=∏i=1Kpi(t)ni(t)−1withpi(t)=ni(t)N $$\begin{array}{} \displaystyle W \left\lbrace \vec{\bf{n}}(t) \right\rbrace = \frac{N!}{\prod\limits_{i=1}^{K} n_i (t)!} = \left [ \prod\limits_{i=1}^{K} \left (\frac{n_i(t)}{N} \right)^{n_i(t)} \right ]^{-1} = \left [ \prod\limits_{i=1}^{K} \left (p_i (t) \right)^{n_i(t)} \right ]^{-1} \quad with \quad p_i(t)=\frac{n_i(t)}{N} \end{array} $$(2)

Boltzmann postulated that W {n⃗(t)} is related to entropy, which is a macroscopic quantity of the system:

S(n→(t))=kln⁡W(n→(t)) $$\begin{array}{} \displaystyle S(\vec{\bf{n}}(t)) = k \ln W(\vec{\bf{n}}(t)) \end{array} $$(3)

Using (2), he obtained the relation between entropy and probability:

S(n→(t))N=−k∑i=1Kpi(t)ln⁡pi(t) $$\begin{array}{} \displaystyle \frac{S(\vec{\bf{n}}(t))}{N} = - k \sum\limits_{i=1}^{K} p_i (t)\ln p_i (t) \end{array} $$(4)

The time evolution of the probability distribution is governed by the Boltzmann’s equation. When it is introduced in (4), it can be proved that the entropy increases until it obtains its maximum value for a stationary distribution, which is the Maxwell-Boltzmann distribution. However, it is possible to obtain this result without invocating the Boltzmann equation, using that in the equilibrium state the entropy of the system obtains its maximum value under the restrictions:

∑i=1Kpi(t)=1andEN=∑i=1Kpi(t)εi $$\begin{array}{} \displaystyle \sum\limits_{i=1}^{K} p_i (t)=1 \quad and \quad \frac{E}{N}=\sum\limits_{i=1}^{K} p_i (t)\varepsilon_i \end{array} $$(5)

Then maximizing (4) with the lateral conditions (5), it is found that the stationary distribution is given by:

p~i=1Ze−βεi;withZ=∑i=1Ke−βεi $$\begin{array}{} \displaystyle \tilde{p}_i = \frac{1}{Z} e ^{-\beta\varepsilon_i}; \quad with \quad Z=\sum\limits_{i=1}^{K} e ^{-\beta\varepsilon_i} \end{array} $$(6)

The value of parameter β=1kT $\begin{array}{} \beta=\frac{1}{kT} \end{array} $ is determined by the thermodynamic information that the internal energy of the ideal gas is given by E=32NkT, $\begin{array}{} E=\frac{3}{2}NkT, \end{array} $ and (6) reduces to the Maxwell-Boltzmann distribution.

In this section we discuss the multifractal decomposition of the unitary interval 𝕀. Any real number x ∈ 𝕀, is expressed in s-base as:

x=∑n=1∞znsn;zn=0,1,…,s−1 $$\begin{array}{} \displaystyle x=\sum\limits_{n=1}^{\infty}\frac{z_n}{s^n};\quad z_n=0,1,\ldots,s-1 \end{array} $$(7)

Let n_i(x, K) denote the number of times the digit i ∈ (0, 1, …, s − 1) occurs among the first K digits of x. Then, the frequency in which this digit appears in x is given by:

limK→∞ni(x,K)K=limK→∞fi(x,K)=φi(x),i=0,1,…,s−1 $$\begin{array}{} \displaystyle \lim_{ K \to \infty} \frac{n_i(x,K)}{K} = \lim_{ K \to \infty} f_i (x,K)={\boldsymbol\varphi}_i(x), \quad i=0,1,\ldots,s-1 \end{array} $$(8)

where 0≤φi≤1,∑i=0s−1φi=1; $\begin{array}{} 0 \leq {\boldsymbol\varphi}_i \leq 1, \sum\limits_{i=0}^{s-1} {\boldsymbol\varphi}_i =1; \end{array} $ thus, x has associated a frequency vector:

