Multifractal analysis for conformal graph directed Markov systems

We derive the multifractal analysis of the conformal measure (or equivalently, the invariant measure) associated to a family of weights imposed upon a (multi-dimensional) graph directed Markov system (GDMS) using balls as the filtration. This analysis is done over a large subset of the limit set. In particular, it coincides with the limit set when the GDMS under scrutiny satisfies a boundary separation condition. It also applies to more general situations such as real or complex continued fractions.


Introduction
Multifractal formalism origins from physics and mathematics (among others, see [6], [2], [3] and [4]). In this latter paper, strong hints of parallels between multifractal theory and the theory of statistical physics were suggested. Some of the first rigorous mathematical results on multifractals can be found in [1] and [15]. Since then, many papers have been written on this subject (for instance, see [10], [11], [12] and [14]). In particular, Pesin [13] developed a general framework in which multifractal formalism can be derived.
We now briefly describe our setting. Let µ be a Borel probability measure on a metric space X. The measure µ is said to have local dimension α at a point x ∈ X if lim r→0 log µ(B(x, r)) log r = α.
For each number α ≥ 0, let X µ (α) be the set of points x ∈ X where the measure µ has local dimension α, and let f µ (α) be the Hausdorff dimension of the set X µ (α). The map α → f µ (α) is called the (fine Hausdorff) multifractal spectrum of the measure µ.
The multifractal analysis of equilibrium states for a natural potential function and a natural family of Hölder continuous weights was performed in [5] for infinite conformal iterated function systems and in [9] for infinite conformal graph directed Markov systems. In both cases, the authors used at every point in the limit set of the given system the natural filtration generated by the initial blocks of the word that encodes the point. In other terms, the analysis was carried out using cylinders.
Research of the first author was supported by NSERC (Natural Sciences and Engineering Research Council of Canada). Research of the second author was supported in part by the NSF Grant DMS 0700831. Part of the work has been done while the second author was visiting the Max Planck Institute in Bonn, Germany. He wishes to thank the institute for its support.
Aiming to give the multifractal analysis a transparent geometrical meaning, we shall derive in the sequel the multifractal analysis for cofinitely regular graph directed Markov systems (GDMSs) using as the filtration a base of balls centred at the given point. Until now, the question of the analysis of balls, which has already been solved in the case of finite systems, remained open for infinite systems.
We conduct our analysis over the full set of parameters on which it can be expected to hold. Moreover, we perform it on a large and dynamically significant subset J r of the limit set J. In fact, under a mild boundary separation condition (for example, the separation condition) the sets J r and J coincide. An additional geometric flavor of our analysis results from the fact that we concentrate on a geometrically meaningful family of Hölder continuous weights. All our results apply to a large class of GDMSs, one-and multi-dimensional alike, including real and complex continued fractions.
Let us describe the content of the paper more precisely. In section 2, we recall the basic definition of conformal graph directed Markov system. In section 3, we describe the Hölder families of weights F and F q,t we shall work with and study the properties of the pressure P (q, t) and temperature T (q) they determine. In section 4, we carry out the multifractal analysis of the conformal measure m F (or equivalently, the invariant measure µ F ) associated to a family of weights F . This analysis is done over a subset J r of J and conducted by means of balls. In particular, we show that for each α there is an auxiliary measure that witnesses the Hausdorff dimension of the set J r,µ (α) and that the f r,µ (α) curve is the Legendre transform of the temperature function T (q). In section 5, we derive the multifractal analysis of the conformal measure m F (or equivalently, the invariant measure µ F ) under additional conditions on the GDMS. In subsection 5.1, we observe that J r = J for all GDMSs which satisfy a boundary separation condition. Real continued fractions with the digit 1 deleted are an important example of such systems. In subsection 5.2, we derive the multifractal analysis over J under three conditions and show that there are families of one-dimensional conformal iterated function systems that meet these three conditions. Real continued fractions (with or without the digit 1) are a good example of such a family.

