Localized asymptotic behavior for almost additive potentials

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of $\phi_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} \phi_n(x)/n=\xi(x)\}$, where $\xi$ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\R^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.


Introduction
We say that (X, T ) is a topological dynamical system (TDS) if X is a compact metric space and T is a continuous mapping from X to itself. We denote by M(X, T ) the set of invariant probability measures on (X, T ).
We say that Φ = (φ n ) ∞ n=1 is almost additive if φ n is continuous from X to R and there is a positive constant C(Φ) > 0 such that Typical examples are the additive potential given by the sequence of Birkhoff sums (S n ϕ = n−1 k=0 ϕ • T k ) n≥1 of a continuous function ϕ : X → R, and more generally sequences of the form (log S n M ) n≥1 , where (S n M ) n≥1 is the sequence of Birkhoff products (M • T n−1 ) · · · (M • T ) · M associated with a continuous function M from X to the set of positive square matrices.
To our best knowledge no result is known for dim H E Φ (ξ) for non constant ξ. We are going to give an answer to this question when (X, T ) is a topologically mixing subshift of finite type endowed with a metric associated with a negative almost additive potential, and then transfer our result to geometric realizations on Moran sets like those studied in [2], the main examples being C 1 conformal repellers and C 1 conformal iterated function systems (see section 3 for precise definitions and statements). In the setting outlined above, if d = 1 and ξ takes its values in L Φ , we find the natural variational formula dim H E Φ (ξ) = max h µ (T ) X log DT dµ : µ ∈ M(X, T ), Φ * (µ) ∈ ξ(X) . Another application concerns harmonic measure. Let us consider here the special case of the set J = C ×C ⊂ R 2 , where C is the middle third Cantor set. The harmonic measure on J is the probability measure ω such that for each x ∈ J and r > 0, ω(B(x, r)) is the probability that a planar Brownian motion started at ∞ attains J for the first time at a point of B(x, r) (see Section 3.4 for more general examples and a reference). For x ∈ J, one defines the local dimension of ω at x as d ω (x) = lim r→0 + log ω(B(x, r))/log r whenever this limit exists. Let I stand for the set of all possible local dimensions for ω. By using the fact that ω is a Gibbs measure, we prove that if ξ : J → R + is continuous and ξ(J) ⊂ I, then the set E ω (ξ) = {x ∈ J : d ω (x) = ξ(x)} is dense in J and the following variational formula holds: dim H E ω (ξ) = sup{dim H E ω (α) : α ∈ ξ(J)}, where E ω (α) = {x ∈ J : d ω (x) = α}.
Our approach necessitates to revisit the case where ξ is constant. This brings out an interesting new property of the Hausdorff spectrum α → dim H E Φ (α). We call this property weak concavity; it is between concavity and quasi-concavity. This structure turns out to be crucial in establishing our results on fixed points in the asymptotic average. The paper is organized as follows. In Section 2 we give basic definitions and recalls about thermodynamic formalism, and then state our main results on subshift of finite type. In Section 3 we give the geometric realizations. The other sections are dedicated to the proofs.

2.
Definitions, and main results on subshifts of finite type Section 2.1.1 introduces some additional notions related to almost additive potentials, Section 2.1.2 introduces the metrics we will put on topologically mixing subshifts of finite types, while Section 2.1.3 recalls the variational principle for almost additive potentials. Then Section 2.2 introduces two fundamental dimension functions in the multifractal analysis of almost additive potentials, as well as a notion of weak concavity. Finally Section 2.3 provides our main results on topologically mixing subshifts of finite types.

