Multifractal spectra and multifractal zeta-functions

We introduce multifractal zetafunctions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zeta-functions.


Introduction
Measures with widely varying intensity are called multifractals and have during the past 20 years been the focus of enormous attention in the mathematical literature. Loosely speaking there are two main ingredients in multifractal analysis: the multifractal spectrum and the Renyi dimensions. One of the main goals in multifractal analysis is to understand these two ingredients and their relationship with each other. It is generally believed by experts that the multifractal spectrum and the Renyi dimensions of a measure encode important information about the measure, and it is therefore of considerable importance to find explicit formulas for these quantities. In [29,[37][38][39] the authors used the zeta-function technique introduced and pioneered by M. Lapidus et al in the intriguing books [27,28] in order to find explicit formulas for the Renyi dimensions of a self-similar measure. At this point we note that it is generally believed that analysing the multifractal spectrum of a measure is considerably more difficult and challenging than analysing its Renyi dimensions, and the main purpose of this paper is to address the substantially more difficult problem of finding explicit formulas for the multifractal spectrum of a selfsimilar measure similar to the explicit formulas for its Renyi dimensions found in [29,[37][38][39]. In particular, and as a first step in this direction, we introduce 22 V. Mijović, L. Olsen AEM multifractal zeta-functions providing precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of selfconformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these, and more general multifractal spectra, equal the abscissae of convergence of the associated zeta-functions.

The first ingredient in multifractal analysis: multifractal spectra
For a Borel measure μ on R d with support equal to K and a positive number α, let us consider the set Δ μ (α) of those points x in R d for which the measure μ(B(x, r)) of the ball B(x, r) with center x and radius r behaves like r α for small r, i.e. the set If the intensity of the measure μ varies very widely, it may happen that the sets Δ μ (α) display a fractal-like character for a range of values of α. In this case it is natural to study the Hausdorff dimension of the sets Δ μ (α) as α varies. We therefore define the multifractal spectrum of μ by where dim H denotes the Hausdorff dimension. Here and below we use the following convention, namely, we define the Hausdorff dimension of the empty set to be −∞, i.e. we put dim H ∅ = −∞.
One of the main problems in multifractal analysis is to study this and related functions. The function f μ (α) was first explicitly defined by the physicists Halsey et al. in 1986 in their seminal paper [16].
The multifractal spectrum f μ is defined using the Hausdorff dimension. There is an alternative approach using "box-counting" arguments leading to the coarse multifractal spectrum. Namely, for a Borel probability measure μ on R d with support equal to K and a real number α, the coarse multifractal spectrum is defined as follows. For positive real numbers r > 0 and δ > 0, we write N μ,δ (α; r) = sup |I| (B(x i , δ)) i∈I is a finite family of balls such that: x i ∈ K for all i.

The second ingredient in multifractal analysis: Renyi dimensions
Renyi dimensions quantify the varying intensity of a measure by analyzing its moments at different scales. Formally, Renyi dimensions are defined as follows. Let μ be a Borel measure on R d . For E ⊆ R d , q ∈ R and δ > 0, we define the q-moment M μ,δ (q; E) of μ on E at scale δ by M μ,δ (q; E) = sup i∈I μ(B(x i , δ)) q (B(x i , δ)) i∈I is a finite family of balls such that: If E equals the support supp μ of μ, then we will use the following shorter notation M μ,δ (q) = M μ,δ (q; supp μ) , τ μ (q) = τ μ (q; supp μ) , τ μ (q) = τ μ (q; supp μ).
We note that the q-moment M μ,δ (q; E) is closely related to the box dimension dim B E of E. Indeed, if we let M δ (E) denote the greatest number of pairwise disjoint balls of radii δ with centers in E, then it follows from the definition of (1. 6) It is also possible to define an integral version of the q-moments M μ,δ (q; E). Namely, for E ⊆ R d , q ∈ R and δ > 0, we define the integral q-moment V μ,δ (q) of μ on E at scale δ by where B(E, δ) = {x ∈ R d | dist(x, E) ≤ δ} and L d denotes the Lebesgue measure in R d . We now define the lower and upper integral Renyi spectra T μ (·; E), T μ (·; E) : R → [−∞, ∞] of μ by As above, if E equals the support supp μ of μ, then we will use the following shorter notation V μ,δ (q) = V μ,δ (q; supp μ) , T μ (q) = T μ (q; supp μ) , T μ (q) = T μ (q; supp μ).
As above, we note that the integral q-moment V μ,δ (q; E) is also closely related to the Minkowski volume of E and the box dimension dim B E of E. Namely, if we let V δ (E) denote the δ approximate Minkowski volume of E, i.e.

