Multifractal spectra and multifractal zeta-functions

We introduce multifractal zeta-functions 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 zetafunctions.


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 Ol4,Ol5,Ol6] the authors used the zeta-function technique introduced and pioneered by M. Lapidus et al in the intriguing books  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 self-similar measure similar to the explicit formulas for its Renyi dimensions found in Ol4,Ol5,Ol6]. In particular, and as a first step in this direction, we introduce multifractal zeta-functions providing precise information of very general classes 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.
1.1. 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 ∆ µ (α) of those points x in R d for which the measure µ(B(x, δ)) of the ball B(x, δ) with center x and radius δ behaves like δ α for small δ, i.e. the set ∆ µ (α) = x ∈ K lim rց0 log µ(B(x, r)) log r = α .
If the intensity of the measure µ varies very widely, it may happen that the sets ∆ µ (α) display a fractal-like character for a range of values of α. In this case it is natural to study the Hausdorff dimensions of the sets ∆ µ (α) as α varies. We therefore define the 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 of the empty set to be −∞, i.e. we put 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 [HaJeKaPrSh].
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: and define the r-approximate coarse multifractal spectrum f c µ (α; r) of µ by f c µ (α; r) = lim inf δց0 log N µ,δ (α; r) − log δ .
(1.4) Finally, the coarse multifractal spectrum f c µ (α) of µ is defined by f c µ (α) = lim rց0 f c µ (α; r) (1.5) (it is clear that this limit exists since f c µ (α; r) is a monotone function of r). We note that it is easily seen that f µ (α) ≤ f c µ (α) , and that this inequality may be strict, see, for example, [Fa1].
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 the box dimension that dim B E = lim δ→0 log M δ (E) − log δ (provided the limit exists) and we clearly have ( 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 T µ (q; E) = lim inf δց0 log V µ,δ (q; E) − log δ , T µ (q; E) = lim sup δց0 log V µ,δ (q; E) − log δ .
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. V δ (E) = L d ( B(E, δ) ), then it is well-known that dim B E = lim δ→0 log( 1 r d V δ (E)) − log δ (provided the limit exists) and we clearly have (1.7) 1.3. The Multifractal Formalism. Based on a remarkable insight together with a clever heuristic argument, it was suggested by theoretical physicists Halsey et al. [HaJeKaPrSh] 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) We can now state the "Multifractal Formalism".
for all α.
During the past 20 years there has been an enormous interest in verifying the Multifractal Formalism and computing the multifractal spectra of measures in the mathematical literature.  [Fa2,Pe] and the references therein. Summarizing the previous paragraph somewhat more succinctly, previous work has almost entirely concentrated on the following problem: Previous work: Previous work has concentrated on finding the limiting behaviour of the following ratios, namely, log M µ,δ (q) − log δ and log N µ,δ (α; r) − log δ .
Indeed, computing the Renyi dimensions τ µ (q) and τ µ (q) involves analysing the limiting behaviour of log Iµ,r (q) − log r , and computing the coarse multifractal spectrum f c µ (α; r) involves analysing the limiting behaviour of Due to the importance of the quantities M µ,δ (q) and N µ,δ (α; r) it is clearly desirable not only to find expressions for the limiting behaviour of 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 zeta-functions. Indeed, renewal theory techniques were introduced and pioneered by Lalley [La1,La2,La2] in the 1980's, and later investigated further by Gatzouras [Ga], Winter [Wi] and most recently Kesseböhmer & Kombrink [KeKo], 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, despite, or perhaps 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 [LapPea1,LapPea2,LapPeaWi, 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 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 the q-moments M µ,δ (q) and the associated Renyi dimenions 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 [Le-VeMe,Ol4] (see also the surveys [Ol5,Ol6]) the authors introduced a one-parameter family of multifractal zeta-functions and established explicit formulas for the integral q-moments V µ,δ (q) expressing V µ,δ (q) as a sum involving the residues of these zeta-functions, and in [Ol1] 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 zeta-functions [LapRo,LapLe-VeRo]. However, the multifractal zetafunctions in [LapRo,LapLe-VeRo] serve very different purposes and are significantly different from the multifractal zeta-functions introduced in [Le-VeMe, Ol2,Ol4]. 