2.1.2.
Weak Gibbs metric on subshifts of finite type. Let (Σ A , T ) be a topologically mixing subshift of finite type over the alphabet {1, · · · , m}, where A is a m×m matrix with entries 0 and 1 such that A p 0 > 0 for some p 0 ∈ N and T is the shift map. We shall endow Σ A with a metric d Ψ naturally associated with a potential Ψ ∈ C − aa (Σ A , T ). This kind of metrics have been considered in [21] and [23] associated with negative additive potentials in order to transfer to the symbolic side the study of some C 1 hyperbolic dynamics.
Let Σ A,n be the set of the admissible words of length n and let Σ A, * := n≥0 Σ A,n . For w ∈ Σ A, * and w = w 1 · · · w n , we denote the length of w by |w| = n. Given w ∈ Σ A, * ∪ Σ A with |w| ≥ n, we denote w 1 · · · w n by w| n . Given u ∈ Σ A, * and v ∈ Σ A, * ∪ Σ A , if u j = v j for j = 1, · · · , |u|, then we say u is a prefix of v and write u ≺ v. For u = u 1 · · · u n ∈ Σ A,n , u * stands for u| n−1 . For x, y ∈ Σ A, * ∪ Σ A such that x = y, x ∧ y stands for the common prefix of x and y of maximal length. Given w ∈ Σ A,n , the cylinder [w] is defined as For Φ ∈ C aa (Σ A , T ) and w ∈ Σ A,n we define The proof is elementary and we omit it.
is a closed ball of Σ A with radius e −n } (n ≥ 0).

It is clear that {[w]
: w ∈ B n (Ψ)} is a covering of Σ A for each n ≥ 0.
If we take Ψ = (−n log m) n≥1 , it is ready to check that d Ψ (x, y) = m −|x∧y| , which is the standard metric on Σ A . We denote this special metric by d 1 .

2.1.3.
Recalls on the thermodynamic formalism. The thermodynamic formalism for almost additive potentials has been studied in several works [13,2,19,17,3,25,4,11]. For our purpose, we only need to consider the subshift of finite type case. Let (Σ A , T ) be a topologically mixing subshift of finite type. Given Φ ∈ C aa (Σ A , T ), the topological pressure can be defined as φ n (x)).
For instance, this set contains the sequence of Birkhoff sums of any Hölder continuous function when Σ A is endowed with a metric d Ψ (see [7]).
2.2. Two dimension functions; weak concavity. Let us recall what is the range of those α such that E Φ (α) = ∅.
Now we introduce two functions which will turn out to take the same values on L Φ and provide the Hausdorff and packing dimensions of the sets E Φ (α). They correspond to different point of views to estimate these dimensions, namely box-counting of balls intersecting E Φ (α) and variational principle for entropy like (1.2). The proofs of the propositions stated in this section are given in Section 4.
Proposition 3. For any Ψ ∈ C − aa (Σ A , T ), the limit  For any Φ ∈ C aa (Σ A , T, d) and any α ∈ L Φ , we have The function Λ : L Φ → R is upper semi-continuous.
We will prove that Λ(α) is the Hausdorff dimension of E Φ (α) for all α ∈ L Φ . The function Λ has more regularity than upper semi-continuity. To make this precise we need several standard notations from convex analysis. Given A ⊂ R d , the affine hull of A is the smallest affine subspace of R d containing A and is denoted by aff(A). For a convex set A, we define ri(A), the relative interior of A as

a convex set and h :
A → R be a function. If there exists c ≥ 1 such that for any α, β ∈ A, we can find then we call h a weakly concave function on A. Note that if c = 1, we go back to the usual concept of concave function. Also, Proposition 4. The function Λ : L Φ → R is bounded, positive and weakly concave. It is continuous on any closed interval I ⊂ L Φ and on ri(A), where A ⊂ L Φ is any convex set. Consequently it is continuous on ri(L Φ ). If moreover L Φ is a convex polyhedron, then Λ is continuous on L Φ . Assume I = [α 0 , α 1 ] ⊂ L Φ and α max ∈ I such that Λ(α max ) = max{Λ(α) : α ∈ I}, then Λ is decreasing from α max to α j , j = 0, 1.

Remark 1.
Large deviations spectra for the Hausdorff dimension estimation of sets like E Φ (α) have been considered since the first studies of multifractal properties of Gibbs or weak Gibbs measures and then extended to the study of Birkhoff averages [12,32,10,31,30,29,5,26,14,15,6,20]. Until now, in the situations where such a spectrum may be non-concave [6,4,20], no description of its regularity like that of Proposition 4 had been given. Moreover, the methods used in the papers mentioned above seem not adapted to provide this information.