The multifractal formalism
Based on a remarkable insight together with a clever heuristic argument, it was suggested by theoretical physicists Halsey et al. [16] that the multifractal spectra f μ and f c μ can be computed using the Renyi dimensions. This result is known as the "Multifractal Formalism" in the physics literature. More precisely, the "Multifractal Formalism" says that the multifractal spectra equal the Legendre transform of the Renyi dimensions. Recall that if ϕ : R → R is a real valued function, then the Legendre transform ϕ * : R → [−∞, ∞] of ϕ is defined by ϕ * (x) = inf y (xy + ϕ(y)). (1.8) can be computed directly from these expressions. We will now describe our strategy for analysing the quantities M μ,δ (q) and N μ,δ (α; r). Very loosely speaking, the quantities M μ,δ (q) and N μ,δ (α; r) "count" the number of balls B(x, δ) satisfying certain conditions. There are two distinct and widely used techniques for analysing the asymptotic behaviour of such (and similar) "counting functions", namely, (1) using ideas from renewal theory or (2) using the Mellin transform and the residue theorem to express the "counting functions" as sums involving the residues of suitably defined zetafunctions. Indeed, renewal theory techniques were introduced and pioneered by Lalley [19][20][21] in the 1980's, and later investigated further by Gatzouras [15], Winter [48] and most recently Kesseböhmer and Kombrink [18], in order to analyse the asymptotic behaviour of the "counting function" M δ (E) = M μ,δ (0, E) = M μ,δ (0) for self-similar sets E (see (1.6)) and similar "counting functions" from fractal geometry. However, while renewal theory techniques are powerful tools for analysing the asymptotic behaviour of "counting functions", they do not yield "explicit" formulas. This is clearly unsatisfactory and it would be desirable if "explicit" expressions could be found. However, in spite of the difficulties, the problem of finding "explicit" formulas of "counting functions" in fractal geometry has recently attracted considerable interest. In particular, Lapidus and collaborators [22][23][24]27,28] have with spectacular success during the past 20 years pioneered the use of applying the Mellin transform to suitably defined zeta-functions in order to obtain explicit formulas for the Minkowski volume V δ (E) = V μ,δ (0, E) = V μ,δ (0) of self-similar fractal subsets E of the line (see (1.7)).
It would clearly be desirable if similar formulas could be found for the multifractal quantities M μ,δ (q) and N μ,δ (α; r) of self-similar (and more general) multifractal measures μ. In multifractal analysis it is generally believed that analysing the q-moments M μ,δ (q) and the associated Renyi dimensions τ * μ (α) and τ * μ (α) is less difficult than analysing the "counting function" N μ,δ (α; r) and the associated multifractal spectra f μ and f c μ . Indeed, in [29,37] (see also the surveys [38,39]) the authors introduced a one-parameter family of multifractal zeta-functions and established explicit formulas for the integral qmoments V μ,δ (q) expressing V μ,δ (q) as a sum involving the residues of these zeta-functions, and in [34] the asymptotic behaviour of the q-moments M μ,δ (q) were analysed using techniques from renewal theory. In addition, we note that Lapidus and collaborators have introduced various intriguing multifractal zetafunctions [25,26]. However, the multifractal zeta-functions in [25,26] serve very different purposes and are significantly different from the multifractal zetafunctions introduced in [29,35,37]. The purpose of this paper is to address the significantly more difficult and challenging problem of performing a similar analysis of the multifractal spectrum "counting function" N μ,δ (α; r). In particular, the final aim is to introduce a class of multifractal zeta-functions allowing us to derive explicit formulas for the "counting function" N μ,δ (α; r) expressing N μ,δ (α; r) as a sum involving the residues of these zeta-functions. As a first step in this direction, in this work we introduce multifractal zetafunctions providing precise information of very general classes of multifractal spectra, including, for example, the spectra f μ and f c μ of self-similar multifractal measures μ. More precisely, we prove that the multifractal spectra equal the abscissae of convergence of the associated zeta-functions. It is our hope that a more careful analysis of these zeta-functions will provide explicit formulas for the "counting function" N μ,δ (α; r) allowing us to express N μ,δ (α; r) as a sum involving the residues of these zeta-functions; this will be explored in [32]. In order to illustrate the ideas involved we now consider a simple example.

An example illustrating the ideas: self-similar measures
To illustrate the above ideas in a simple setting, we consider the following example involving self-similar measures. Recall, that self-similar measures are defined as follows. Let (S 1 , . . . , S N ) be a list of contracting similarities S i : R d → R d and let r i denote the similarity ratio of S i . Also, let (p 1 , . . . , p N ) be a probability vector. Then there is a unique Borel probability measure μ on R d such that see [10,17]. The measure μ is called the self-similar measure associated with the list (S 1 , . . . , S N , p 1 , . . . , p N ). If the so-called Open Set Condition (OSC) is satisfied, then the multifractal spectra f μ and f c μ are given by the following formula. Namely, if the OSC is satisfied and if we define β : then it follows from, [6,42] that where β * denotes the Legendre transform of β (recall, that the definition of the Legendre transform is given in (1.8)). For α ∈ R, we are now attempting to introduce a "natural" self-similar multifractal zeta-function ζ sim α whose abscissa of convergence equals f μ (α). To do this we first introduce the following notation. Write Σ * = {i = i 1 · · · i n | n ∈ N , i j ∈ {1, . . . , N} } i.e. Σ * is the set of all finite strings i = i 1 · · · i n with n ∈ N and i j ∈ {1, . . . , N}. For a finite string i = i 1 · · · i n ∈ Σ * of length n, we write |i| = n, and we write r i = r i1 · · · r in and p i = p i1 · · · p in . With this notation, we can now motivate the introduction of a "natural" multifractal zeta-function as follows. Namely, since f μ (α) measures the size of the set of points x for which lim δ 0 log μ(B(x,δ)) log δ = α and since log μ(B(x,δ)) log δ has the same form as log p i log r i , it is natural to define the self-similar multifractal zeta-function ζ sim α by for those complex numbers s for which the series converges absolutely. An easy and straightforward calculation (which we present below) shows that the abscissa of convergence σ ab ( ζ sim α ) of ζ sim μ is less than f μ (α), i.e.
However, it is also clear that we, in general, do not have equality in (1.12).
because the series (1.11) that defines ζ sim α (s) is obtained by summing over the empty set). Since it also follows from [6,10,42] log ri ), we therefore conclude that: It follows from the above discussion that while the definition of ζ sim α (s) is "natural", it does not encode sufficient information for us to recover the multifractal spectra f μ (α) and f c μ (α). The reason for the strict inequality in (1.13) is, of course, clear: even though there are no strings i ∈ Σ * for which the ratio log p i log r i equals α if α ∈ (min i log pi log ri , max i log pi log ri ) \ { log p i log r i | i ∈ Σ * }, there are nevertheless many sequences (i n ) n of strings i n ∈ Σ * for which the ratios log p in log r in converge to α. In order to capture this, it is necessary to ensure that those strings i for which the ratio log p i log r i is "close" to α are also included in the series defining the multifractal zeta-function. For this reason, we modify the definition of ζ sim α and introduce a self-similar multifractal zeta-function obtained by replacing the original small "target" set {α} by a larger "target" set I (for example, we may choose the enlarged "target" set I to be a nondegenerate interval). In order to make this idea precise we proceed as follows. For a closed interval I, we define the self-similar multifractal zeta-function ζ sim ζ sim I (s) = ζ sim α (s). We can now proceed in two equally natural ways. Either, we can consider a family of enlarged "target" sets shrinking to the original main "target" {α}; this approach will be referred to as the shrinking target approach. Or, alternatively, we can consider a fixed enlarged "target" set and regard this as our original main "target"; this approach will be referred to as the fixed target approach. We now discuss these approaches in more detail.
(1) The shrinking target approach. For a given (small) "target" {α}, we consider the following family [α − r, α + r] r>0 of enlarged "target" sets [α − r, α + r] shrinking to the original main "target" {α} as r 0, and attempt to relate the limiting behaviour of the abscissa convergence of ζ sim [α−r,α+r] to the multifractal spectrum f μ (α) at α. In order to make this idea formal we proceed as follows. For each α ∈ R and for each r > 0, we define the zeta-function The next result, which is an application of one of our main results (namely Theorem 3.6), shows that the multifractal zeta-functions ζ sim α (·; r) encode sufficient information for us to recover the multifractal spectra f μ (α) and f c μ (α) by letting r 0.
(2) The fixed target approach. Alternatively we can keep the enlarged "target" set I fixed and attempt to relate the abscissa of convergence of the multifractal Vol. 91 (2017) Multifractal spectra and multifractal zeta-functions 31 zeta-function ζ sim I associated with the enlarged "target" set I to the values of the multifractal spectrum f μ (α) for α ∈ I. Of course, inequality (1.13) shows that if the "target" set I is "too small", then this is not possible. However, if the enlarged "target" set I satisfies a mild non-degeneracy condition, namely condition (1. 16), guaranteeing that I is sufficiently "big", then the next result, which is also an application of one of our main results (namely Theorem 3.6), shows that this is possible. More precisely the result shows that if the enlarged "target" set I satisfies condition (1.16), then the multifractal zeta-function ζ sim I associated with the enlarged "target" set I encode sufficient information for us to recover the suprema sup α∈I f μ (α) and sup α∈I f c μ (α) of the multifractal spectra f μ (α) and f c μ (α) for α ∈ I.
where σ ab ζ sim I denotes the abscissa of convergence of the zeta-function ζ sim I . We emphasise that Theorems 1.1 and 1.2 are presented in order to motivate this work and are special cases of the substantially more general theory of multifractal zeta-functions developed in this paper.
The next section, i.e. Sect. 2, describes the general framework developed in this paper and list our main results. In Sect. 3 we will discuss a number of examples, including, mixed and non-mixed multifractal spectra of self-similar and self-conformal measures, and multifractal spectra of Birkhoff ergodic averages.