Preliminaries on Graph Directed Markov Systems
Let us first describe the setting of conformal graph directed Markov systems introduced in [9]. Graph directed Markov systems are based on a directed multigraph (V, E, i, t) and an associated incidence matrix A. The multigraph consists of a finite set V of vertices, a countable (finite or infinite) set of edges, and two functions i, t : E → V that indicate for each directed edge e ∈ E its initial vertex i(e) and its terminal vertex t(e), respectively. The matrix A : E × E → {0, 1} is an edge incidence matrix and thus tells which edges may follow a given edge. Moreover, it respects the multigraph, that is, if A e 1 e 2 = 1 then t(e 1 ) = i(e 2 ). It is thereafter natural to define the set of all one-sided infinite A-admissible words The set of all subwords of E ∞ A of length n ∈ IN will be denoted by E n A , whereas the set of all finite subwords will be denoted by E * A = ∪ n∈IN E n A . From a dynamical point of view, we will consider the left shift map σ : E ∞ A → E ∞ A which drops the first letter of each word. A graph directed Markov system (GDMS) consists of a directed multigraph and an edge incidence matrix together with a set of non-empty compact subsets {X v } v∈V of a common Euclidean space IR d , a number 0 < s < 1, and for every e ∈ E a one-to-one contraction ϕ e : X t(e) → X i(e) with Lipschitz constant at most s.
A GDMS is called iterated function system (IFS) provided that A ef = 1 if and only if t(e) = i(f ) and that V is a singleton.
For ω ∈ E n A , n ∈ IN, we define Note that the functions i and t extend naturally to E * A by setting i(ω) := i(ω 1 ) and t(ω) = t(ω |ω| ). The main object of our interest will be the limit set J of S. This set is the image of the symbolic space E ∞ A under a coding map π. Indeed, given any ω ∈ E ∞ A , the sets ϕ ω|n (X t(ω|n) ) form a decreasing sequence of non-empty compact sets whose diameters converge to zero. Therefore their intersection is a singleton, and we denote its element by π(ω). This defines the coding map π : E ∞ A → X, where X := v∈V X v is the disjoint union of the compact sets X v . Clearly, π is a continuous function when E ∞ A is equipped with the topology generated by the cylinders [e] n = {ω ∈ E ∞ A | ω n = e}, e ∈ E, n ∈ IN. Hence the limit set of the GDMS S is ϕ ω|n (X t(ω|n) ).
Recall also (cf. section 4.2 in [9]) that a GDMS S = {ϕ e } e∈E is called conformal (and thereafter a CGDMS) if the following conditions are satisfied.
(i) For every v ∈ V , the set X v is a compact, connected subset of IR d which is the closure of its interior (i.e. X v = Int IR d (X v )); (ii) (Open set condition (OSC)) For all e, f ∈ E, e = f , IR d and such that for every e ∈ E with t(e) = v, the map ϕ e extends to a C 1 conformal diffeomorphism of W v into W i(e) ; (iv) (Cone property) There exist γ, l > 0 such that for every v ∈ V and every x ∈ X v there is an open cone Con(x, γ, l) ⊂ Int(X v ) with vertex x, central angle γ, and altitude l; (v) There are two constants L ≥ 1 and α > 0 such that for every e ∈ E and every pair of points x, y ∈ W t(e) , where |ϕ ′ e (x)| denotes the norm of the derivative of ϕ e at x and (ϕ ′ e ) −1 is the supremum norm taken over W i(e) .
Remark 2.1. According to Proposition 4.2.1 in [9], condition (v) is automatically satisfied (with α = 1) when d ≥ 2. This condition is also fulfilled if d = 1, the alphabet E is finite and all the ϕ e 's are of class C 1+ε .
The following useful fact has been proved in Lemma 4.2.2 of [9].
An immediate consequence of this lemma is the famous bounded distortion property.
(v') (Bounded Distortion Property (BDP)) There exists a constant K ≥ 1 such that A and every x, y ∈ W t(ω) .
Recall that a CGDMS S satisfy the Strong Open Set Condition (SOSC) if J ∩ Int(X) = ∅, that is, ∪ v∈V (J v ∩ Int(X v )) = ∅. Recall further that a matrix A is finitely primitive if there exists a finite set Ω ⊂ E * A of words of the same length such that for all e, f ∈ E there is a word ω ∈ Ω for which eωf ∈ E * A . From this point on we assume that all the systems we deal with satisfy those two properties.