2.3.
Main results on topologically mixing subshift of finite type.
Throughout this subsection we fix Φ ∈ C aa (Σ A , T, d) and Ψ ∈ C − aa (Σ A , T ). We work on the metric space (Σ A , d Ψ ). If E ⊂ (Σ A , d Ψ ), dim H E, dim P E, dim B E stand for its Hausdorff, packing and box dimensions respectively. To not assuming additional regularity assumption for Φ and Ψ is natural, since this flexibility on Ψ makes it possible to describe a larger class of geometric realizations of the next results, and there is no special reason to considers the sets E Φ (ξ) under restrictions on Φ. However, the proofs will use extensively approximations of almost additive potentials by Hölder potentials.
and the function D is weakly concave.
(1) In [6,4], assuming that The argument is strongly based on the differentiability of the related pressure functions in these cases.
(2) In [20], the authors consider the case of additive potentials Φ and Ψ, and work under the assumption that Ψ corresponds to a Hölder potential. They show D(α) = E(α) for all α ∈ L Φ . Here we work under weaker assumptions, i.e. both Φ and Ψ are almost additive. Also, we use a different method to compute the function D(α), namely concatenation of Gibbs measures. Such a method has been used successfully in [23] to deal with the special sets E Ψ (α) when Ψ is additive ( i.e. taking Φ = Ψ; notice that in this case the spectrum is always concave). Here, we need to refine such an approach in order to remove some delicate points in our geometric application to attractors of C 1 conformal iterated function systems.

Remark 4.
(1) The proof of Theorem 2.3 uses the weak concavity of the spectrum D. It also requires to concatenate Gibbs measures in a more elaborated way than to determine D.
(2) In fact we shall prove a slightly more general result than Theorem 2.3(1): (3) An extension of Theorem 2.3(4) is given in the final remark of Section 3.4.

Geometric results
In this section we show how the main results of the previous section can be applied to multifractal analysis on conformal repellers and on attractors of conformal IFS satisfying the strong open set condition. Such sets fall in the Moran-like geometric constructions considered in [2,29]. At first we describe this kind of construction (Section 3.1). Then we state the geometric results deduced from Theorems 2.2 and 2.3 (Section 3.2). We give our application to fixed points in the asymptotic average for dynamical systems in R d in Section 3.3. Finally, we give an application to the local scaling properties of weak Gibbs measures in Section 3.4, special example of which is the harmonic measure on planar conformal Cantor sets.
3.1. General setting of geometric realization. Let (Σ A , T ) be a topologically mixing subshift of finite type over the alphabet {1, · · · , m} and Ψ ∈ C − aa (Σ A , T ). Let X be R d ′ or be a connected, d ′ -dimensional C 1 Riemannian manifold. Consider a family of sets {R w : w ∈ Σ A, * }, where each R w ⊂ X is a compact set with nonempty interior. We assume that this family of compact sets satisfies the following conditions: (2) For any integer n > 0, the interiors of distinct R w , w ∈ Σ A,n are disjoint.
(3) Each R w contains a ball of radius r w and is contained in a ball of radius r w .
(4) There exists a constant K > 1 and a negative sequence η n = o(n) such that for every w ∈ Σ A, * , Notice that Ψ max < 0, then Let J = n≥0 w∈Σ A,n R w . We call J the limit set of the family {R w : w ∈ Σ A, * }. We can define the coding map χ : It is clear that χ is continuous and surjective when Σ A is endowed with standard metric d 1 and J is endowed with the induced metric ρ from X.
We say that J is a Moran type geometric realization of (Σ A , d Ψ ).
For this kind of construction we have the following useful observation: In this paper we consider two classes of Moran type geometric realizations of Σ A .
(1) Topologically mixing C 1 conformal repeller (J, g). We refer the book [29] for the definitions and the basic properties related to conformal repellers. It is well known that in this case (J, g) has a Markov partition {R 1 , · · · , R m }. For each w = w 1 · · · w n , define R w := R w 1 ∩ g −1 (R w 2 ) ∩ · · · ∩ g −n+1 (R wn ). Define ψ(x) = − log |g ′ (χ(x))| and Ψ = (S n ψ) ∞ n=1 . By the definition of R w and the property of Markov partition, the condition (1) and (2) are checked directly. (3) and (4) are stated in [29] (Proposition 20.2), except that for (4) we have an additional term exp(η |w| ) = exp(− Ψ |w| ) (see Section 2.1.1 for the definition of Ψ |w| ). This is because we only assume g to be continuous rather than Hölder continuous. Thus J is a Moran type geometric realization of (Σ A , d Ψ ) for some primitive matrix A and the potential Ψ. Moreover in this case we have χ • T = g • χ.
(2) Attractors of C 1 conformal IFS satisfying the strong open set condition (SOSC) (see [28] for details). Let {f 1 , · · · , f m } be such an IFS and denote by J its attractor. Define (1) and (2) hold for {R w : w ∈ Σ A, * }. Moreover, arguments similar to those used to prove Proposition 20.2 in [29] show that (3) and (4) also hold. Thus, {R w : w ∈ Σ A, * } is a Moran type geometric realization of (Σ A , d Ψ ) with potential Ψ. Notice that here Σ A is the full shift Σ m . By the uniqueness of the attractor it is easy to verify that the attractor J is the limit set of the family {R w : w ∈ Σ A, * }.