Main definitions: the zeta-functions ζ U,Λ C (·) and ζ U,Λ C (·; r)
In this section we describe the framework developed in this paper and list our main results. We first recall and introduce some useful notation. For an infinite string i = i 1 i 2 . . . ∈ Σ N and a positive integer n, we will write i|n = i 1 · · · i n . In addition, for a positive integer n and a finite string i = i 1 · · · i n ∈ Σ n with length equal to n, we will write |i| = n, and we let [i] denote the cylinder generated by i, i.e.
Also, let S : Σ N → Σ N denote the shift map. Finally, we denote the family of Borel probability measures on Σ N by P(Σ N ) and we equip P(Σ N ) with the weak topology. The multifractal zeta-function framework developed in this paper depend on a space X and two maps U and Λ satisfying various conditions. We will now introduce the space X and the maps U and Λ.
(1) First, we fix a metric space X.
(3) Finally, we fix a function Λ : Σ N → R satisfying the following three conditions: (C1) The function Λ is continuous; (C2) There are constants c min and c max with −∞ < c min ≤ c max < 0 such that c min ≤ Λ ≤ c max ; (C3) There is a constant c with c ≥ 1 such that for all positive integers n and all i, j ∈ Σ N with i|n = j|n, we have Condition (C2) is clearly motivated by the hyperbolicity condition from dynamical systems, and Condition (C3) is equally clearly motivated by the bounded distortion property from dynamical systems.
Associated with the space X and the maps U and Λ, we now define the following multifractal zeta-functions.
Definition. (The zeta-functions ζ U,Λ C and ζ U,Λ C (·; r) associated with the space X and the maps U and Λ) For a finite string i ∈ Σ n , let (2017) Multifractal spectra and multifractal zeta-functions 33 and for a positive integer n and an infinite string i ∈ Σ N , let L n : Σ N → P(Σ N ) be defined by For C ⊆ X, we define the zeta-function ζ U,Λ C associated with the space X and the maps U and Λ by for those complex numbers s for which the series converges absolutely, and for r > 0 and C ⊆ X, we define the zeta-function ζ U,Λ C (·; r) associated with the space X and the maps U and Λ by for those complex numbers s for which the series converges absolutely and Next, we formally define the abscissa of convergence (of a zeta-function).
Definition. (Abscissa of convergence) Let ( a i ) i∈Σ * be a family of positive numbers and define the (zeta-)function ζ by for those complex numbers s for which the series converges. The abscissa of convergence of ζ is defined by Our main results, i.e. Theorems 2.1 and 2.2 below, relate the abscissa of converge of the zeta-functions ζ U,Λ C (·; r) and ζ U,Λ C to various multifractal quantities, including, the coarse multifractal spectrum associated with the space X and the maps U and Λ. In order to state Theorems 2.1 and 2.2 we will now define the coarse multifractal spectra.
Definition. (The coarse multifractal spectra associated with the space X and the maps U and Λ) For i = i 1 · · · i n ∈ Σ * , we let i = i 1 · · · i n−1 ∈ Σ * denote the "parent" of i. Next, for i ∈ Σ * and δ > 0, we write . We define the lower and upper r-approximate coarse multifractal spectrum associated with the space X and the maps U and Λ by and we define the lower and upper coarse multifractal spectrum associated with the space X and the maps U and Λ by Below we state our main results. As suggested by the discussion in Sect. 1.4, we will attempt to relate the abscissae of convergence of the multifractal zetafunctions ζ U,Λ C and ζ U,Λ C (·; r) to various multifractal spectra using two different but equally natural approaches: the shrinking target approach or the fixed target approach. The shrinking target approach is discussed in Sect. 2.2 and the fixed target approach is discussed in Sect. 2.3.