Infinite systems naturally break into two main classes called irregular and regular systems. This dichotomy can be determined from the existence of a conformal measure or, equivalently, the existence of a zero of the topological pressure function. Recall that the topological pressure P (t), t ≥ 0, is defined as follows. For every n ∈ IN, set Then If the functionζ : E ∞ A → IR is given by the formulã ζ(ω) = log |ϕ ′ ω 1 (π(σ(ω)))|, then P (t) = P (tζ), where P(tζ) is the classical topological pressure of the function tζ when E is finite (so the space E ∞ is compact), and is understood in the sense of [5] and [9] when E is infinite. The finiteness parameter θ of the system is defined by inf{t ≥ 0 : P (1) (t) < ∞} = inf{t ≥ 0 : P (t) < ∞}. It is easy to show that the pressure function is non-increasing on [0, ∞), that it is (strictly) decreasing, continuous and convex on [θ, ∞), and that P (d) ≤ 0. Of course, P (0) = ∞ if and only if E is infinite. The following characterization of the Hausdorff dimension HD(J) of the limit set J was proved in [9], Theorem 4.2.13. For every F ⊂ E, we write S| F for the subsystem {ϕ e } e∈F of S, and J F for the limit set of S| F . Theorem.
If P (t) = 0, then t is the only zero of the function P (t) and t = HD(J).
A system S was called regular provided there is some t ≥ 0 such that P (t) = 0. In fact, a system is regular if and only if it admits a conformal measure. Recall that a Borel probability measure m is said to be t-conformal provided m(J) = 1 and for every ω ∈ E * A and for every Borel set B ⊂ X t(ω) m(ϕ ω (B)) = B |ϕ ′ ω | t dm, and for all incomparable words ω, τ ∈ E * A m ϕ ω (X t(ω) ) ∩ ϕ τ (X t(τ ) ) = 0.
There are natural subclasses of regular systems. Among others, a system is called cofinitely regular provided every non-empty cofinite subsystem S ′ = {ϕ e } e∈E ′ (i.e. E ′ is a cofinite subset of E) is regular. A finite system is clearly cofinitely regular, and it was shown in [9], Theorem 4.3.4 that an infinite system is cofinitely regular exactly when the pressure is infinite at the finiteness parameter.
Theorem. An infinite system S is cofinitely regular if and only if P (θ) = ∞ ⇔ P (1) Throughout the rest of this paper, all systems under investigation are assumed to be cofinitely regular.

Hölder families of functions, Pressure and Temperature
We adopt the notational convention that families of functions shall be denoted by uppercase letters, while their members will be denoted by lowercase letters. Moreover, functions and measures associated with the symbolic space E ∞ A will wear a tilde ∼, with the notable exception of the shift map σ.
Throughout this paper, let S = {ϕ e : X t(e) → X i(e) | e ∈ E} be a cofinitely regular CGDMS satisfying the Strong Open Set Condition (SOSC) and having an underlying finitely primitive edge incidence matrix A. Let θ be the finiteness parameter of S. Let u > θ. Let be a bounded Hölder family of functions. Let β be the order of that family. Hölder of order β means that (cf. section 3.1 in [9]) Bounded simply means that Ψ := sup e∈E ψ e < ∞.
Denote byψ : E ∞ A → IR the potential function (also called amalgamated function) induced by the family Ψ, which is defined byψ According to Lemma 3.1.3 in [9], the functionψ is bounded and Hölder continuous of order β. Let also Log = log |ϕ ′ e | : X t(e) → IR e ∈ E . By Lemma 4.2.2 in [9], the family Log is Hölder of order α log s. Moreover, for any t > θ, the family tLog is summable, that is In particular, uLog is summable. Denote byζ : E ∞ A → IR the amalgamated function induced by the family Log, that is,ζ (ω) = log |ϕ ′ ω 1 (π(σω))|. Thenζ is a summable Hölder continuous function of order α log s. It follows from the above definitions and properties that the family F = Ψ+uLog, that is, where f e = ψ e + u log |ϕ ′ e |, is a summable Hölder family of functions of order γ = max{β, α log s}. The amalgamated functionf : E ∞ A → IR induced by the family F satisfies f =ψ + uζ and is a summable Hölder continuous function of order γ. Note also that F andf are bounded above by sup Ψ := sup e∈E sup x∈X t(e) ψ e (x). Recall that the topological pressure P (F ) of F is defined by By considering the family F −P (F ), we may assume without loss of generality that P (F ) = 0.