3.2.
Multifractal analysis on Moran type geometric realizations. We are going to conduct multifractal analysis on Moran type geometric realizations, thus we need a dynamics g on J so that (J, g) is a factor of some (Σ A , T ). For C 1 conformal repellers, there is such a natural dynamic. For the attractor of a C 1 conformal IFS, there is no such one in general, the difficulty coming from those points having several codings. However, under the SOSC, we can naturally define such a g by removing a "negligible" part of J: Let {f 1 , · · · , f m } be a C 1 conformal IFS satisfying the SOSC. Let V be an open set such that the SOSC holds. By [28], such an open set always exists as soon as the mappings f i are C 1+ǫ and the OSC holds. Define Z ∞ := w∈Σ A, * f w (∂V ) and Z ∞ := χ −1 ( Z ∞ ). We have the following lemma (proved in Section 8): By the previous lemma we can define the mapping g : Let J be a Moran type geometric realization of (Σ A , d Ψ ). We set J = J when J is a C 1 conformal repeller and J = J \ Z ∞ when J is the attractor of a C 1 conformal IFS satisfying the SOSC.

Given a sequence of functions
We must redefine these objects because until now they were defined for compact dynamical systems, while J may be not compact.
When J is a conformal repeller the system (J, g) is naturally a TDS.
When J is the attractor of a C 1 conformal IFS satisfying the SOSC, if φ is a continuous function from J to R d , it generates the additive potential Φ = (S nφ ) ∞ n=1 on (Σ A , T ), whereφ = φ • χ, and it also defines Φ = (S n φ) ∞ n=1 on ( J, g). Then for α ∈ R d we have Theorem 3.2. Let J be a Moran type geometric realization of (Σ A , d Ψ ). If J is a C 1 conformal repeller, let Φ ∈ C aa (J, g, d) and define Φ as above. If J is the attractor of a C 1 conformal IFS satisfying the SOSC, let φ be a continuous map from J to R d , and define the additive potential Remark 5. For the case of conformal repellers, the connection between Theorem 3.2 and the other works [6,20,4] is similar to that done in Remark 3(1) and (2).
For the set E Φ (ξ) we have the following result: Theorem 3.3. Let J be a Moran type geometric realization of (Σ A , d Ψ ), which is either a C 1 conformal repeller or the attractor of a C 1 conformal IFS satisfying the SOSC. Let Φ and Φ be the same as in Theorem 3.2. Let ξ : J → R d be continuous and 3.3. Application to fixed points in the asymptotic average for dynamical sys- We are interested in the Hausdorff dimension of the set of all such points: If ξ stands for the identity map on J and Φ stands for the additive potential associated with the potential ξ, in our setting we have F(J, g) = E Φ (ξ).
The set L Φ is contained in the convex hull of J, and it contains the set of the fixed points of g. An example of trivial situation is provided by the unit circle endowed with dynamic g(z) = z 2 in C. There, F(J, g) = {1}. How about general conformal repellers and attractors of conformal IFS? This question is non trivial in general. We are going to describe a class of conformal IFS, namely self-similar generalized Sierpinski carpets, for which the situation is non trivial and we have a complete answer.
We consider a special self-similar IFS We assume further the SOSC fulfills. Let x j stand for the unique fixed point of f j and let J be the attractor of this IFS. Notice that the mappings f j have no rotation part, thus the convex hull of J satisfies Co(J) = Co{x 1 , · · · , x m } =: ∆, and is a convex polyhedron. We further assume that Co(J) has dimension d (otherwise we can define this IFS in a smaller affine subspace).
Let W stand for the open set such that the SOSC holds. It is ready to see that V := W ∩ ∆ is also an open set such that SOSC holds. We can define the dynamics g on J = J \ Z ∞ , where Z ∞ is defined as in the previous subsection. Now we have the following result whose proof is given in Section 8.
Moreover if the point at which D Φ attains its maximum belongs to J, then F( J , g) is of full Hausdorff dimension.
We have the following corollary, in which the lower bound for the Hausdorff dimension follows directly from Theorem 3.4 and the upper bound follows from standard estimates based on the bounds provided in Section 5.1.