φ→(x)=(φ0(x),φ1(x),…,φs−1(x)) $$\begin{array}{} \displaystyle \vec{{\varphi}} (x)= ({\boldsymbol\varphi}_0(x), {\boldsymbol\varphi}_1(x), \ldots, {\boldsymbol\varphi}_{s-1}(x) ) \end{array} $$(9)

Let M(φ⃗) be the set of points in 𝕀 with the same frequency vector. For obtaining the multifractal decomposition of 𝕀, we separate the unitary interval in the different sets M(φ⃗), and using the Eggleston’s theorem [4], we evaluate their Hausdorff Dimension. This can be done using a multiplicative process that consists of dividing 𝕀 in s-cylinders of first order:

Cz1=[z1s,z1s+1s);z1∈(0,1,…,s−1) $$\begin{array}{} \displaystyle C_{z_1} = \Bigg [ \frac{z_1}{s},\frac{z_1}{s} + \frac{1}{s} \Bigg ); \quad z_1 \in (0,1,\ldots,s-1) \end{array} $$(10)

Then, we divide each C_z1 in s-cylinders of second order:

Cz1z2=[z1s+z2s2,z1s+z2s2+1s2);z1,z2∈(0,1,…,s−1) $$\begin{array}{} \displaystyle C_{z_1z_2} = \Bigg [ \frac{z_1}{s}+\frac{z_2}{s^2},\frac{z_1}{s} + \frac{z_2}{s^2}+\frac{1}{s^2} \Bigg ); \quad z_1,z_2 \in (0,1,\ldots,s-1) \end{array} $$(11)

obtaining S² of 2-cylinders. Repeating K times this procedure on each cylinder, we obtain S^K cylinders:

Cz1z2…zK=[∑n=1Kznsn,∑n=1Kznsn+1sn);z1,z2,…,zK∈(0,1,…,s−1) $$\begin{array}{} \displaystyle C_{z_1z_2 \ldots z_K} = \Bigg [ \sum\limits_{n=1}^{K}\frac{z_n}{s^n},\sum\limits_{n=1}^{K}\frac{z_n}{s^n}+ \frac{1}{s^n} \Bigg ); \quad z_1,z_2,\ldots,z_K \in (0,1,\ldots,s-1) \end{array} $$(12)

Each K-cylinder is characterized by the sequence σ_K = z₁z₂ … z_K, we group them by the frequency vector

f→=(f0,f1,…,fs−1) $$\begin{array}{} \displaystyle \vec{\bf{f}} =(f_0, f_1, \ldots, f_{s-1}) \end{array} $$

where f_r is the frequency that shows that digit r = (0, 1, …, s − 1) occurs in σ_K. The number of K-cylinders with the same frequency vector f⃗ is given by

W(f→)=K!n0(K)!n1(K)!…ns−1(K)!=f0f0(K)(K)f1f1(K)(K)…fs−1fs−1(K)(K)−K $$\begin{array}{} \displaystyle W(\vec{\bf{f}}) = \frac{K!}{n_0(K)!n_1(K)! \ldots n_{s-1}(K)!} = \left [f_0^{f_0(K)}(K)f_1^{f_1(K)}(K) \ldots f_{s-1}^{f_{s-1}(K)}(K) \right ]^{-K} \end{array} $$(13)

On the other hand, each K-cylinder has a diameter:

Λ(Cz1z2…zK)=ΛK=s−K $$\begin{array}{} \displaystyle \Lambda(C_{z_1z_2 \ldots z_K}) = \Lambda_K = s^{-K} \end{array} $$(14)

due to the fact that when K → ∞, each K-cylinder goes to a point x of the unitary interval with a frequency vector given by (8). The Hausdorff dimension of M(φ⃗) is

DimHM(φ→)=−limK→∞ln⁡W(f→)ln⁡ΛK=−1ln⁡s∑j=0s−1φjlnφj $$\begin{array}{} \displaystyle \text{Dim}_H \,M(\vec{{\varphi}}) = - \lim_{ K \to \infty} \frac{\ln W(\vec{\bf{f}})}{\ln \Lambda_K} = - \frac{1}{\ln s}\sum\limits_{j=0}^{s-1} {\boldsymbol\varphi}_j \ln {\boldsymbol\varphi}_j \end{array} $$(15)

This relation is the Eggleston’s theorem [4]. In the Appendix A, we show how (15), can be found using the definition of Hausdorff measure applied for dyadic intervals [6]; we find the value D for which this measure is non-singular, and it corresponds with the Hausdorff dimension of the set.