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 zeta-functions providing precise information of very general classes of multifractal spectra, including, for example, the spectra f µ and f c µ of selfsimilar 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 [MiOl]. In order to illustrate the ideas involved we now consider a simple example.
1.4. 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 [Fa1,Hu]. 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 if the OSC is satisfied and if we define for all α ∈ R 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 δց0 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 straight forward calculation (which we present below) shows that the abscissa of convergence σ ab ( ζ sim log ri ], then it is easily seen that that for all i ∈ Σ * , we have log p i log r i = α, whence σ ab ( ζ sim α ) = −∞, and inequality (1.12) is therefore trivially satisfied. On the other hand, if α ∈ [min i log pi log ri , max i log pi log ri ], then it follows from [CaMa, Fa1,Pa] that there is a (unique) q ∈ R with f µ (α) = f c µ (α) = αq + β(q). Hence, for each ε > 0, we have (using the fact that i p q i r However, it is also clear that we, in general, do not have equality in (1.12). Indeed, the set { log p i log r i | i ∈ Σ * } is clearly countable (because Σ * is countable) and if α ∈ R \ { log p i log r i | i ∈ Σ * }, then σ ab ( ζ α ) = −∞ (because the series (1.11) that defines ζ sim α (s) is obtained by summing over the empty set). Since it also follows from [CaMa, Fa1,Pa] log ri ), we therefore conclude that: for all except at most countably many α ∈ (min (1.13) It follows from the above discussion that while the definition of ζ sim α (s) is "natural", it is not does not encode sufficient information allowing 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 there are nevertheless many sequences (i n ) n of strings i n ∈ Σ * for which the ratios log p in log r in converges 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 non-degenerate 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 for those complex numbers s for which the series converges absolutely. Observe that if I = {α}, then 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 enlarge "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 ց 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 ζ sim α (·; r) by ζ sim α (s; r) = ζ sim [α−r,α+r] (s) The next result, which is an application of one of our main results (see Theorem 3.6), shows that the multifractal zeta-functions ζ sim α (·; r) encode sufficient information allowing us to recover the multifractal spectra f µ (α) and f c µ (α) by letting r ց 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 zeta-function ζ sim I associated with the enlarger "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 enlarger "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 (see Theorem 3.6), shows that this is possible. More precisely the result shows that if the enlarger "target" set I satisfies condition (1.16), then the multifractal zeta-function ζ sim I associated with the enlarger "target" set I encode sufficient information allowing 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.
Theorem 1.2. Fixed targets. Assume that the list (S 1 , . . . S N ) satisfies the OSC and let µ be the self-similar measure defined by (1.9). For a closed interval I, let ζ sim I be defined by (1.14). If where σ ab ζ sim I denotes the abscissa of convergence of the zeta-function ζ sim I .
We emphasise that Theorem 1.1 and Theorem 1.2 are presented in order to motive this work and are special cases of the substantially more general and abstract theory of multifractal zeta-function developed in this paper.
The next section, i.e. Section 2, describes the general framework developed in this paper and list our main results. In Section 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.