First main result: the shrinking target approach: finding lim
For a given "target" C, we consider the following family B(C, r) r>0 of enlarged "target" sets B(C, r) shrinking to the original main "target" C as r 0, and attempt to relate the limiting behaviour of the abscissa of convergence of the zeta-function ζ U,Λ C (·; r) = ζ U,Λ B(C,r) to the coarse multifractal spectrum f U,Λ (C) and other multifractal quatities. Our first main result, i.e. Theorem 2.1 below, shows that this approach is possible. More precisely, Theorem 2.1 shows that the abscissa of convergence of the zeta-function ζ U,Λ C (·; r) converges as r 0, and that this limit equals the coarse multifractal spectrum of C. We also show that the limit can be obtained by a variational principle involving the supremum of the entropy of all shift invariant Borel probability measures μ ∈ P(Σ N ) with Uμ ∈ C. In Sect. 3 we show that in many important cases the limit lim r 0 σ ab ζ U,Λ C (·; r) equals the traditional multifractal spectra.
Vol. 91 (2017) Multifractal spectra and multifractal zeta-functions 35 Theorem 2.1. (Shrinking targets) Let X be a metric space and let U : P(Σ N ) → X be continuous with respect to the weak topology. Let C ⊆ X be a closed subset of X.
(1) The lower coarse multifractal spectrum associated with the space X and the maps U and Λ: we have (2) The variational principle: we have here P S (Σ N ) denotes the family of shift invariant Borel probability measures on Σ N and h(μ) denotes the entropy of μ ∈ P S (Σ N ).
In order to prove Theorem 2.1 it suffices to prove the following three inequalities: Inequality (2.1) is proven in Sect. 5 using techniques from the theory of large deviations. Inequality (2.2) is proven in Sect. 6 using techniques from ergodic theory. Finally, inequality (2.3) follows directly from the definitions and is proved in Sect. 7.

Second main result: the fixed target approach: finding σ ab ζ U,Λ C
Alternatively, instead of choosing a family of "target" sets that shrinks to the given "target" C, we can keep the given "target" set C fixed and attempt to relate the abscissa of convergence of the multifractal zeta-function ζ U,Λ C associated with the "target" set C to the values of the coarse multifractal spectrum f U,Λ (C). Of course, the example in Sect. 1.4 shows that if the "target" set C is "too small", then this is not possible. However, if the coarse multifractal spectrum f U,Λ satisfies a continuity condition at C guaranteeing that the interior of C is "sufficiently big", then our second main result, i.e. Theorem 2.2 below, shows that this approach is possible. More precisely, Theorem 2.2 shows that if the coarse multifractal spectrum f U,Λ is inner continuous at C (the definition of inner continuity will be given below), then the abscissa of convergence of the zeta-function ζ U,Λ C equals the coarse multifractal spectrum of C. In analogy with Theorem 2.1, we also show that the abscissa of convergence of ζ U,Λ C can be obtained by a variational principle involving the supremum of the entropy of all shift invariant Borel probability measures μ ∈ P(Σ N ) with Uμ ∈ C. However, before stating Theorem 2.2, we first define the continuity condition that the coarse multifractal spectrum f U,Λ is required to satisfy.
Definition. (Inner continuity) Let P (X) denote the family of subsets of X and for C ⊆ X and r > 0, write We say that a function Φ : We can now state Theorem 2.2. (1) The lower coarse multifractal spectrum associated with R M and the maps U and Λ: we have (2) The variational principle: we have here P S (Σ N ) denotes the family of shift invariant Borel probability measures on Σ N and h(μ) denotes the entropy of μ ∈ P S (Σ N ).
Theorem 2.2 follows easily from Theorem 2.1 and is proved in Sect. 8.

Euler product
We will now prove that the multifractal zeta-function ζ U,Λ C has a natural Euler product. We begin with a definition.

Definition. (Composite and prime)
A finite string i ∈ Σ * is called composite (or peiodic) if there are u ∈ Σ * and a positive integer n > 1 such that i = u · · · u where u is repeated n times. A finite string i ∈ Σ * is called prime if it is not composite.

Applications: multifractal spectra of measures and multifractal spectra of ergodic Birkhoff averages
We will now consider several applications of Theorems 2.1 and 2.2 to multifractal spectra of measures and ergodic averages. In particular, we consider the following examples: • Section 3.1: Multifractal spectra of self-conformal measures.