On the other hand, the relation (15) establishes a non-trivial connection between Hausdorff dimension and the Shannon entropy, which is discussed in the Appendix B.

When a statistical measure is assigned to each point of 𝕀, it is decomposed in subsets J_α conformed by points with the same pointwise dimension. A simple case is when we introduce the Bernoulli measure μ on the unitary interval with probability vector p⃗ = (p₀, p₁, …, p_s−1); assigning to each digit j belongs to x a probability p_j, it introduces a singular measure that can be characterized by the pointwise dimension of μ at x [5]:

dμ(x)=limr→0ln⁡μ(B(x,r))ln⁡r $$\begin{array}{} \displaystyle \text{d}_\mu (x) = \lim_{ r \to 0} \frac{\ln \mu(B(x,r))}{\ln r} \end{array} $$(16)

where B(x, r) is a ball of radius r centered in x; this quantity can be expressed in terms of the K-cylinders as:

dμ(x)=limK→∞ln⁡μ(Cz1z2…zK)ln⁡Λ(Cz1z2…zK)whenx=limK→∞Cz1z2…zK $$\begin{array}{} \displaystyle \text{d}_\mu (x) = \lim_{ K \to \infty} \, \frac{\ln \mu (C_{z_1z_2 \ldots z_K})}{\ln \Lambda (C_{z_1z_2 \ldots z_K})} \quad \text{when} \quad x=\lim_{ K \to \infty} C_{z_1z_2 \ldots z_K} \end{array} $$(17)

The μ measure of the K-cylinder is given by

μ(Cz1z2…zK)=pz1pz2…pzK=p0f0(K)p1f1(K)…ps−1fs−1(K)K $$\begin{array}{} \displaystyle \mu (C_{z_1z_2 \ldots z_K}) = p_{z_1}p_{z_2} \ldots p_{z_K}= \left[ p_0^{f_0(K)}p_1^{f_1(K)} \ldots p_{s-1}^{f_{s-1}(K)} \right]^K \end{array} $$(18)

The x pointwise dimension is obtained introducing (18) and (14) into (17):

dμ(x)=limK→∞−1ln⁡s∑j=0s−1fj(K)ln⁡pj=−1ln⁡s∑j=0s−1φj(x)ln⁡pj $$\begin{array}{} \displaystyle \text{d}_\mu (x) = \lim_{ K \to \infty} \, -\frac{1}{\ln s} \sum\limits_{j=0}^{s-1} f_j (K) \ln p_j = -\frac{1}{\ln s}\sum\limits_{j=0}^{s-1} {\boldsymbol\varphi}_j(x)\ln p_j \end{array} $$(19)

We note that all the points that belong to M(φ⃗) have the same pointwise dimension. However, there are an infinite number of sets M(φ⃗) with the same value of the pointwise dimension, because given a particular value of d_μ (x) = α and the normalization condition ∑j=0s−1φj(x)=1, $\begin{array}{} \sum\limits_{j=0}^{s-1} {\boldsymbol\varphi}_j(x)=1, \end{array} $ we cannot determine the s components of the frequency vector.

As each M(φ⃗) is a fractal with the Hausdorff dimension given by Eggleston’s theorem, then J_α, the set of points with d_μ (x) = α, is a multifractal:

Jα={x∣dμ(x)=α}=⋃M(φ→)such thatφ→⋅ln⁡p→=α $$\begin{array}{} \displaystyle J_{\alpha} = \Bigl\lbrace x \mid \text{d}_\mu (x) =\alpha \Bigr\rbrace = \bigcup M(\vec{{\varphi}}) \quad \text{such that} \quad \vec{{\varphi}} \cdot \ln \vec{\bf{p}} = \alpha \end{array} $$(20)

where was defined: a→⋅ln⁡b→=∑j=0s−1ajln⁡bj $\begin{array}{} \vec{\boldsymbol{a}} \cdot \ln \vec{\boldsymbol{b}} = \sum\limits_{j=0}^{s-1} a_j \ln b_j \end{array} $ as the Hausdorff dimension satisfies that [6]

ifM=⋃MnthenDimH(M)=supDimHMn $$\begin{array}{} \displaystyle \text{if} \quad M=\bigcup M_n \quad \text{then} \quad \text{Dim}_H(M)=\sup\,\text{Dim}_H M_n \end{array} $$(21)