Statements of main results.
2.1. 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. Fix a positive integer N . Let Σ = {1, . . . , N } and for a positive integer n, write i.e. Σ n is the family of all strings i = i 1 . . . i n of length n with i j ∈ {1, . . . , N } and Σ * is the family of all finite strings i = i 1 . . . i m with m ∈ N and i j ∈ {1, . . . , N }. Also write i.e. Σ N is the family of all infinite strings i = i 1 i 2 . . . with i j ∈ {1, . . . , N }. 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 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 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 where B(C, r) = {x ∈ X | dist(x, C) ≤ r} denotes the closed r neighborhood of C.
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 ζ(s) = i a s i for those complex numbers s for which the series converges. The abscissa of convergence of ζ is defined by Our main results, i.e. Theorem 2.1 and Theorem 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 Theorem 2.1 and Theorem 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 and . 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 Section 1.4, we will attempt to relate the abscissae of convergence of the multifractal zeta-functions 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 Section 2.2 and the fixed target approach is discussed in Section 2.3.

2.2.
First main result. The shrinking target approach: finding lim rց0 σ ab ζ U,Λ C (·; r) . 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 ց 0, and attempt to relate the limiting behaviour of the abscissa 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 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 ց 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 U µ ∈ C. In Section 3 we show that in many important cases the limit lim rց0 σ ab ζ U,Λ C (·; r) equals the traditional multifractal spectra.
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 Section 5 using techniques from the theory of large deviations. Inequality (2.2) is proven in Section 6 using techniques from ergodic theory. Finally, inequality (2.3) follows directly from the definitions and is proved in Section 7.
2.3. 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 zetafunction ζ U,Λ C associated with the "target" set C to the values of the multifractal spectrum coarse multifractal spectrum f U,Λ (C). Of course, the example in Section 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 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 U µ ∈ 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.
Theorem 2.2. Fixed targets. Fix a positive integer M . Let U : P(Σ N ) → R M be continuous with respect to the weak topology. Let C ⊆ R M be a closed subset of R M and assume that f U,Λ is inner continuous at C.
(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 Section 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 is 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.
Theorem 2.3 shows that ζ U,Λ C has an Euler product. In Theorem 2.3 we use the following notation, namely, if f is a holomorphic function that does not attain the value 0, then we let Lf denote the logarithmic derivative of f , i.e. Lf = f ′ f . We can now state Theorem 2.3.
Theorem 2.3. Euler product. Let X be a metric space and let U : P(Σ N ) → X be continuous with respect to the weak topology. Assume that (1) For complex numbers s with Re(s) > σ ab ( ζ U,Λ C ), the product Theorem 2.3 is proved in Section 9.
3. Applications: multifractal spectra of measures and multifractal spectra of ergodic Birkhoff averages We will now consider several of applications of Theorem 2.1 and Theorem 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.
3.1. Multifractal spectra of self-conformal measures. Since our examples are formulated in the setting of self-conformal (or self-similar) measures we begin be recalling the definition of selfconformal (and self-similar) measures. A conformal iterated function system with probabilities is a list It follows from [Hu] 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 , (S i ) i=1,... ,N ; in particular, if each map S i is a contracting similarity, then the set K is called the self-similar set associated with the list V , X , .. ,N is a probability vector then it follows from [Hu] 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 V , X , .. ,N . We will frequently assume that the list V , X , (S i ) i=1,... ,N satisfies the Open Set Condition defined below. Namely, the list Next, we define the natural projection map π : Σ N → K. However, we first make the follwing 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 . ∈ 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 rց0  [Mo]) have shown that the set of divergence points, i.e. the set of points x for which the limit lim rց0 log µB(x,r) log r does not exist, typically is highly "visible" and "observable", namely it has full Hausdorff dimension. More precisely, it follows from [BaSc] that if the OSC is satisfied and t denotes the Hausdorff dimension of K, then Hausdorff measure restricted to K. This suggests that the set ∆ µ has a surprising rich and complex fractal structure, and in order to explore this more carefully Olsen & Winter [OlWi1,OlWi2] 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 rց0 f (r) for the set of accumulation points of f as r ց 0, i.e. acc rց0 ϕ(r) = x ∈ M x is an accumulation point of f as r ց 0 .  [Ca,Vo] for earlier but related results in a slightly different setting).

In [OlWi1] Olsen & Winter introduced and investigated the generalised Hausdorff multifractal spec
Theorem B [LiWuXi, Mo,OlWi1]. 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 Theorem 2.1 and Theorem 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.
Theorem 3.1. Multifractal zeta-functinons for multifractal spectra of self-conformal measures. 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-conformal multifractal zeta-function by 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 for q ∈ R. Let C be a closed subset of R. Then the following hold: In particular, if α ∈ R, then we have lim rց0 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 (2.1) If C is an interval and (2.2) If C is an interval and • C ∩ − β ′ (R) = ∅ and the OSC is satisfied, then we have