Multifractal spectra of self-conformal measures
Since our examples are formulated in the setting of self-conformal (or selfsimilar) measures we begin be recalling the definition of self-conformal (and self-similar) measures. A conformal iterated function system is a list is a contractive similarity map, i.e. there exists It follows from [17] that there exists a unique non-empty compact set K with K ⊆ X such that The set K is called the self-conformal set associated with the list V , X, ..,N is a probability vector then it follows from [17] that there is a unique probability measure μ with supp μ = K such that The measure μ is called the self-conformal measure associated with the list if each map S i is a contracting similarity, then the measure μ is called the self-similar measure associated with the list V , X , ..,N . We will frequently assume that the list V , X , Next, we define the natural projection map π : Σ N → K. However, we first make the following definitions. Namely, for i = i 1 · · · i n ∈ Σ * , write The natural projection map π : Σ N → K is now defined by Finally, we collect the definitions and results from multifractal analysis of self-conformal measures that we need in order to state our main results. We first recall, that the Hausdorff multifractal spectrum f μ of μ is defined by Vol. 91 (2017) Multifractal spectra and multifractal zeta-functions 39 for α ∈ R where dim H denotes the Hausdorff dimension. In the late 1990's Patzschke [42], building on works by Cawley & Mauldin [6] and Arbeiter & Patzschke [1], succeeded in computing the multifractal spectra f μ (α) assuming the OSC. In order to state Patzschke's result we make the following definitions. Define Φ, Λ : . . ∈ Σ N , and for q ∈ R, let β(q) be the unique real number such that 0 = P β(q)Λ + qΦ ; here, and below, we use the following standard notation, namely if ϕ : Σ N → R is a Hölder continuous function, then P (ϕ) denotes the pressure of ϕ. Also, recall that the Legendre transform is defined in (1.8). We can now state Patzschke's result.
Theorem A.
[P] Let μ be defined by (3.2) and α ∈ R. If the OSC is satisfied, then we have Of course, in general, the limit lim r 0 log μB(x,r) log r may not exist. Indeed, recently Barreira and Schmeling [4] (see also Olsen and Winter [40,41], Xiao, Wu and Gao [49] and Moran [31]) have shown that the set of divergence points, i.e. the set Δ μ = x ∈ K the expression log μB(x, r) log r diverges as r 0 of points x for which the limit lim r 0 log μB(x,r) log r does not exist, is typically highly "visible" and "observable", namely it has full Hausdorff dimension. More precisely, it follows from [4] that if the OSC is satisfied and t denotes the Hausdorff dimension of K, then x ∈ K the expression log μB(x, r) log r diverges as r 0 = ∅ provided μ is proportional to the t-dimensional Hausdorff measure restricted to K, and dim H x ∈ K the expression log μB(x, r) log r diverges as r 0 = dim H K provided μ is not proportional to the t-dimensional Hausdorff measure restricted to K. This suggests that the set Δ μ has a surprisingly rich and complex fractal structure, and in order to explore this more carefully Olsen and Winter [40,41] introduced various generalised multifractal spectra functions designed to "see" different sets of divergence points. In order to define these spectra we introduce the following notation. If M is a metric space and ϕ : (0, ∞) → M is a function, then we write acc r 0 f (r) for the set of accumulation points of f as r 0, i.e. acc r 0 ϕ(r) = x ∈ M x is an accumulation point of f as r 0 .
In [40] Olsen and Winter introduced and investigated the generalised Hausdorff multifractal spectrum F μ of μ defined by Note that the generalised spectrum is a genuine extension of the traditional multifractal spectrum f μ (α), namely if C = {α} is a singleton consisting of the point α, then clearly F μ (C) = f μ (α). There is a natural divergence point analogue of Theorem A. Indeed, the following divergence point analogue of Theorem A was first obtained by Moran [31] and Olsen and Winter [40], and later in a less restrictive setting by Li, Wu and Xiong [30] (see also [5,46] for earlier but related results in a slightly different setting).
Theorem B. [30,31,40] Let μ be defined by (3.2) and let C be a closed subset of R. If the OSC is satisfied, then we have As a first application of Theorems 2.1 and 2.2 we obtain a zeta-function whose abscissa of convergence equals the generalised multifractal spectrum F μ (C) of a self-conformal measure μ. The is the content of the next theorem.
For a closed set C ⊆ R, we define the self-conformal multifractal zeta-function by Vol. 91 (2017) Multifractal spectra and multifractal zeta-functions 41 For a closed set C ⊆ R and r > 0, we define the self-conformal multifractal zeta-function by and if α ∈ R and C = {α} is the singleton consisting of α, then we write ζ con C (s; r) = ζ con α (s; r), i.e. we write for q ∈ R. Let C be a closed subset of R. Then the following hold: In particular, if α ∈ R, then we have lim r 0 σ ab ζ con α (·; r) = β * (α).
(1.2) If the OSC is satisfied, then we have In particular, if the OSC is satisfied and α ∈ R, then we have Proof. This follows immediately from the more general Theorem 3.2 in Sect. 3.2 by putting M = 1.

Mixed multifractal spectra of self-conformal measures
Recently mixed (or simultaneous) multifractal spectra have generated an enormous interest in the mathematical literature, see [3,31,35,36]. Indeed, previous results (Theorems A and B) only considered the scaling behaviour of a single measure. Mixed multifractal analysis investigates the simultaneous scaling behaviour of finitely many measures. Mixed multifractal analysis thus combines local characteristics which depend simultaneously on various different aspects of the underlying dynamical system, and provides the basis for a significantly better understanding of the underlying dynamics. We will now make these ideas precise.
The mixed multifractal spectrum f μ μ μ of the list μ μ μ = (μ 1 , . . . , μ M ) is defined by Of course, it is also possible to define generalised mixed multifractal spectra designed to "see" different sets of divergence points. Namely, we define the generalised mixed Hausdorff multifractal spectrum F μ μ μ of the list μ μ μ = (μ 1 , . . . , μ M ) by Again we note that the generalised mixed multifractal spectrum is a genuine extension of the traditional mixed multifractal spectrum clearly F μ (C) = f μ (α α α). Assuming the OSC, the generalised mixed multifractal spectrum F μ (C) can be computed [31,35]. In order to state the result from [31,35], The generalised mixed multifractal spectra f μ μ μ and F μ μ μ are now given by the following theorem.
Theorem C. [31,35] In particular, if the OSC is satisfied and α α α ∈ R M , then we have As a second application of Theorems 2.1 and 2.2 we obtain a zeta-function whose abscissa of convergence equals the generalised mixed multifractal spectrum F μ μ μ (C) of a list μ μ μ of self-conformal measures. This is the content of the next theorem.  For a closed set C ⊆ R M and r > 0, we define the self-conformal multifractal zeta-function by (

1.2) If the OSC is satisfied, then we have
We will now prove Theorem 3.2. Recall that the function Λ : Σ N → R is defined by and note that if i ∈ Σ * , then Hence, for C ⊆ R M we have In order to prove Theorem 3.2, we first prove the following three auxiliary results, namely, Propositions 3.3-3.5.