Thus, the Hausdorff dimension of J_α is given by:

D(α)=DimH(Jα)=supDimHM(φ→)withφ→⋅ln⁡p→=α $$\begin{array}{} \displaystyle \text{D}(\alpha)=\text{Dim}_H(J_\alpha)=\sup\,\text{Dim}_H M(\vec{{\varphi}}) \quad \text{with} \quad \vec{{\varphi}} \cdot \ln \vec{\bf{p}} = \alpha \end{array} $$(22)

The statistical multifractal decomposition of 𝕀 consists of grouping the points x in subsets with the same value of the pointwise dimension, and each subset J_α is characterized by its Hausdorff dimension D(α).

We determine D(α) using the Eggleston’s theorem and the extremal principle given by (22); this procedure is similar to the maximum entropy principle used by Boltzmann for obtaining the stationary distribution characterizing the equilibrium state. The Hausdorff dimension D(α) is determined by the frequency distribution φ⃗^* that maximizes:

DimHM(φ→)=−1ln⁡s∑r=0s−1φrlnφr $$\begin{array}{} \displaystyle \text{Dim}_H \, M(\vec{{\varphi}}) = - \frac{1}{\ln s}\sum\limits_{r=0}^{s-1} {\boldsymbol\varphi}_r \ln {\boldsymbol\varphi}_r \end{array} $$

with lateral conditions:

α=−1ln⁡s∑j=0s−1φjln⁡pj;∑j=0s−1φj=1 $$\begin{array}{} \displaystyle \alpha = -\frac{1}{\ln s}\sum\limits_{j=0}^{s-1} {\boldsymbol\varphi}_j \ln p_j; \quad \sum\limits_{j=0}^{s-1} {\boldsymbol\varphi}_j = 1 \end{array} $$(23)

Following the usual maximizing procedure we find that:

φr∗=Pr(q)=prqZqwhereZq=∑r=0s−1prq $$\begin{array}{} \displaystyle {\varphi}_r^*=P_r(q)=\frac{p_r^q}{Z_q} \quad \text{where} \quad Z_q=\sum\limits_{r=0}^{s-1} p_r^q \end{array} $$(24)

The q parameter is determined by the equation:

α(q)=∑r=0s−1Pr(q)[−ln⁡prln⁡s]=P→(q)⋅(−ln⁡p→ln⁡s) $$\begin{array}{} \displaystyle \alpha(q)=\sum\limits_{r=0}^{s-1}P_r(q) \Bigl [ -\frac{\ln p_r}{\ln s} \Bigr] = \vec{\bf{P}}(q) \cdot \Bigl(-\frac{\ln \vec{\bf{p}} }{\ln s} \Bigr) \end{array} $$(25)

The Hausdorff dimension of J_α is found using (24) in (15):

D(q)=D(α(q))=−1ln⁡s∑r=0s−1Pr(q)ln⁡Pr(q) $$\begin{array}{} \displaystyle \text{D}(q) = \text{D}(\alpha(q)) = - \frac{1}{\ln s} \sum\limits_{r=0}^{s-1}P_r(q) \ln P_r(q) \end{array} $$(26)

The dimension spectra for the statistical multifractal decomposition of the unitary interval is found when the q parameter is eliminated for (25) and (26). In thermodynamics, the entropy is one of the relevant functions, but there are several functions that contain the same thermodynamic information, they are the thermodynamic potentials, we show that in multifractals, a similar situation occurs. From (24) we have:

ln⁡Pr(q)=qln⁡pr(q)−ln⁡Zq $$\begin{array}{} \displaystyle \ln P_r(q) = q \ln p_r(q) - \ln Z_q \end{array} $$(27)

Using this result in (26), we have:

D(q)=qα(q)−τ(q) $$\begin{array}{} \displaystyle \text{D}(q) = q\alpha(q) - \tau (q) \end{array} $$(28)

where:

τ(q)=−1ln⁡sln⁡Zq $$\begin{array}{} \displaystyle \tau (q) = - \frac{1}{\ln s}\ln Z_q \end{array} $$(29)

We proceed to show that D(α) and τ (q) conform a Legendre transform pair. The derivative of (29) is given by:

dτ(q)dq=−1ln⁡s1Zq∑r=0s−1prqln⁡pr=∑r=0s−1Pr(q)(−ln⁡prln⁡s)=α(q) $$\begin{array}{} \displaystyle \frac{d\tau (q)}{dq} = - \frac{1}{\ln s}\frac{1}{Z_q} \sum\limits_{r=0}^{s-1}p_r^q \ln p_r = \sum\limits_{r=0}^{s-1}P_r(q) \Bigl ( -\frac{\ln p_r}{\ln s} \Bigr) =\alpha (q) \end{array} $$(30)

Considering that q = q(α), the derivative of (28) with respect to α is:

dDdα=q+αdqdα−dτdqdqdα=q $$\begin{array}{} \displaystyle \frac{d D}{d\alpha} = q + \alpha \frac{dq}{d\alpha} - \frac{d \tau}{dq}\frac{dq}{d\alpha} =q \end{array} $$(31)

This result infers that dD = qdα, therefore d(D − qα) = −dτ = −α dq, which implies (30) and hence τ (q) and -D(α) are the Legendre transform of each other.

In the statistical multifractal decomposition of the unitary interval, we focused our attention to the Hausdorff dimension of the sets J_α, which is a geometrical property of these sets. However, they also have statistical properties, because they are the support of the q-measures, in such way that given a q value, this measure is supported by only one of the J_α sets. Given a probability vector p⃗, a set of probability vectors P⃗(q) can be constructed, these vectors have the ability to scan the structure of the multifractal decomposition [8]. We obtained the escort probability vector P⃗(q) in the Boltzmann scheme for multifractals, they are equivalent to the Maxwell-Boltzmann distribution in statistical physics, where the q value plays the rule of the inverse of the temperature. The statistical q-measures are Bernoulli measures in the unit interval generated by P⃗(q), which is defined by (24), i.e.

P→(q)=(P0(q),P1(q),…,Ps−1(q));withPr(q)=prqZq $$\begin{array}{} \displaystyle \vec{\bf{P}}(q)=\Bigl ( {\bf P}_0(q),{\bf P}_1(q), \ldots, {\bf P}_{s-1}(q) \Bigr);\quad \text{with} \quad {\bf P}_r(q)=\frac{p^q_r}{Z_q} \end{array} $$(32)

where q is any real, this vector is called the escort distribution of p⃗ = (p₀, p₁, …, p_s−1) of q-order [8].

The q-measure μ_q assigns to each digit j belongs to x a probability P_j(q), then a K-cylinder has the q-measure:

μq(Cz1z2…zK)=Pz1(q)Pz2(q)…PzK(q)=μq(Cz1z2…zK)ZqK $$\begin{array}{} \displaystyle \mu_q \Bigl ( C_{z_1z_2 \ldots z_K} \Bigr )= {\bf P}_{z_1}(q){\bf P}_{z_2}(q) \ldots {\bf P}_{z_K}(q)= \frac{\mu^q \Bigl ( C_{z_1z_2 \ldots z_K} \Bigr )}{Z^K_q} \end{array} $$(33)

We define a K_α-cylinder by the following property:

limK→∞ln⁡μ(Cz1z2…zKα)ln⁡Λ(Cz1z2…zKα)=−1ln⁡slimK→∞ln⁡μ(Cz1z2…zKα)K=α $$\begin{array}{} \displaystyle \lim_{ K \to \infty} \,\frac{\ln \mu \Bigl ( C^\alpha_{z_1z_2 \ldots z_K} \Bigr )}{\ln \Lambda \Bigl ( C^\alpha_{z_1z_2 \ldots z_K} \Bigr )}=-\frac{1}{\ln s} \, \lim_{ K \to \infty} \, \frac{\ln \mu \Bigl ( C^\alpha_{z_1z_2 \ldots z_K} \Bigr )}{K} = \alpha \end{array} $$(34)