Proof
This follows immediately from the more general Theorem 3.2 in Section 3.2 by putting M = 1.
3.2. Mixed multifractal spectra of self-conformal measures. Recently mixed (or simultaneous) multifractal spectra have generated an enormous interest in the mathematical literature, see [BaSa,Mo,Ol2,Ol3]. Indeed, previous result (Theorem A and Theorem 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. For m = 1, . . . , M , let (p m,1 , . . . , p m,N ) be a probability vector, and let µ m denote the self-conformal measure associated with the list V , X , The mixed multifractal spectrum f µ µ µ of the list µ µ µ = (µ 1 , . . . , µ M ) is defined by log µ 1 (B(x, r)) log r , . . . , log µ M (B(x, r)) log r = α α α for α α α ∈ R M . 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 extensions of the traditional mixed multifractal spectrum F µ (α α α), namely, if C = {α α α} is a singleton consisting of the point α α α, then clearly F µ (C) = f µ (α α α). Assuming the OSC, the generalised mixed multifractal spectrum F µ (C) can be computed [Mo,Ol2]. In order to state the result from [Mo,Ol2], we introduce the following definitions. Define Λ, Φ m : Σ N → R for m = 1, . . . , M by Λ(i) = log |DS i1 (πSi)| and Φ m (i) = log p m,i1 for i = i 1 i 2 . . . ∈ Σ N , and write Φ Φ Φ = (Φ 1 , . . . , Φ M ). Define β : R M → R by 0 = P β(q)Λ + q|Φ Φ Φ for q ∈ R M ; recall that if ϕ : Σ N → R is a Hölder continuous map, then P (ϕ) denotes the pressure of ϕ. Also, for x, y ∈ R M , we let x|y denote the usual inner product of x and y, and if ϕ : R M → R is a function, we define the Legendre transform ϕ * : The generalised mixed multifractal spectra f µ µ µ and F µ µ µ are now given by the following theorem.
Theorem C. [Mo,Ol2]. Let µ 1 , . . . , µ M be defined by (3.3) and let C ⊆ R M be a closed set. Put µ µ µ = (µ 1 , . . . , µ M ). If the OSC is satisfied, then we have In particular, if the OSC is satisfied and α α α ∈ R M , then we have As a second application of Theorem 2.1 and Theorem 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. The is the content of the next theorem.
For a closed set C ⊆ R M , we define the self-conformal multifractal zeta-function by For a closed set C ⊆ R M and r > 0, we define the self-conformal multifractal zeta-function by for q ∈ R M . Let C be a closed subset of R M . Then the following hold: (1.1) We have lim rց0 σ ab ζ con C (·; r) = sup α α α∈C β * (α α α) .
(1.2) If the OSC is satisfied, then we have lim rց0 σ ab ζ con C (·; r) = sup (2.2) If C is convex and • C ∩ − ∇β(R M ) = ∅ and 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 It is well-known that Λ satisfies Conditions (C1)-(C3) in Section 2.1. Also, a straight forward calculation shows that sup k∈ 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.

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.
(4) Let C be a closed subset of R M . If C is convex and

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, [Fa2] or [Pa]. It is not difficult to see that the desired result follows from this and the fact that the function Λ : Σ N → R defined by Λ(i) = log |DS i1 (πSi)| for i = i 1 i 2 . . . ∈ Σ N satisfies Conditions (C1)-(C3) in Section 2.1.
(2) Fix r > 0. Let (∆ n ) n be the sequence from (1). Since ∆ n → 0, we can find a positive integer N r such that if n ≥ N r , then ∆ n ≤ r. Consequently, using (3.10) in Part (1), for s ∈ R, we have (3.14) The desired results follow immediately from inequalities (3.13) and (3.14).
We can now prove Theorem 3.2.
3.3. Multifractal spectra of self-similar measures. Due to important role self-similar measures play in fractal geometry, it is instructive to note the following special case of Theorem 3.1.
Theorem 3.6. Multifractal zeta-functinons for multifractal spectra of self-similar measures. Assume that the maps S 1 , . . . , S N are contracting similarities and let r i denote the contraction ratio of S i . For i = i 1 . . . i n ∈ Σ * , let r i = r i1 · · · r in . 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 for q ∈ R. Let C be a closed subset of R. Then the following hold: In particular, if α ∈ R, then we have lim rց0 σ ab ζ sim 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 (2.1) If C is an interval and  log ri = ∅ and the OSC is satisfied, then we have log µ(B(x, r)) log r ⊆ C .

