The thermodynamic formalism for almost-additive sequences

We study the nonadditive thermodynamic formalism for the 
class of almost-additive sequences of potentials. We define the 
topological pressure $P_Z(\Phi)$ of an almost-additive sequence 
$\Phi$, on a set $Z$. We give conditions which allow us to 
establish a variational principle for the topological pressure. We 
state conditions for the existence and uniqueness of equilibrium 
measures, and for subshifts of finite type the existence and 
uniqueness of Gibbs measures. Finally, we compare the results for 
almost-additive sequences to the thermodynamic formalism for the 
classical (additive) case [10] [11] [3], 
the sequences studied by Barreira [1], 
Falconer [5], and that of Feng and Lau [7], [6].

1. Introduction. In [7] and [6], Feng and Lau developed a thermodynamic formalism for some sequences of functions on subshifts of finite type in the context of multifractal formalism associated to certain iterated function systems with overlaps. These sequences of functions are not additive and fall into the category of sequences introduced in this paper, called almost-additive. We give conditions under which their thermodynamic formalism can be generalized to this class of sequences on an arbitrary compact metric space, with respect to a continuous map f .
In section 3, we examine some properties of almost-additive sequences. We note that an almost-additive sequence may not be close to any additive sequence (see Section 2, Examples 2 and 4).
In this paper we study the nonadditive thermodynamic formalism for the class of almost-additive sequences. Several examples of almost-additive sequences are given 2 ANNA MUMMERT in Section 2, among which is the sequence studied by Feng and Lau. The thermodynamic formalism of the example sequences has been studied (see Ruelle [10], Sinai [11], Bowen [3], Barreira [1], Falconer [5], Feng and Lau [7], [6]). We compare these previous results to our thermodynamic formalism.
Let W m (U) be the set of all m-strings. For U ∈ W m (U) denote m(U ) = m and define the set A set Γ ⊂ ∪ m≥1 W m (U) is said to be a cover of Z if Z ⊂ U ∈Γ X(U ).
where the infimum is taken over all Γ ⊂ W n (U) covering Z.
Definition 2. The topological pressure of an almost-additive sequence Φ, on a compact f -invariant set Z ⊂ X, is given by lim n→∞ 1 n Z n (Φ, Z, U).
When necessary, we write P f,Z (Φ) = P Z (Φ) for the topological pressure of Φ with respect to the function f .
If the almost-additive sequence of functions Φ satisfies lim diam(U )→0 lim n→∞ γ n (Φ, U) then the limits in the definition of topological pressure exist. In Section 4, we show that this is the definition given by Barreira [1].
We study the variational principle in Section 5. Let M (X, f ) be the set of finvariant Borel probability measures on X. We obtain the following general estimate of the topological pressure for any almost-additive sequence of functions. To obtain the inverse inequality, and in particular the variational principle, the sequence of functions must satisfy an additional condition.
Theorem 2. Suppose that Φ is an almost-additive sequence of functions satisfying property (1), and the functions c 1 (n) and c 2 (n) satisfy ALMOST-ADDITIVE SEQUENCES 3 Then Condition (2) is a strong requirement on the sequence Φ. It is satisfied by some almost-additive sequences but not by the sequence studied by Feng and Lau in [7] and [6]. For these sequences we provide another condition which guarantees the variational principle. This requirement works well in the case when the system is a subshift of finite type. The extra structure provided by the subshift allows us to develop a thermodynamic formalism with conditions different than (2).
Let (X, σ) be a subshift of finite type and Φ an almost-additive sequence satisfying (1). The topological pressure on a compact σ-invariant set Z ⊂ X is given by the following formula where Γ is the unique cover of Z by n-cylinders.
Theorem 3. Suppose that (X, σ) is a mixing subshift of finite type and Φ an almostadditive sequence satisfying (1). Also, suppose that there exists γ such that for every Then Remark. After this paper was finished, I became aware of a preprint by Barreira Nonadditive Thermodynamic Formalism: Equilibrium and Gibbs Measures [2]. Building on some ideas of his work one can actually drop requirement (2) and prove the following result.
Theorem 4. Suppose that Φ is an almost-additive sequence of functions satisfying property (1). Then At the end of Section 5, we show how to modify the proof of Theorem 2 to obtain this result.
We explore the existence and uniqueness of equilibrium measures in Section 6. The existence of equilibrium measures is, in itself, an interesting problem. We obtain the following result on existence of such measures without requiring the topological pressure to exist.
A map f is called expansive if there exists an > 0 so that for any points x, y ∈ X with ρ(f k (x), f k (y)) < for all k ∈ Z then x = y.
Theorem 5. Suppose that f is an expansive homeomorphism of X. Then for any almost-additive sequence of functions Φ there exists an equilibrium measure µ Φ on X.
Corollary 1. Suppose that Φ is an almost-additive sequence satisfying (1) and (2), and that f is an expansive homeomorphism of X. Then there exists a measure µ Φ on X such that We note that some authors would define an equilibrium measure to be a measure µ Φ satisfying (4).
Let C n be an n-cylinder in a subshift of finite type X.
for any n > 0, C n ⊂ Σ A and x ∈ C n .
In Section 6.1 we obtain the following results on existence and uniqueness of equilibrium measures for a mixing subshift of finite type (X, σ).
Theorem 6. Suppose that Φ is an almost-additive sequences of functions on a mixing subshift of finite type (X, σ) satisfying (1) and (3). Then there exists a unique Gibbs measure on X. Moreover, this measure is the unique equilibrium measure for Φ.
For an additive sequence, if ϕ 1 is Hölder continuous, then condition (3) holds, and Theorem 6 gives the classical result on the existence of equilibrium measures for subshifts of finite type.
For almost-additive sequences, which are not additive, condition (3) holds for sequences with γ n (Φ, U) ≤ bα n for every n, for some positive constants b and α ∈ (0, 1). We note that this condition is satisfied by sequences for which each ϕ n is α-Hölder continuous with common constant C, as required for the sequences studied by Feng and Lau [7], [6].
A diffeomorphism f : X → X is called Axiom A if the set Ω(f ) of non-wandering points is hyperbolic and is the closure of the periodic points. The spectral decomposition theorem gives that Ω(f ) = Ω 1 · · · Ω s , where the Ω i are pairwise disjoint closed sets with a. f (Ω i ) = Ω i and f |Ωi is topologically transitive, b. Ω i = Z i,1 · · · Z i,ni , with the Z i,j pairwise disjoint closed sets with f (Z i,j ) = Z i,j+1 and f |Zi,j topologically mixing. The map f is expansive on its hyperbolic set; thus by Theorem 5 there exists at least one equilibrium measure for Φ. We show the following theorem in Section 6.2.
Theorem 7. Suppose that Ω is a basic set for an Axiom A diffeomorphism f and Φ an almost-additive sequence of functions satisfying conditions (1) and (3). Then there exists a unique equilibrium measure µ Φ for Φ.
In the course of proving Theorem 7, we show the variational principle without requiring Φ to satisfy (2).