The set of all K_α-cylinders contains all the points of 𝕀 with d_μ (x) = α, therefore this set conforms the cover C_K(J_α) of the set J_α. For large values of K we have that:

μ(Cz1z2…zKα)≈[Λ(Cz1z2…zKα)]α=s−αK $$\begin{array}{} \displaystyle \mu \Bigl ( C^\alpha_{z_1z_2 \ldots z_K} \Bigr ) \approx \Bigl [ \Lambda \Bigl ( C^\alpha_{z_1z_2 \ldots z_K} \Bigr ) \Bigr]^\alpha =s^{-\alpha K} \end{array} $$(35)

Using (35) and (29) in (33), we find that for large K, the q-measure of a K_α-cylinder is given by:

μq(Cz1z2…zKα)≈s−K(qα−τ(q)) $$\begin{array}{} \displaystyle \mu_q \Bigl ( C^\alpha_{z_1z_2 \ldots z_K} \Bigr ) \approx s^{-K(q\alpha - \tau(q))} \end{array} $$(36)

On the other hand, for large K, the number of K_α-cylinders goes as:

N(Cz1z2…zKα)≈[Λz1z2…zK]−D(α)=sKD(α) $$\begin{array}{} \displaystyle N \Bigl ( C^\alpha_{z_1z_2 \ldots z_K} \Bigr ) \approx \Bigl [ \Lambda_{z_1z_2 \ldots z_K} \Bigr]^{-D(\alpha)} = s^{KD(\alpha)} \end{array} $$(37)

Then, the q-measure of the cover C_K(J_α) for large K is given by

μq[CK(Jα)]≈[s−K]qα−τ(q)−D(α) $$\begin{array}{} \displaystyle \mu_q \Bigl [ C_K(J_\alpha) \Bigr] \approx \Bigl [ s^{-K} \Bigr]^{q\alpha - \tau(q)-D(\alpha)} \end{array} $$(38)

As 0 ≤ μ_q [C_K(J_α)] ≤ 1 and s^−K < 1, for all values of α is satisfied the inequality:

τ(q)≤qα−D(α) $$\begin{array}{} \displaystyle \tau(q) \leq q\alpha - D(\alpha) \end{array} $$(39)

The q-measure of the set J_α is given by:

μq(Jα)=limK→∞[s−K]qα−τ(q)−D(α)=δ(α−α∗) $$\begin{array}{} \displaystyle \mu_q (J_\alpha) = \lim_{ K \to \infty} \, \Bigl [ s^{-K} \Bigr]^{q\alpha - \tau(q)-D(\alpha)} = \delta(\alpha - \alpha^*) \end{array} $$(40)

Thus, the q-measure of J_α is null for all values of α ≠ α^* and is one for α = α^*, which is defined by the relation:

τ(q)=qα∗−D(α∗) $$\begin{array}{} \displaystyle \tau(q) = q\alpha^* - D(\alpha^*) \end{array} $$(41)

Then, the set J_α^* is the support of the q-measure. From (39) and (41), we conclude that:

τ(q)=inf[qα−D(α)] $$\begin{array}{} \displaystyle \tau(q) = \inf \, \Bigl [ q\alpha - D(\alpha) \Bigr] \end{array} $$(42)

where the infimum is taken with respect to α; thus, (42) defines τ(q) as the Legendre transform of −D(α). We note that this result is obtained from (40), which is the generalization of the random weighted curdling proposed by Mandelbrot [9] and [10].