Proposition 3.3. Let U and Λ be defined by (3.5) and (3.4), respectively. For
Proof. This result is folklore for M = 1. The proof of Proposition 3.3 for an arbitrary positive integer can (with some modifications) be modelled on the argument for M = 1. However, for the sake of brevity we have decided to omit the proof.

Proposition 3.4. Let U and Λ be defined by (3.5) and (3.4), respectively. Let
C be a closed subset of R M . If C is convex and Proof. Note that it follows from Theorem 2.1 and Proposition 3.3 that if W is a closed subset of R M , then (1) There is a sequence (Δ n ) n with Δ n > 0 and Δ n → 0 such that for all closed subsets C of R M and for all n ∈ N, i ∈ Σ n and u ∈ Σ N , we have (2) For all closed subsets W of R M and all r > 0, we have σ ab ζ U,Λ W (·; r) ≤ σ ab ζ con B(W,2r) , (3.11) σ ab ζ con B(W,r) ≤ σ ab ζ U,Λ W (·; 2r) . Proof. (1) It is well-known that there is a constant c 0 > 0 such that for all i ∈ Σ * and all u ∈ Σ N , we have 1 c0 ≤ diam K i |DS i (πu)| ≤ c 0 , see, for example, [11] or [42]. It is not difficult to see that the desired result follows from this and the fact that the function Λ : (2) Fix r > 0. Let (Δ n ) n be the sequence from (1). Since Δ n → 0, we can find a positive integer N r such that if n ≥ N r , then Δ n ≤ r. Consequently, using (3.10) in Part (1) (3.14) The desired results follow immediately from inequalities (3.13) and (3.14).

Multifractal spectra of self-similar measures
Due to the important role self-similar measures play in fractal geometry, it is instructive to note the following special case of Theorem 3.1.
Let (p 1 , . . . , p N ) be a probability vector, and let μ denote the self-conformal measure associated with the list V , X ,

For a closed set C ⊆ R, we define the self-similar multifractal zeta-function by
For a closed set C ⊆ R and r > 0, we define the self-similar multifractal zeta-function by and if α ∈ R and C = {α} is the singleton consisting of α, then we write ζ C (s; r) = ζ α (s; r), i.e. we write Olsen AEM for q ∈ R. Let C be a closed subset of R. Then the following hold: In particular, if α ∈ R, then we have (1.2) If the OSC is satisfied, then we have In particular, if the OSC is satisfied and α ∈ R, then we have Proof. Theorem 3.6 follows immediately from Theorem 3.1.
It is, of course, also possible to formulate a version of Theorem 3.2 for a finite list of self-similar measures. However, for sake of brevity we have decided not to do this.

Multifractal spectra of ergodic Birkhoff averages
We first fix γ ∈ (0, 1) and define the metric d γ on Σ N by d γ (i, j) = γ max{n | i|n=j|n} ; throughout this section, we equip Σ N with the metric d γ and continuity and Lipschitz properties of functions f : Σ N → R from Σ N to R will Vol. 91 (2017) Multifractal spectra and multifractal zeta-functions 51 always refer to the metric d γ . Multifractal analysis of Birkhoff averages has received significant interest during the past 10 years, see, for example, [2,[12][13][14]33,36,41]. The multifractal spectrum F erg f of ergodic Birkhoff averages of a continuous function f : Σ N → R is defined by for α ∈ R; recall that the projection map π : Σ N → R d is defined in Sect. 3.1 and that S : Σ N → Σ N denotes the shift map. One of the main problems in multifractal analysis of Birkhoff averages is the detailed study of the multifractal spectrum F erg f . For example, Theorem D below is proved in different settings and at various levels of generality in [12][13][14]33,36,41]. Before we can state our result we introduce the following notation. If (x n ) n is a sequence of real numbers, then we write acc n x n for the set of accumulation points of (x n ) n , i.e. acc n x n = x ∈ R x is an accumulation point of (x n ) n .
Also, recall that P S (Σ N ) denotes the family of shift invariant Borel probability measures on Σ N and that h(μ) denotes the entropy of μ ∈ P S (Σ N ). We can now state Theorem D.
Theorem D. [12][13][14]33,36,41] In particular, if the OSC is satisfied and α ∈ R, then we have As a third application of Theorem 2.1 we obtain a zeta-function whose abscissa of convergence equals the multifractal spectrum F erg f of ergodic Birkhoff averages of a Lipschitz function f . This is the content of the next theorem.
and if α ∈ R and C = {α} is the singleton consisting of α, then we write ζ C (s; r) = ζ α (s; r), i.e. we write Then the following hold: In particular, if α ∈ R, then we have (2) If the OSC is satisfied, then we have In particular, if the OSC is satisfied and α ∈ R, then we have We will now prove Theorem 3.7. Recall that the function Λ : Σ N → R is defined by  (3.21) and note that if i ∈ Σ * , then Hence, for C ⊆ R we have In order to prove Theorem 3.7, we first prove the following auxiliary result, namely, Proposition 3.8. (1) There is a sequence (Δ n ) n with Δ n > 0 for all n and Δ n → 0 such that for all closed subsets C of R and for all n ∈ N, i ∈ Σ n and u ∈ Σ N , we have (2) We have .
It is not difficult to see that the desired result follows from (3.23).
(2) This statement follows from Part (1) by an argument very similar to the proof of Part (2) and Part (3) in Proposition 3.5, and the proof is therefore omitted.
We can now prove Theorem 3.7.

Proof of Theorem 3.7.
(1) This statement follows immediately from Theorem 2.1 and Proposition 3.8.
(2) This statement follows immediately from Part (1) using Theorem 2.2 and Theorem D.

Preliminary results
The purpose of this short section is to prove  (5) For k ∈ Σ N and a real number α, the following two statements are equivalent: n log s k|n → α. Proof. Statements (1), (2) and (4) follow easily from the definitions. Statement (3) follows from (1) and (2), and statement (5) follows from (4).