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 self-similar measures. However, for sake of brevity we have decided not to do this.
3.4. 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 always refer to the metric d γ . Multifractal analysis of Birkhoff averages has received significant interest during the past 10 years, see, for example, [BaMe,FaFe,FaFeWu,FeLaWu,Oli,Ol3,OlWi2]. 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 Section 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 [FaFe,FaFeWu,FeLaWu,Oli,Ol3,OlWi2]. Before we can state 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. [FaFe,FaFeWu,FeLaWu,Oli,Ol3,OlWi2]. Let f : Σ N → R be a Lipschitz function. Define Λ : Σ N → R by Λ(i) = log |DS i1 (πSi)| for i = i 1 i 2 . . . ∈ Σ N . Let C be a closed subset of R. If the OSC is satisfied, then 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.
Theorem 3.7. Multifractal zeta-functinons for multifractal spectra of of ergodic Birkhoff averages. Let f : Σ N → R be a Lipschitz function.
For i ∈ Σ * , let and write i = iii . . . ∈ Σ N . For a closed set C ⊆ R M , we define the self-similar multifractal zetafunction of f by ζ erg 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: (1) We have 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 (∆ n ) n with ∆ n > 0 for all n and ∆ 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 lim rց0 σ ab ζ erg C (·; r) = lim rց0 σ ab ζ U,Λ C (·; r) .

Proof
(1) Let Lip(f ) denote the Lipschitz constant of f . It is clear that for all n ∈ N, i ∈ Σ n and u ∈ Σ N , we have . (3.23) 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 Proposition 4.1 establishing various auxiliary results needed for the proof of Theorem 2.1. Let c min and c max be the constants from the Condition (C2) in Section 2.1 and write s min = e cmin , s max = e cmax . (4.1) we can now state and prove Proposition 4.1. Recall, that for i ∈ Σ n , the number s i is defined by Proposition 4.1. Let c be the constant from Condition (C3) in Section 2.1. Let i, j ∈ Σ * . ( (4) For k ∈ Σ N and a positive integer n, we have exp (5) For k ∈ Σ N and a real number α, the following two statements are equivalent: (ii) 1 n log s k|n → α.

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 [Va] 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 condistions hold: (i) For each closed subset K of X, we have lim sup n 1 a n log P n (K) ≤ − inf x∈K I(x) ; (ii) For each open subset G of X, we have lim inf n 1 a n log P n (G) ≥ − inf x∈G I(x) .
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 < ∞ .
(ii) We have lim M→∞ lim sup n 1 a n log {M≤F } exp(a n F ) dP n = −∞ .
(Observe that the Conditions (i)-(ii) are satisfied if F is bounded.) Then the following statements hold.
(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
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 Section 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 We now prove the following two claims.
This completes the proof of Claim 2.
Combining Claim 1 and Claim 2 shows that Let c be the constant from Condition (C3) in Section 2.1, and notice that it follows from Proposition 4.1 that if i ∈ Σ N and n is a positive integer, then we have s t i|n ≤ c |t| exp( t n−1 k=0 ΛS k ( i|n ) ). We conclude from this and (5.2) that Next, we observe that it follows from [El] 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) of Theorem 5.1 that lim sup 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 (I − F ) − inf ν∈PS (Σ N ) (I(ν) − F (ν)). As the set {U ∈ B(C, r)} = U −1 (B(C, 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.
Lemma 5.3. Let X be a metric space and let f, g : X → R be upper semi-continuous functions with f, g ≥ 0. Then f g is upper semi-continuous.

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 semi-continuous function. Let K 1 , K 2 , . . . ⊆ X be non-empty compact subsets of X with K 1 ⊇ K 2 ⊇ . . . . Then

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 Finally, letting ε ց 0 gives the desired result.
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.