Equivalently, P (f) = 0 by Proposition 3.1.4 in [9], where Since S is a CGDMS with an underlying finitely primitive matrix and F is a summable Hölder family of functions, Theorem 3.2.3 and Proposition 4.2.5 in [9] assert that there exists a unique F -conformal measure m F supported on J. In other words, for every ω ∈ E * A and for every Borel set B ⊂ X t(ω) Moreover, m F =mf • π −1 , wheremf is the unique eigenmeasure of the conjugate Perron-Frobenius operator L * f . The existence and the uniqueness ofmf is guaranteed by Corollary 2.7.5(a) in [9]. The measuremf is also a Gibbs state forf according to Corollary 2.7.5(b). Furthermore,f admits a unique σ-invariant Gibbs stateμf according to Corollary 2.7.5(c). This Gibbs state is completely ergodic. It is also the unique equilibrium measure forf by Theorem 2.2.9. As required by that theorem, note thatf ∈ L 1 (μf ), forζ ∈ L 1 (μf ) as shown in the following calculation: where 0 < δ < u − θ and −∞ < B := inf{ ϕ ′ e δ log ϕ ′ e : e ∈ E} < 0, and where we used the bounded distortion property (explaining the presence of K ≥ 1), the fact thatmf is a Gibbs state forf (hence the presence of C ≥ 1), that P (f) = 0, and that lim x→0 + x δ log x = 0 (by means of B). Since any two Gibbs states forf are boundedly equivalent, the measuresμf andmf are boundedly equivalent and thus the measures µ F :=μf •π −1 and m F =mf •π −1 are boundedly equivalent. For this reason, µ F is called the S-invariant version of m F . Now, for every (q, t) ∈ IR 2 define the family F q,t := qF + tLog = qΨ + (qu + t)Log, and its corresponding amalgamated functioñ It is easy to see that the F q,t 's are Hölder families of functions of order γ (the order of F ) and that thef q,t 's are Hölder continuous functions of order γ (likef). Moreover, F q,t andf q,t are summable if and only if qu + t > θ. Hence for every (q, t) ∈ IR 2 such that qu + t > θ, there exists a unique F q,t -conformal measure m Fq,t supported on J. This measure is such that m Fq,t =mf q,t • π −1 , wheremf q,t is the unique eigenmeasure of the conjugate Perron-Frobenius operator L * fq,t . The measuremf q,t is a Gibbs state forf q,t . Furthermore,f q,t admits a unique completely ergodic σ-invariant Gibbs stateμf q,t . This Gibbs state is also the unique equilibrium measure forf q,t . As the Gibbs statesμf q,t andmf q,t are boundedly equivalent, so are the measures µ Fq,t :=μf q,t • π −1 and m Fq,t =mf q,t • π −1 . Note thatf q,t ∈ L 1 (μ q,t ), for ζ ∈ L 1 (μ q,t ) as a calculation similar to (3.1) shows. Furthermore, where P ((qu + t)Log) = P (qu + t) (cf. section 2). We now state fundamental properties of the pressure as a function of the two variables q and t.
Theorem 3.1. Let S = {ϕ e : X t(e) → X i(e) | e ∈ E} be a cofinitely regular CGDMS satisfying the Strong Open Set Condition (SOSC) and having an underlying finitely primitive edge incidence matrix A. Let θ be the finiteness parameter of S, and let u > θ. Then the pressure function (q, t) → P (q, t) := P (F q,t ) = P (f q,t ), (q, t) ∈ IR 2 satisfies the following properties.
(a) P (q, t) < ∞ if and only if qu + t > θ; is the Lyapunov exponent ofμ q,t ; (g) t → P (q, t) is convex (and thereby continuous) on the interval (θ − qu, ∞); Proof. (a) This follows from (3.2) and the fact that P (t) := P (t Log) < ∞ if and only if t > θ.
If q 1 u + t 1 ≤ θ, then P (q 1 , t 1 ) = ∞ and the statement thus holds. So suppose that q 1 u + t 1 > θ. Then the n-th partition function off q 2 ,t 2 satisfies (d) This also follows from (3.3) by setting q 1 = q 2 = q, t 1 > θ and t 2 = t and letting t → ∞.