The relations (41) and (42) imply that α^*, satisfies the following conditions:

∂∂α[qα−D(α)]α∗=q−dD(α)dαα∗=0⇒dD(α)dαα∗=q $$\begin{array}{} \displaystyle \left. \frac {\partial}{\partial \alpha} \Bigl [q\, \alpha - D(\alpha) \Bigr ] \right|_{\alpha^*} = q - \left. \frac {d D(\alpha)}{d \alpha} \right|_{\alpha^*} =0 \Rightarrow \left. \frac {d D(\alpha)}{d \alpha} \right|_{\alpha^*} = q \end{array} $$(43)

d2D(α)dα2α∗<0 $$\begin{array}{} \displaystyle \left. \frac {d^2 D(\alpha)}{d \alpha^2} \right|_{\alpha^*} \lt 0 \end{array} $$(44)

Taking the derivative of (41) and using (43) and (29), we obtain that

α∗=dτ(q)dq=−1ln⁡sdln⁡Zqdq=α(q) $$\begin{array}{} \displaystyle \alpha^* = \frac {d \tau(q)}{dq} = - \frac{1}{\ln s} \frac{d \ln Z_q}{dq} = \alpha (q) \end{array} $$(45)

Therefore, the q-measure is concentrated in the set J_α(q) with α(q), which is given by (45), and it can be rewritten as the following average on P⃗(q):

α(q)=∑i=0s−1Pi(q)(−ln⁡piln⁡s)=〈−ln⁡piln⁡s〉q=−P→(q)⋅ln⁡p→ln⁡s $$\begin{array}{} \displaystyle \alpha(q) = \sum\limits_{i=0}^{s-1} {\bf P}_i(q) \Bigl ( -\frac{\ln p_i}{\ln s} \Bigr) = \Bigl \langle -\frac{\ln p_i}{\ln s} \Bigr \rangle _q = - \frac{\vec{{\bf P}}(q) \cdot \ln \vec{\bf{p}}}{\ln s} \end{array} $$(46)

The Hausdorff dimension of J_α(q) is obtained by using (46) into (41):

D(α(q))=qα(q)−τ(q)=−1ln⁡sP→(q)⋅ln⁡P→(q) $$\begin{array}{} \displaystyle D(\alpha(q)) = q \alpha(q) - \tau(q) = - \frac{1}{\ln s} \vec{{\bf P}}(q) \cdot \ln \vec{{\bf P}}(q) \end{array} $$(47)

When a Bernoulli statistical measure characterized by a probability vector p⃗ is introduced in a fractal, it is decomposed into sets J_α, which are multifractal. The determination of their Hausdorff dimension D(α) requires to use an extremal property of the Hausdoff dimension, similar to the maximum entropy principle. D(α) is determined in terms of a probability distribution P⃗(q), we find that each set J_α is an statistical attractor set where the q-measure defined in terms of P⃗(q) is concentrated.

As a consequence of the singular behaviour of the q-measure on the sets J_α, given by (40), we obtain the following:

τ(q) and −D(α) are Legendre transform of each other and,

An important case of (40) is when q = 1. The 1-measure is generated by the probability vector p⃗, the statistical attractor or the curdling set is conformed by the points with their pointwise dimension is identical with the Hausdorff dimension of the set, and it is given by Shannon entropy of p⃗, i.e.

α(q=1)=D(q=1)=−1ln⁡s∑j=0s−1pjln⁡pj=−p→⋅ln⁡p→ln⁡s $$\begin{array}{} \displaystyle \alpha(q=1) = D(q=1) = - \frac{1}{\ln s} \sum\limits_{j=0}^{s-1} p_j \ln p_j = - \frac{\vec{\bf{p}} \cdot \ln \vec{\bf{p}}}{\ln s} \end{array} $$(48)

Then, D(q = 1) is the Hausdorff dimension of the measure theoretical support of p⃗, which was found by Billingsley [6] in his work about the Hausdorff dimension in probability theory. Chhabra and Jensen [11] applied this result to P⃗(q) and found (47), after using heuristic arguments introduces (46), and with these expressions they found an alternative method for obtaining the singularity spectrum. On the other hand, Mandelbrot [10] introduced the curdling set for explaining the energy dissipation in fully developed turbulence using a multiplicative cascade process and identified this set with the Besicovitch fractal [9] extended the Mandelbrot suggestions, Feder [7] obtained and showed that (48) characterizes the set where the 1-measure is concentrated. The result (40) can be showed for a singular measure, and obtain an unified description of the multifractal decomposition can be obtained, which relates the Halsey et al [2] and Chhabra and Jensen [11] methods for obtaining the spectral singularity (see sections 5 to 7 of reference [12]).