Proof
(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 U µ ∈ B(C, r), then we have − h(µ) where we have used the fact that Λ dµ < 0 because Λ < 0. This implies that if µ ∈ P S (Σ N ) with U µ ∈ B(C, r), then We deduce from this inequality and Theorem 5.2 that lim sup This completes the proof of (1).
(2) It follows immediately from Part (1) that (5.10) 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 [Wa,Theorem 8.2]) with h ≥ 0 and the function f : continuous) with f ≥ 0, we conclude from Lemma 5.3 that the function Φ : is upper semi-continuous. Lemma 5.4 applied to Φ therefore implies Combining (5.12) and (5.13) gives (5.14) 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 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 = k 1 k 2 . . . and l = l 1 l 2 . . . such that k 1 = l 1 , i = uk and j = ul). We will always equip Σ N with the metric d Σ N defined by for i, j ∈ Σ N . In the results below, we will always compute the Hausdorff dimension of a subset of Σ N with respect to the metric d Σ N . Note that when Σ N is equipped with the metric d Σ N , then Lemma 6.1. Let (X, d) be a metric space and let U : P(Σ N ) → X be continuous with respect to the weak topology. Let C be a closed subset of X and r > 0.
(1) There is a positive integer M r such that if k ≥ M r , u ∈ Σ k and k, l ∈ Σ N , then (2) There is a positive integer M r such that if m ≥ M r , then

Proof
(1) For a function f : and define the metric L in P(Σ N ) by we note that it is well-known that L is a metric and that L induces the weak topology. Since U : P(Σ N ) → X is continuous and P(Σ N ) is compact, we conclude that U : P(Σ N ) → X is uniformly continuous. This implies that we can choose δ > 0 such that all measures µ, ν ∈ P(Σ N ) satisfy the following implication: Next, choose a positive integer M r such that 1 M r (1 − s max ) < δ ; (6.4) recall, that s max is defined in (4.1). 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( U L k (uk) , U L 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( U L k (uk) , U L 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 U L k i ∈ B(C, r 2 ) for all k ≥ m. We must now prove that U L k [i|k] ⊆ B(C, r) for all k ≥ m. We therefore fix k ≥ m and j ∈ [i|k]. We must now prove that U L 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 (6.5) However, since k ≥ m ≥ M r and u ∈ Σ k , we conclude that d( U L k (uk) , U L k (ul) ) ≤ r 2 . Also, since k ≥ m, we deduce that U L k i ∈ B(C, r 2 ), whence dist( U L 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 [Fa1] 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, [Ed].
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 for all r > 0. Fix a positive integer m. We now prove that ,r) [i] (6.9) for all 0 < δ < s m min and all r > 0. Indeed, fix j ∈ ∆ 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 ∈ ∆ m (r) therefore implies that U L |j0| [j 0 ] = U L k0 [j|k 0 ] ⊆ B(C, r) This shows that j 0 ∈ Π U,Λ δ (C, r), whence j ∈ [j|k 0 ] = [j 0 ] ⊆ ∪ i∈Π U,Λ δ (C,r) [i]. This proves (6.9). Inclusion (6.9) shows that for all 0 < δ < s m min , the family ( for all r > 0. Since (6.10) holds for all m, we conclude that for all r > 0. Combining (6.8) and (6.11) now shows that for all r > 0. Finally, letting r ց 0 in (6.12) completes the proof.
In order to statement 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. Lemma 6.3. Let µ ∈ P S (Σ N ) with supp µ = Σ N . (Here supp µ denotes the topological support of µ.) Then there exists a sequence (µ n ) n of probability measures on Σ N satisfying the following three conditions.
(2) For each n, the measure µ n is ergodic. ( Proof Fix a positive integer n. Since supp µ = Σ N , we deduce that µ[i] > 0 for all i ∈ Σ * . Hence, for m ∈ N and i 1 . . . i m ∈ Σ m , we can define p n,i1...im by for n < m. (6.13) Since clearly i p n,i = 1 and i p n,i1...imi = p n,i1...im for all m and all i 1 . . . i m ∈ Σ m , there exists a (unique) probability measure µ n on Σ N such that for all m and all i 1 . . . i m ∈ Σ m (cf. [Wa,p. 5]).
Claim 1. We have µ n → µ weakly. Proof of Claim 1. It follows from definition (6.13) that µ n [i] = µ[i] for all i ∈ Σ n . This clearly implies that µ n → µ weakly. This completes the proof of Claim 1.
Claim 2. For each n, there is a Hölder continuous function ϕ n : Σ N → R such that the following conditions hold.
(1) P (ϕ n ) = 0 , (2) The measure µ n is a Gibbs state of ϕ n . Proof of Claim 2. We first note that µ n is shift invariant. Indeed, since µ is shift invariant, a small calculation shows that i µ n [ii] = µ n [i] for all i ∈ Σ * . This implies that µ n (S −1 [i]) = µ n [i] for all i ∈ Σ * , whence µ n (S −1 B) = µ n (B) for all Borel sets B.
Next we show that µ n is a Gibbs state for a Hölder continuous function. Define ϕ n : Σ N → R by The map ϕ n is clearly Hölder continuous, and it follows from the definition of µ n that 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 a 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.
This completes the proof of Claim 4.
The proof now follows from Claim 1, Claim 3 and Claim 4.
The next auxiliary result provides a formula for the upper Hausdorff dimension of is a probability measure. If µ is a probability measure on Σ N , we define the upper Hausdorff dimension of µ by (Recall that dim H denotes the Hausdorff dimension.) The next result provides a formula for the upper Hausdorff dimension of an ergodic probability measure on Σ N . This result is folklore and follows from the Shannon-MacMillan-Breiman theorem and the ergodic theorem. However, for sake of completeness we have decided to include the short proof.
Proposition 6.4. Let µ be an ergodic probability measure on Σ N .