Observe that in the proof of part (e) of the above theorem P (θ) ≥ q inf Ψ suffices to guarantee the existence of a zero for the pressure function t → P (q, t). The assumption of cofinite regularity on S ensures that the pressure function t → P (q, t) has a zero for every q ∈ IR.
Corollary 3.2. For all q ∈ IR there exists a unique T (q) ∈ (θ −qu, ∞) such that P (q, T (q)) = 0. The function T (q) is called the temperature function.
In order to allege notation, let Now, let q ∈ IR. Recall thatζ ∈ L 1 (μ q ) and thusf ∈ L 1 (μ q ). Thereafter, let By the variational principle for pressure (cf. Theorems 2.1.6-2.1.8 in [9]), note that Hence α(q) ≥ 0. Moreover, Finally, we study some basic properties of the temperature function T (q). (d) The function q → T (q), q ∈ IR, is convex, meaning that T ′′ (q) ≥ 0 for all q ∈ IR. This function is not strictly convex if and only ifμf is equal toμ −HD(J)ζ .
Proof. (a) By Proposition 2.6.13 in [9], ∂P ∂t (q, t) = ζ dμ q,t = −χμ q,t (σ) < 0 for every (q, t) ∈ IR 2 such that qu + t > θ. In particular, this is true for all pairs (q, T (q)). Since T (q) is uniquely determined by the condition P (q, T (q)) = 0, it follows from Theorem 2.6.12 in [9] and the implicit function theorem that T is real-analytic on IR.
(c) It follows from the fact that P (q, T (q)) = 0 for all q ∈ IR and from Proposition 2.6.13 in [9] that Hence Having already observed that α(q) ≥ 0, we thus know that T ′ (q) ≤ 0. In order to prove that T ′ (q) < 0, we need to show that f dμ q = 0. But since f dμ q ≤ −hμ q (σ) ≤ 0 by (3.4), it suffices to show that hμ q (σ) > 0. This follows immediately from Theorem 2.5.2 in [9]. (d) Lemma 4.9.5 in [9] (with ∆ 1 replaced by IR) and its proof carry over to the current setting without any change.
4. Multifractal analysis of the conformal measure m F over a subset of J Let S = {ϕ e : X t(e) → X i(e) | e ∈ E} be a cofinitely regular CGDMS satisfying SOSC and having an underlying edge incidence matrix A which is finitely primitive. Let θ be the finiteness parameter of S. Let also F be a family of functions of the form F = Ψ + uLog such that P (F ) = 0, where Ψ is a bounded Hölder family of functions and u > θ.
We shall now develop the multifractal analysis of the conformal measure m F (or equivalently, the invariant measure µ F ) associated to the family F . We shall conduct this analysis by means of balls and we shall restrict ourselves to a subset J r of the limit set J of S. As we shall see later, J r is often a fairly large subset of J. By definition, J r is the set of points of J which are coded by the set of infinite admissible words The words in this set code points of the limit set that behave tamely when a multifractal analysis is carried out using balls, for infinitely many of their iterates are positively separated from the boundary of the phase space. The conformality of the measure m F can then be used at those iterates to estimate the local dimension of m F from above. Before going any further, we observe that E ∞ r is a set of full measure.
. Thus, by ergodicity ofμ, we have thatμ(E ∞ r ) is 0 or 1. We shall now show that this latter possibility always prevails. Since S satisfies SOSC, there exists It follows from Birkhoff's Ergodic Theorem that the set of infinite admissible words whose iterates' images visit infinitely many times the ball B(x, r) has measure 1, that is,μ τ ∈ E ∞ A σ n τ ∈ π −1 (B(x, r)) for infinitely many n's = 1. Thereforeμ(E ∞ r ) = 1. (The same conclusion can be drawn by means of Poincaré's Recurrence Theorem.) This lemma tells us thatμf (E ∞ r ) = 1 andμf q,t (E ∞ r ) = 1 for all (q, t) ∈ IR 2 such that qu + t > θ. In particular,μ q (E ∞ r ) = 1 for all q ∈ IR. This implies immediately that µ F (J r ) =μf • π −1 (π(E ∞ r )) = 1, that is, the set E ∞ r is a set of full µ F -measure. Similarly, µ Fq,t (J r ) = 1 for all (q, t) ∈ IR 2 such that qu + t > θ. In particular, µ q (J r ) = 1 for all q ∈ IR.