m j } 1≤j≤N , satisfying SOSC and living respectively in R d j . Denote by J j , 1 ≤ j ≤ N , their respective attractors as well as Both the previous results yield the result presented in the introduction of the paper: To see this, for a fixed integer m ≥ 2 let g m : Then χ is continuous and surjective. Consider the IFS {f j : j = 0, · · · , m − 1} defined as f j (x) = (x + j)/m. It is seen that the SOSC holds with V = (0, 1). Let By the law of large number applied to the measure of maximal entropy we get D Φ (1/2) = 1. We conclude by noticing that . Next we consider concrete examples of carpets in the unit square.
Heterogeneous carpets in the unit square. In order to fully illustrate our purpose, we consider an IFS S 0 = {f 1 , · · · , f N } in R 2 made of contractive similitudes without rotations such that the squares f i ([0, 1] 2 ) form a tiling of [0, 1] 2 . All these situations have been determined in [9]. In this way, ]0, 1[ 2 can be chosen as the open set such that the SOSC holds, and the boundaries of the sets f i (]0, 1[ 2 ) have big intersections. The picture on the left of Figure 1 give an example of this kind of IFS. This IFS contains 15 dilation maps, and the dynamics on this attractor is highly non trivial. We consider the case of the regular tiling associated with the The simpler situation is that of S 2 . In this case, G = (1/2, 1/2), the center of symmetry of J S 2 is the fixed point of f 1,1 and it belongs to L Φ . Moreover, it is obvious that the uniform measure (or Parry measure) on J S 2 is carried by the set E Φ (G). This yields the result by Theorem 3.4, and dim H F( J S 2 , g S 2 ) = log 5/ log 3.
In the case of S 1 , the point G is still the center of symmetry of J S 1 , so D Φ reaches its maximum at G. However, G does not belong to J S 1 . Since Φ is Hölder continuous and the tiling is regular, we know that D Φ is strictly concave. By using the symmetry, one deduces that the restriction of D Φ to J S 1 reaches its maximum at any of the four points (1/3, 1/3),

3.4.
Localized results for weak Gibbs measures. Let {f 1 , · · · , f m } be a homogenous self-similar IFS in C satisfying the strong separation condition, that is, each function f j has the form f j (z) = a j z + b j where 0 < ρ = |a j | < 1, and there exists a topological closed disk D such that f j (D) ⊂ D and the f j (D) are pairwise disjoint. There is a natural coding map χ : Σ m → J. Moreover if we define ψ(x) ≡ log ρ for x ∈ Σ m , and Ψ = (S n ψ) ∞ n=1 , then χ : (Σ m , d Ψ ) → (J, | · |) is a bi-Lipschitz homeomorphism.
Let φ : J → R be continuous and defineφ = φ • χ. By subtracting a constant potential if necessary, we can assume P (T,φ) = 0. There exists a weak Gibbs measureμ on Σ m (see [22]), i.e. a probability measure such that the exists positive sequence (C n ) n≥1 such that for all x ∈ Σ m and n ≥ 1 , with lim n→∞ log(C n )/n = 0 (if φ is Hölder continuous, then C n is bounded and µ is a Gibbs measure). In particular, n log ρ in the sense that either both the limits do not exist, either they exist and are equal. Define µ := χ * (μ) and µ is called a weak Gibbs measure associated with φ. By the bi-Lipschitz property of χ and the strong separate condition, we can easily conclude that d µ (y) = lim n→∞ S n φ(y)/(n log ρ) for any y ∈ J.
By applying Theorem 3.3 for d = 1, we have the following property regarding the local property of weak Gibbs measure: Corollary 2. Let µ be the weak Gibbs measure associated with φ. Then the set of all possible local dimension for µ is the interval L Φ / log ρ. Assume ξ : J → R is continuous Now let ω stand for the harmonic measure on J. It is well known that (see for example the survey paper [24]) there exists a Hölder continuous function φ : J → R such that w ≍ µ, where µ is the equilibrium state of φ.
Final remark. At least when d = 1, it is not difficult to extend the results obtained in this paper by considering Υ = (γ n ) n≥1 ∈ C + aa (Σ A , T ) and the more general level sets E Φ/Υ (ξ) = {x ∈ Σ A : lim n→∞ φ n (x)/γ n (x) = ξ(x)}; when ξ is constant, such sets have been considered in the contexts examined in [6,4]. The formula is that if the continuous function ξ takes values in the set When Υ = −Ψ, this can be applied to the local dimension of Gibbs measures associated with Hölder potentials ϕ on any C 1 conformal repeller of a map f , since in this case we know from [29] that such a measure is doubling so that the local dimension is directly related to the asymptotic behavior of S n ϕ/S n (− log Df ). Consequently, Corollary 3 can be extended to harmonic measure on more general conformal repellers (see [24]).
it is easy to prove that |u| ≤ |v 1 | + |v 2 |. We will use this basic fact several times.