Proof
Since µ is ergodic, it follows from the Shannon-MacMillan-Breiman theorem that Next, for each i ∈ Σ N and r > 0, let n i,r denote the unique integer such that s i|n i,r < r ≤ s i|n i,r . It follows from the definition of the metric d Σ N on Σ N (see (6.1) and (6.2)) that B(i, r) = [i|n i,r ]. Also, if we let c denote the constant from Condition (C3) in Section 2.1, then it follows from Proposition 4.1 that s i|n i,r < r ≤ s i|n i,r ≤ c smin s i|n i,r . Combining these facts, we now deduce from (6.18) that 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 rց0 log µ(B(i,r)) log r (see, for example, [Fa2]), and it therefore follows immediately from (6.19) that dim H µ = µ-ess sup i lim inf rց0 The final auxiliary result says that the map C → f U,Λ (C) is upper semi-continuous. 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 E, F ∈ F (X), write ∆(E, F ) = min sup x∈E dist(x, F ) , sup y∈F dist(y, E) (6.21) and define D by D = min(1, ∆) . (6.22) Lemma 6.5. Let X be a metric space and let U : P(Σ N ) → X be continuous with respect to the weak topology. Equip F (X) with the Hausdorff metric D. Then the function f U,Λ : F (X) → R is upper semicontinuous, i.e. for each C ∈ F (X) and each ε > 0, there exists a real number ρ > 0 such that if F ∈ F (X) and D(F, C) < ρ, then
We now claim that if F ∈ F (X) and D(F, C) < ρ, then To prove this, let F ∈ F (X) with D(F, C) < ρ. It follows from Claim 1 and (6.23) that if 0 < r < ρ, then Since this inequality holds for all 0 < r < ρ, we finally conclude that f U, We can now state and prove the main result in this section, namely, Theorem 6.6 providing a proof of inequality (2.2).
Theorem 6.6. 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. We have Let ε > 0. Next, fix µ ∈ P S (Σ N ) with U µ ∈ C. We will now prove that Let F (X) be denied as in (6.20), i.e. F (X) = {F ⊆ X | F is closed and non-empty}, and and equip F (X) with the Hausdorff metric D, see (6.21) and (6.22). It follows from Lemma 6.5 that the function f U,Λ : F (X) → R is upper semi-continuous, and we can therefore choose ρ ε > 0 such that: if F ∈ F (X) and D(F, C) < ρ ε , then (6.26) Next, observe that we can choose an S-invariant probability measure γ on Σ N such that supp γ = Σ N . For t ∈ (0, 1), we now write µ t = (1 − t)µ + tγ ∈ P S (Σ N ). As U is continuous with U µ ∈ C and µ t → µ weakly as t ց 0, there exists 0 < t ε < 1 such that for all 0 < t < t ε , we have dist(U µ t , C) < ρ ε .
Claim 1. For all 0 < t < t ε , we have Proof of Claim 1. Using the fact that the entropy map h : P S (Σ) → R is affine (cf. [Wa]) we conclude that However, since Λ is continuous and µ t,n → µ t weakly (by (6.28)), we conclude that Λ dµ t,n → Λ dµ t . We deduce from this and the fact that h(µ t,n ) → h(µ t ) (by (6.30)) that (6.34) Combining (6.33) and (6.34) now yields Also, since µ t,n is ergodic (by (6.29)), it follows from Proposition 6.4 that dim H µ t,n = − h(µt,n) log N , and we therefore conclude from (6.35) that This completes the proof of Claim 1.
Proof of Claim 2. It follows immediately from the ergodicity of µ t,n and the ergodic theorem that µ t,n ({i ∈ Σ N | lim m L m i = µ t,n }) = 1. Hence This completes the proof of Claim 2.
Since µ ∈ P S (X) with U µ ∈ C was arbitrary, it follows immediately from (6.25) that Finally, letting ε ց 0 gives the desired result.