In fact, J r contains a rich family of subsets of full measure. To define these subsets, we proceed as follows. For every ω ∈ E ∞ A and r ≥ 0, let {n j (ω, r)} be the increasing sequence of all positive integers n such that σ n ω ∈ π −1 (X i(σ n ω) \B(∂X i(σ n ω) , r)). This sequence may be empty, non-empty and finite, or infinite depending on ω and r. However, for every ω ∈ E ∞ r the sequence {n j (ω, r)} is infinite for every 0 ≤ r < r max (ω), where r max (ω) := lim sup n→∞ dist π(σ n ω), ∂X i(σ n ω) > 0. Now, for every R ≥ 0 define the completely invariant set We claim that these subsets of E ∞ r have all full measure. Lemma 4.2. For every R ≥ 0 small enough, we have thatμ(E ∞ rr (R)) = 1 for all ergodic, σinvariant Borel probability measureμ on E ∞ A with suppμ = E ∞ A and such that ζ dμ > −∞.
Proof. Let 0 ≤ R < D, where D = sup x∈J dist(x, ∂X) := sup v∈V sup x∈Jv dist(x, ∂X v ). SOSC guarantees that D > 0. Letμ be an ergodic, σ-invariant Borel probability measure on E ∞ A with suppμ = E ∞ A and such that ζ dμ > −∞. Since E ∞ rr (R) is completely σinvariant, the ergodicity ofμ forcesμ(E ∞ rr (R)) to equal 0 or 1. We shall now prove that this latter possibility always prevails. Since 0 Applying Birkhoff's Ergodic Theorem twice (once with the characteristic function of the set π −1 (X v \B(∂X v , R)) and once with the potential functionζ), we obtain that the set has measure 1. Writing n j instead of n j (ω, R) to allege notation, we have for every ω in this set that It follows immediately thatμ(E ∞ rr (R)) = 1.
As an immediate corollary, we obtain that the completely invariant set is a set of full measure. We now recall a few basic definitions from multifractal analysis.
Let µ be a Borel probability measure on X. The pointwise or local dimension d µ (x) of µ at x ∈ X is the power law behaviour (if any) of µ(B(x, r)) for small r > 0, that is, (B(x, r)) log r .
We further define the lower and upper dimensions of µ at x ∈ X by respectively. Denote the set of points of J r at which the local dimension of a measure µ is equal to α by J r,µ (α) = {x ∈ J r | d µ (x) = α}. Denote the Hausforff dimension of J r,µ (α) by f r,µ (α) = HD(J r,µ (α)). Now for every α ≥ 0, let We shall now prove that for every q ∈ IR the measure µ q confers full measure to the set of points of J r where the local dimension of the measure m F is α(q).
Theorem 4.4. The following statements hold.
(c) Let q ∈ IR. Using part (a) with α = α(q) and part (b), we deduce that Let us now remind the reader about the Legendre transform. Let k be a strictly convex function on an interval I (hence k ′′ > 0 wherever this second derivative exists). The Legendre transform of k is the function l defined by l(p) = max{p x − k(x)} wherever the maximum exists. It can be proved that the domain of l is either a point, an interval or a half-line. It can further be shown that l is strictly convex and that the Legendre transform is involutive. We then say that the functions k and l form a Legendre transform pair. The following theorem (see [16]) gives a useful characterization of a Legendre transform pair. We shall now prove that f r,m F (α) and T (q) form a Legendre transform pair. Recall that T ′ (q) = −α(q) by Theorem 3.3.
Like (4.2), the conformality of m q and the bounded variation principle forf q give where B(f q ) is a constant of bounded variation (see Lemma 2.3.1 in [9]). Hence such that y = π(σ n j τ ), and for such y and τ we have S ω|n j F (y) = S n jf (τ ). The conformality of m F , the fact that B(π(σ n j ω), R) ⊂ Int(X t(ωn j ) ), and the OSC then give where B(f ) is a constant of bounded variation forf (see the bounded variation principle for Hölder continuous potentials, Lemma 2.3.1 in [9]). Thus, Using (4.5) and (4.6), we deduce that As d mq (x) ≤ qα(q) + T (q) for every x ∈ J r,m F (α(q)), we deduce that f r,m F (α(q)) ≤ qα(q) + T (q).