Proof of inequality (2.1)
The purpose of this section is to prove Theorem 5.5 providing a proof of inequality (2.1). The proof of (2.1) is based on results from large deviation theory. In particular, we need Varadhan's [45] large deviation theorem (Theorem 5.1.(i) below), and a non-trivial application of this (namely Theorem 5.1.(ii) below) providing first order asymptotics of certain "Boltzmann distributions".
Definition. Let X be a complete separable metric space and let (P n ) n be a sequence of probability measures on X. Let (a n ) n be a sequence of positive numbers with a n → ∞ and let I : X → [0, ∞] be a lower semicontinuous function with compact level sets. The sequence (P n ) n is said to have the large deviation property with constants (a n ) n and rate function I if the following two conditions hold: Theorem 5.1. Let X be a complete separable metric space and let (P n ) n be a sequence of probability measures on X. Assume that the sequence (P n ) n has the large deviation property with constants (a n ) n and rate function I. Let F : X → R be a continuous function satisfying the following two conditions: (i) For all n, we have exp(a n F ) dP n < ∞. (1) We have lim n 1 a n log exp(a n F ) dP n = − inf x∈X (I(x) − F (x)).
(2) For each n define a probability measure Q n on X by Q n (E) = E exp(a n F ) dP n exp(a n F ) dP n .
Then the sequence (Q n ) n has the large deviation property with constants (a n ) n and rate function ( Proof. Statement (1) Proof. We start by introducing some notation. If i ∈ Σ * , then we define i ∈ Σ N by i = ii . . .. We also define M n : Σ N → P S (Σ N ) by for i ∈ Σ N ; recall, that the map L n : Σ N → P(Σ N ) is defined in Sect. 2. Furthermore, note that if i ∈ Σ N , then M n i is shift invariant, i.e. M n maps Σ N into P S (Σ N ) as claimed. Next, let P denote the probability measure on Σ N given by Finally, we define F : Observe that since Λ is bounded, i.e. Λ ∞ < ∞, we conclude that F ∞ = |t| Λ ∞ < ∞. Also, for a positive integer n, define probability measures P n , Q n ∈ P(P S (Σ N )) by Vol. 91 (2017) Multifractal spectra and multifractal zeta-functions 57 We now prove the following two claims.
This completes the proof of Claim 2.
Combining Claims 1 and 2 shows that Next, we observe that it follows from [9] that the sequence (P n = P • M −1 n ) n ⊆ P P S (Σ N ) has the large deviation property with respect to the sequence (n) n and rate function I : P S (Σ N ) → R given by I(μ) = log N −h(μ). We therefore conclude from Part (1)  Also, since the sequence (P n = P • M −1 n ) n ⊆ P P S (Σ N ) has the large deviation property with respect to the sequence (n) n and rate function I : P S (Σ N ) → R given by I(μ) = log N − h(μ), we conclude from Part (2) of Theorem 5.1 that the sequence (Q n ) n has the large deviation property with respect to the sequence (n) n and rate function ( r)) is closed, it therefore follows from the large deviation property that lim sup This completes the proof.
We will now use Theorem 5.2 to prove Theorem 5.5 providing a proof of inequality (2.1). However, we first prove two small lemmas. Proof. Since f and g are upper semi-continuous with f, g ≥ 0, this result follows easily from the definition of upper semi-continuity, and the proof is therefore omitted. Lemma 5.4. Let X be a metric space and let Φ : X → R be an upper semicontinuous function. Let K 1 , K 2 , . . . ⊆ X be non-empty compact subsets of X with Proof. First note that it is clear that inf n sup x∈Kn Φ(x) ≥ sup x∈∩nKn Φ(x). We will now prove the reverse inequality, namely, inf n sup x∈Kn Φ(x) ≤ sup x∈∩nKn Φ(x). Let ε > 0. For each n, we can choose x n ∈ K n such that Φ(x n ) ≥ sup x∈Kn Φ(x) − ε. Next, since K n is compact for all n and K 1 ⊇ K 2 ⊇ · · · , we can find a subsequence (x n k ) k and a point x 0 ∈ ∩ n K n such that x n k → x 0 . Also, since K n1 ⊇ K n2 ⊇ · · · , we conclude that sup x∈Kn 1 Φ(x) ≥ sup x∈Kn 2 Φ(x) ≥ · · · , whence inf k sup x∈Kn k Φ(x) = lim sup k sup x∈Kn k Φ(x).
We can now state and prove Theorem 5.5.
Theorem 5.5. Let X be a metric space and let U : P(Σ N ) → X be continuous with respect to the weak topology. Let C ⊆ X be a closed subset of X and r > 0.
(1) For brevity write We must now prove that if t > u, then Let t > u and write ε = t−u 3 > 0. It follows from the definition of u that if μ ∈ P S (Σ N ) with Uμ ∈ B(C, r), then we have − h(μ) We deduce from this inequality and Theorem 5.2 that 62 V. Mijović This completes the proof of (1).
(2) It follows immediately from Part (1) that lim sup Vol. 91 (2017) Multifractal spectra and multifractal zeta-functions 63 Also, the function r → sup μ∈PS (Σ N ) , Uμ∈B(C,r) − h(μ) Λ dμ is clearly increasing, and it therefore follows that lim sup Next, since the function U : P(Σ N ) → X is continuous, we conclude that the set U −1 B(C, 1 k ) is closed, and it therefore follows that the set K k = P S (Σ N ) ∩ U −1 B(C, 1 k ) is compact. Also, since the entropy function h : P S (Σ N ) → R is upper semi-continuous (see [47,Theorem 8.2]) with h ≥ 0 and the function f : Λ dμ is upper semi-continuous. Lemma 5.4 applied to Φ therefore implies that Combining (5.12) and (5.13) gives Finally, the desired result follows by combining (5.10), (5.11) and (5.14).