Proofs of Propositions 3 and 4
4.1. Proof of Proposition 3. We need some facts gathered in the two following lemmas. We omit their simple proofs based on elementary using of the almost additivity of Φ and the continuity of the φ n .
(4) If u ≺ v are such that u ∈ B n 1 (Ψ) and v ∈ B n 2 (Ψ), then Let us start the proof of Proposition 3. The hard part is (2.7). At first we will show that log f (α, n, ǫ), as a sequence of n, has a kind of subadditivity property. Due to this subadditivity, by a standard procedure, we get the desired equality of the two limits. The proof is an adaption of that given in [15] (see Proposition 5) and [20] (see Proposition 4.3). However instead of d 1 and an additive potential φ considered in [15], here we consider d Ψ and an almost additive potential Φ, so the proof is more involved.
We get the desired subadditivity by taking β n = m ac 1 (n)+b .
• Coincidence of two limits. Next we show that Note that these limits exist since f (α, n, ǫ) is a non-increasing function in the variable ǫ. Denote by θ the left-hand side limit. Then for any δ > 0, there exists ǫ 0 > 0 such that lim inf n→∞ log f (α, n, ǫ 0 )/n < θ + δ.
• Results about D(Ψ). By essentially repeating the same proof as above (in fact it is much easier), we can show lim inf n→∞ log #B n (Ψ) n = lim sup n→∞ log #B n (Ψ) n .
Since each x ∈ E is a local cone point of E when E is ri(A), or E is a closed interval in A, or E is A itself and A is a convex closed polyhedron, the other results follow.

Now we prove Proposition 4. The new point in this proposition is the weak concavity.
In fact when d Ψ = d 1 , as shown in [15], the function Λ is indeed concave. When the more general metric d Ψ is considered, the length of w ∈ B n (Ψ) has fluctuations (see Lemma 4.2 (1)), which destroy the concavity of Λ. However, these fluctuations are controllable, so that a careful analysis yields the weak concavity of Λ.
Assume A ⊂ L Φ is a convex set, and I ⊂ L Φ is a closed interval. By Lemma 4.3, Λ is lower semi-continuous on ri(A) and I. Combining this with the upper semi-continuity yields the continuity on ri(A) and I. Taking A = L Φ we get the continuity on ri(L Φ ). Now assume L Φ is a polyhedron. By Lemma 4.3, Λ is lower semi-continuous on L Φ . This, together with the upper semi-continuity yields the continuity on L Φ .

Proof of Theorem 2.2(1)
By Proposition 2, we have E Φ (α) = ∅ if and only if α ∈ L Φ . For the next statement, our plan is the following: we show that D(α) ≤ Λ(α) ≤ E(α) ≤ D(α). We divide this into three steps corresponding to the next Sections 5.1, 5.2 and 5.3.