All of the above results give us an analog of Theorem 4.9.4 in [9].
Theorem 4.7. Let S = {ϕ e : X t(e) → X i(e) | e ∈ E} be a cofinitely regular CGDMS satisfying SOSC and having an underlying finitely primitive edge incidence matrix A. Let θ be the finiteness parameter of S.Suppose that hμ q (σ)/χμ(σ) > θ. Then the following statements hold.
Proof. (a) Using Theorem 3.3(b), notice that µ 1 = µ F . Thus, by Theorem 4.4(c), we have µ F (J r,m F (α(1))) = 1. Since α(1) = f dμf / ζ dμf and since the measures µ F and m F are boundedly equivalent, we hence have that then E ∞ r = E ∞ A and thus J r = J. Thus, section 4 gives us the multifractal analysis over J. Indeed, denoting the set of points of J at which the local dimension of a measure µ is equal to α by J µ (α) and the Hausforff dimension of J µ (α) by f µ (α), Theorem 4.7 reduces to the following.
Theorem 5.1. Let S = {ϕ e : X t(e) → X i(e) | e ∈ E} be a cofinitely regular CGDMS satisfying SOSC and BSC, and having an underlying finitely primitive edge incidence matrix A. Let θ be the finiteness parameter of S.Suppose that hμ q (σ)/χμ(σ) > θ. Then the following statements hold.
An important family of CGDMSs (in fact, conformal IFSs (CIFSs)) which satisfies the boundary separation condition are real continued fractions with the digit 1 deleted.
Then S is a cofinitely regular CIFS which satisfies both the strong open set condition and the boundary separation condition.

5.2.
Multifractal analysis over J under other conditions. We shall now prove that f m F (α) and T (q) form a Legendre transform pair under some conditions. Recall that T ′ (q) = −α(q) by Theorem 3.3.
Theorem 5.3. Suppose there exists a countable set J 0 ⊂ J such that for every x ∈ J\J 0 there are ω ∈ E ∞ A with π(ω) = x, a constant C = C(x, ω) > 0, an increasing subsequence {n j } j∈IN = {n j (x, ω)} j∈IN of natural numbers and a sequence {r n j } j∈IN = {r n j (x, ω)} j∈IN of positive real numbers such that (1) m F (B(x, r n j )) ≤ C exp(S n jf (ω)) for all j ∈ IN; (2) 1 > r n j ≥ C ϕ ′ ω|n j for all j ∈ IN ; (3) lim j→∞ r n j = 0. Then f m F (α(q)) = qα(q) + T (q) for all q ∈ IR.
In the framework of Theorem 5.3, Theorem 4.7 reduces to the following.
Theorem 5.4. Let S = {ϕ e : X t(e) → X i(e) | e ∈ E} be a cofinitely regular CGDMS satisfying SOSC and BSC, and having an underlying finitely primitive edge incidence matrix A. Let θ be the finiteness parameter of S.Suppose that the conditions of Theorem 5.3 are fulfilled. Suppose also that hμ q (σ)/χμ(σ) > θ. Then the following statements hold.
(a) The number d µ F (x) exists for µ F -a.e. x ∈ J r and d µ F (x) = f dμf ζ dμf .
There are interesting families of CGDMSs (in fact, even of CIFSs) which satisfy the conditions imposed in Theorem 5.3. Among others, let us mention real continued fractions over the interval [0, 1]. In fact, there is a larger class of one-dimensional CIFSs for which conditions (1)-(3) of Theorem 5.3 are fulfilled. By one-dimensional, we simply mean that X is a subinterval of IR.
Theorem 5.6. Let S = {ϕ n : X → X | n ∈ IN} be a one-dimensional CIFS satisfying the following properties.
(i) ϕ ′ n is comparable to ϕ ′ n+1 , that is, there exists a constant C ≥ 1 such that (ii) The generators ϕ n , n ∈ IN, of S either all preserve orientation (i.e. ϕ ′ n > 0 for all n ∈ IN) or all reverse orientation (i.e. ϕ ′ n < 0 for all n ∈ IN). (iii) Either ϕ n+1 > ϕ n for all n ∈ IN or ϕ n+1 < ϕ n for all n ∈ IN. Then all three conditions in Theorem 5.3 are met.