Proof of inequality (2.2)
The purpose of this section is to prove Theorem 6.6 providing a proof of inequality (2.2). We first state and prove a number of auxiliary results. For i, j ∈ Σ N with i = j, we will write i ∧ j for the longest common prefix of i and j (i.e. i ∧ j = u where u is the unique element in Σ * for which there are k, l ∈ Σ N with k = If k ≥ M r , u ∈ Σ k and k, l ∈ Σ N , then it follows from (6.4) that and we therefore conclude from (6.3) that d( UL k (uk) , UL k (ul) ) ≤ r 2 . (2) It follows from (1) that there is a positive integer M r such that if k ≥ M r , u ∈ Σ k and k, l ∈ Σ N , then d( UL k (uk) , UL k (ul) ) ≤ r 2 . We now claim that if m ≥ M r , then In order to prove this inclusion, we fix m ≥ M r and i ∈ Σ N with UL k i ∈ B(C, r 2 ) for all k ≥ m. We must now prove that UL k [i|k] ⊆ B(C, r) for all k ≥ m. We therefore fix k ≥ m and j ∈ [i|k]. We must now prove that UL k j ∈ B(C, r). For brevity write u = i|k. Since j ∈ [i|k] = [u], we can now find (unique) k, l ∈ Σ N such that i = uk and j = ul. We now have However, since k ≥ m ≥ M r and u ∈ Σ k , we conclude that d( UL k (uk) , UL k (ul) ) ≤ r 2 . Also, since k ≥ m, we deduce that UL k i ∈ B(C, r 2 ), 66 V. Mijović, L. Olsen AEM whence dist( UL k i , C ) ≤ r 2 . It therefore follows from (6.5) that This completes the proof. Lemma 6.2. Let X be a metric space and let U : P(Σ N ) → X be continuous with respect to the weak topology. Let C ⊆ X be a closed subset of X. Then recall that dim H denotes the Hausdorff dimension.
Proof. For a subset Ξ of Σ N , we let dim B Ξ denote the lower box dimension of Ξ; the reader is referred to [10] for the definition of the lower box dimension. We will use the fact that dim H Ξ ≤ dim B Ξ for all Ξ ⊆ Σ N , see, for example, [8].
We now introduce the following notation. For brevity write Also, for a positive integer m and a positive real number r > 0, write Observe that if M is any positive integer, then we clearly have for all r > 0. We also observe that it follows from Lemma 6.1 that for each positive number r > 0 there is a positive integer M r such that for all m ≥ M r . It follows from (6.6) and (6.7) that [i] (6.9) for all 0 < δ < s m min and all r > 0. Indeed, fix j ∈ Δ m (r). Now, let k 0 denote the unique positive integer such that if we write j 0 = j|k 0 , then s j0 ≤ δ < s j0 , i.e. s j0 ≈ δ. Since it follows from Proposition 4.1 that s k0 min = s |j0| min ≤ s j0 ≤ δ < s m min , we conclude that k 0 ≥ m, and the fact that j ∈ Δ m (r) therefore implies that UL |j0| . This proves (6.9). Inclusion (6.9) shows that for all 0 < δ < s m min , the family ( for all r > 0. Finally, letting r 0 in (6.12) completes the proof.
In order to state and prove the next lemma we introduce the following notation. Namely, for a Hölder continuous function ϕ : Σ N → R, we will write P (ϕ) for the topological pressure of ϕ. We can now state and prove Lemma 6.3. for all i ∈ Σ N and all m > n. This shows that μ n is the Gibbs state of ϕ n , and that the pressure P (ϕ n ) of ϕ n equals 0, i.e. P (ϕ n ) = 0; cf. [Bo]. This completes the proof of Claim 2.
Claim 3. For each n, the measure μ n is ergodic.
Proof of Claim 3. It follows from Claim 2 that μ n is the Gibbs state of a Hölder continuous function. This implies that μ n is ergodic. This completes the proof of Claim 3.

Claim 4. We have h(μ n ) → h(μ).
Proof of Claim 4. For measurable partitions A, B of Σ, let h(μ; A) and h(μ; A|B) denote the entropy of A with respect to μ, and the conditional entropy of A given B with respect to μ, respectively.
The proof now follows from Claim 1, Claim 3 and Claim 4. log μ(B(i, r)) log r = − h(μ) Λ dμ , (6.19) where μ-ess sup denotes the μ essential supremum. Finally, we note that it is well-known that dim H μ = μ-ess sup i lim inf r 0 log μ(B(i,r)) log r (see, for example, [11]), and it therefore follows immediately from (6.19) that dim H μ = μ-ess sup i lim inf r 0 log μ (B(i,r)) log r The final auxiliary result says that the map C → f U,Λ (C) is upper semicontinuous. In order to state this result we introduce the following notation. For a metric space X, we write F(X) = F ⊆ X F is closed and non-empty (6.20) and we equip F(X) with the Hausdorff metric D; recall, that since X may be unbounded, the Hausdorff distance D is defined as follows, namely, for Proof. Let C ∈ F(X) and ε > 0. Next, it follows from the definition of f U,Λ (C) that we can choose a real number r 0 with 0 < r 0 < 1 such that f U,Λ (C, r 0 ) < f U,Λ (C) + ε. (6.23) Let ρ = r0 2 . We now prove the following claim. Proof of Claim 1. Fix 0 < r < ρ and δ > 0. Since D(F, C) < ρ = r0 2 and r 0 < 1, we first conclude that B(F, r0 2 ) ⊆ B(C, r 0 ). Hence, if i ∈ Π U,Λ δ (F, r), then this and the fact that 0 < r < ρ = r0 2 imply that UL |i| [i] ⊆ B(F, r) ⊆ B(F, ρ) = − h(μ) Λ dμ .
Hence it suffices to prove that σ ab ζ U,Λ C = f U,Λ (C).

Proof of Theorem 2.3
The purpose of this section is to prove Theorem 2.3. It follows from the calculations involved in establishing (9.1) that the product Q U,Λ C (s) converges and that Q U,Λ C (s) = 0 for all s ∈ G. In addition, we deduce from (