Proof of inequality (2.3)
The purpose of this section is to prove Theorem 7.1 providing a proof of inequality (2.3).
Theorem 7.1. 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.
Proof of Claim 1. Indeed, if i = i 1 . . . i m ∈ Σ m with s i ≈ ρ n , then s i ≤ ρ n < sî, whence s i ≤ ρ n . It also follows from Proposition 4.1 that s i = sî im ≥ 1 c sîs im > 1 c ρ n s min = smin cρ ρ n+1 ≥ ρ n+1 where the last inequality is due to the fact that smin cρ ≥ 1 because ρ < min( smin c , δ ε ) ≤ smin c . This completes the proof of Claim 1. Also, for n ∈ N and i ∈ Σ * , the following implication follows from Claim 1: We conclude immediately from (7.3) that However, if i ∈ Π U s,ρ n (C, r), then s i ≈ ρ n , and it therefore follows from Claim 1 that ρ n+1 < s i ≤ ρ n , whence s i ≥ ρ nt ρ |t| . We conclude from this and (7.5) that Finally, since ρ n ≤ ρ < min( smin c , δ ε ) ≤ δ ε , we deduce from (7.1) that ρ −nt = (ρ n ) −t ≤ N U s,ρ n (C, r). This and (7.6) now implies that This completes the proof of Claim 2.

Proof
Let y ∈ B I(C, ε) , r . We must now prove that y ∈ C. Assume, in order to reach a contradiction, that y ∈ C. Since I(C, ε) is a closed, it follows that we can find x ∈ I(C, ε) such that |y − x| = dist y , I(C, ε) . Also, since x ∈ I(C, ε) ⊆ C and y ∈ C, it follows from Lemma 8.2 that there is v ∈ [[x, y]] ∩ ∂C. We now conclude that r ≥ dist y , I(C, ε) [since y ∈ B I(C, ε) , r ] = |y − x| ≥ ε .

Proof of Theorem 2.2
We first note that it follows from Theorem 2.1 that 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.

Proof of Theorem 2.3
For brevity write G = {s ∈ C | Re(s) > σ ab ( ζ U,Λ C )}. Since sup |i|=n 1 log s i → 0 as n → ∞ (because sup |i|=n s i → 0 as n → ∞), we conclude that the series Z 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 (9.1) that for all s ∈ G, we have