Λ(α) ≤ E(α).
Our approach is inspired by that of [15], which deals with the case that d Ψ = d 1 and Φ = (S n φ) n≥1 is additive, where φ : Σ A → R d is continuous.
To show this inequality we need to approximate the almost additive potentials Φ and Ψ by two sequences of Hölder potentials. We describe this procedure as follows.
Given Φ ∈ C aa (Σ A , T, d), for each k ∈ N we define Φ (k) ∈ C aa (Σ A , T, d) as follows. For each w ∈ Σ A,k choose x w ∈ [w]. For any x ∈ [w] define φ k (x) := φ k (x w )/k. Then φ k depends only on the first k coordinates of x ∈ Σ A and is Hölder continuous. Define . Thus Φ (k) is additive and Hölder continuous.
This lemma will be proved at the end of this section.
Let us at first assume the claim holds and finish the proof. Notice that the set of invariant measures ν such that |Φ * (ν) − α| ≤ 5ǫ is compact, so by using the upper semicontinuity of h ν and letting ǫ tend to 0 we can find an invariant measure ν 0 such that Φ * (ν 0 ) = α and Next we show that the claim holds. In the following C = C(Ψ), C i = C i (Ψ) for i = 1, 2. Fix ǫ > 0. For any k ∈ N define Φ (k) and Ψ (k) according to (5.2). By Lemma 5.1, we can find k ∈ N such that For any w ∈ B n (Ψ), we have C 1 n ≤ |w| ≤ C 2 n, thus for n ≥ N 1 /C 1 , we have F (α, n, ǫ, Φ, Ψ) ⊂ F (α, n, 2ǫ, Φ (k) , Ψ), consequently Following [15], we introduce a way to classify the words in F (α, n, 2ǫ, Φ (k) , Ψ), by which we can estimate the cardinality of it effectively.
For any word w ∈ Σ A, * such that |w| ≥ k, we define the counting function θ w : Σ A,k → N as θ w (u) = #{j : w j · · · w j+k−1 = u}, which counts the numbers of times the word u appears in w. It is clear that h θw := u θ w (u) = |w| − k + 1. We call it the height of θ w .
Write Γ(θ) := #T (θ). Then we have In the following we estimate log Γ(θ)/n for each θ ∈ P (n) k . Since it is hard to estimate it directly, we turn to the estimations of log Γ(θ)/h θ and n/h θ .
It is known (see [15]) that for any η > 0, there is a positive integer N = N (η) such that for any w ∈ Σ A,l+k−1 with l > N , there exists a probability vector p ∈ △ + k such that We discard the trivial case where Φ ≡ 0 and fix η > 0 such that η < ǫ/(m k Φ ). Now take any n ≥ max{ such that θ = θ w , then |w| = h θ + k − 1. Fix a p ∈ △ + k as described above. Consider the Markov measure ν p corresponding to p (see [15] for the definition and related properties). For any word v ∈ T (θ) we have .

On the other hand
Combining this with (5.7) and the fact that Ψ ⋆ n /n = o(1) we get Combine (5.6) and (5.8) we get .
Proof of Lemma 5.1. At first we assume Φ ∈ C aa (Σ A , T ).
For n ∈ N, write n = pk + s with 0 ≤ s < k. Write C = C(Φ), by using the almost additivity of Φ for each 0 ≤ j ≤ k − 1 we have , applying the result just proven to each component of Φ we get the result.

Proof of Theorem 2.3
We prove the slighly more general result mentioned in Remark 4 (2). Suppose that ξ is continuous outside a subset E of Σ A , bounded and ξ(Σ A ) ⊂ aff(L Φ ). Also, suppose that dim H E < λ := sup{D(α) : α ∈ ξ(Σ A \ E) ∩ ri(L Φ )}. To prepare the proof of our geometric results, we need the following more general result. Proposition 7. Let Z ⊂ Σ A be a closed set such that µ(Z) = 0 for any Gibbs measure µ fully supported on Σ A . For any δ > 0 such that λ − δ > dim H E, we can construct a Moran subset Θ ⊂ Σ A such that Θ \ E ⊂ E Φ (ξ), dim H Θ ≥ λ − δ and there exists an increasing sequence of integers (g j ) j≥1 such that Tg j x ∈ Z for any x ∈ Θ and any j ≥ 1.
Since D is continuous in ri(L Φ ), we can find η > 0 such that B η := B(α 0 , η) ∩ aff(L Φ ) ⊂ ri(L Φ ) and for any α ∈ B η we have |D(α) − D(α 0 )| < δ/2. Consequently Now we proceed in four steps. The two first steps provide the scheme of the construction of the set Θ and a good measure ρ, a piece of which is supported by Θ. The next two steps complete the construction to ensure that Θ as the required properties and the dimension of ρ restricted to Θ has a Hausdorff dimension larger than or equal to λ − δ.
Step 2: Construction of the Moran set Θ.
Next we want to specify the integer sequence {L j } j≥1 and pick out carefully a Moran set Θ ⊂ [ϑ] such that ρ(Θ) > 0 and Θ has the last property stated in the proposition. We proceed as follows.
Take an integer sequence {L j } n≥1 such that L j ≥ max(N j , N j ) for j ≥ 1 and consider the associated measure ρ constructed in step 1. We define the desired Moran set as By construction, T g j −1 x ∈ Z for any x ∈ Θ and any j ≥ 0. Defineg j = g j − 1, we checked the last property of Θ.
For η ∈ {φ, ψ} and j ≥ 1, let This number is finite since each η j is Hölder continuous. By (7.4) we can find integer sequence M j ր ∞ such that The sequence (L j ) j≥1 can be specified to satisfy the additional properties L j ≥ M j+1 and max(K 1 (j), K 2 (j), K 3 (j)) ≤ ε j g j , Let us check that Θ \ E ⊂ E Φ (ξ). Let x ∈ Θ \ E, n ≥ g 1 and j ≥ 1 such that g j ≤ n < g j+1 . Since n ≥ g j > L j ≥ M j+1 , by (7.9) we have For the second term in the right hand side we have (with g −1 = 0, α k = α x|g k−1 and At first we have |S L k φ k (T g k−1 x) − L k α k | + |S n−g j φ j+1 (T g j x) − (n − g j )α j+1 |.
Thus we have Since L k ≥ M k+1 ≥ M k we also have Thus both terms are o(g j ) as n → ∞. Consequently both terms are o(n).
On the other hand, by construction (recall that α k+1 = α x|g k ) where x x|g k is the special point in [x| g k ] chosen in the construction of the measure ρ. Since x ∈ E, ξ is continuous at x. We have lim k→∞ Osc(ξ, [x| g k ]) + 2 −k−j 0 d 0 = 0.
A slight modification of the above proof with ξ taken constant yields the following proposition: Proposition 8. Assume Z ⊂ Σ A is a closed set such that µ(Z) = 0 for any Gibbs measure µ fully supported on Σ A . For any α ∈ L Φ , we can construct a subset Θ ⊂ E Φ (α) such that dim H Θ ≥ E(α) and there exists an integer sequence g j ր ∞ such that T g j x ∈ Z for any x ∈ Θ and any j ≥ 1. In particular, E(α) ≤ D(α).
(2) If ξ(Σ A ) ⊂ L Φ , the construction of a Moran subset of E Φ (ξ) can be done around any point of Σ A , like in the proof of Proposition 7. The only difference is that in this case the dimension of this set is of no importance. Hence, E Φ (ξ) is dense.

Proofs of results in section 3
We will use the following lemma, which is standard and essentially the same as Lemma 5.1 in [21] (the proof is elementary).
Lemma 8.1. Let X and Y be metric spaces and χ : X → Y a surjective mapping with the following property: there exists a function N : (0, ∞) → N with log N (r)/ log r → 0 when r → 0 such that for any r > 0, the pre-image χ −1 (B) of any r-ball in Y can be covered by at most N (r) sets in X of diameter less than r. Then for any set E ⊂ Y we have dim H E ≥ dim H χ −1 (E).
For the converse inequality, let us check the condition of the above lemma. Let B ⊂ J be a ball of radius r, let n ∈ N such that e −n ≤ r < e 1−n . Define by Lemma 4.2 (2). Thus #G r B ≤ K d ′ (e+2K) d ′ e o(n) . So we conclude that log N (r)/ log r = log #G r B / log r → 0 as r → 0. Thus by lemma 8.1, we can conclude that dim H E ≥ dim H χ −1 (E).