5
Corollary 2. Suppose that (X, f ) is an Axiom A diffeomorphism, Φ is an almostadditive sequence of functions satisfying conditions (1) and (3). Then

2.
Examples of Almost-Additive Sequences of Functions.
2.1. Additive Sequences. Let φ : X → R be a continuous function. The sequence is an additive sequence and thus is almost-additive with c 1 (n) and c 2 (n) identically equal to zero. Such sequences appear in the classical thermodynamic formalism studied by Ruelle [10], Sinai [11], and Bowen [3]. These sequences trivially satisfy properties (1) and (2). Thus we can apply Definition 2 to obtain the classical topological pressure of φ on a compact f -invariant set Z, where the infimum is taken over all Γ ⊂ W n (U) covering Z. Theorem 2 gives the classical variational principle If φ is Hölder continuous then condition (3) holds, and we recover the existence and uniqueness of Gibbs and equilibrium measures under the same conditions as in the classical case.
2.2. Nonadditive Sequences. Consider a sequence satisfying (1) for which there is a continuous function ψ : X → R so that uniformly on X as n → ∞. Set γ n = ||ϕ n − ϕ n−1 • f − ψ|| ∞ . Then for every m ≥ 1 and every x ∈ X, we have that Fix two numbers m, n ≥ 1. One can show the following inequality.
Without loss of generality set ϕ 0 ≡ 0. The above inequality with m = 0 implies that  (7) and (8) we have that The above inequality shows that a sequence satisfying (6) is almost-additive if the function is bounded by a constant C 2 for every n ≥ 1; since c 1 (n) = −c 2 (n), for every n, we would also have the bound −C 2 ≤ c 1 (n), for every n. This condition on the functions c 1 , c 2 is satisfied if the sequence ϕ n − ϕ n−1 • f converges fast enough to the function ψ.
On the other hand, there are sequences which are almost-additive but do not satisfy (6). For example, let which does not converge as n → ∞, where k i are constants depending on α.
Sequences satisfying (6) were shown by Barreira to have a variational principle corresponding to the nonadditive topological pressure (see Barreira [1], and Section 3 below). An almost-additive sequence satisfying (1) and (2) need not satisfy (6). Thus the set of sequences admitting a variational principle as in Barreira [1] is not disjoint from those as in this paper, but neither is either class contained in the other.
For sequences satisfying (1) and (6), which are also almost-additive satisfying (2), we can apply Theorems 2 to obtain the variational principle. Since these sequences satisfy (6), the variational principle can be expressed as For a compact f -invariant set Z, we note that L(Z) = Z.
Suppose that f is an expansive homeomorphism of X. Theorem 5 implies that for sequences satisfying (6) which are also almost-additive there exists an equilibrium measure. Uniqueness of equilibrium measures follows from Theorem 7 for f which are Axiom A, and almost-additive sequences satisfying conditions (1) and (3).

Subadditive Sequences.
Consider (X, f ) = (Σ A , σ), where Σ A is a mixing subshift of finite type with transition matrix A. Let Φ be an almost-additive sequence with c 2 (n) ≡ 0. Assume that the following conditions hold a. a uniform bound |(1/n)ϕ n (x)| ≤ M , b. a Lipschitz condition |(1/n)ϕ n (x) − (1/n)ϕ n (y)| ≤ a|x − y|, and c. bounded variation, i.e. there exists a constant b independent of n such that |ϕ n (x) − ϕ n (y)| ≤ b whenever x, y ∈ C n , for some n-cylinder C n .
Conditions (a)-(c) imply that these sequences satisfy conditions (1) and (3). Thus we can define the topological pressure of the sequence Φ on the set Z (see Definition 2). Theorem 3 gives a corresponding variational principle, and Theorem 6 gives the existence and uniqueness of a Gibbs (and equilibrium) measure.
The thermodynamic formalism for these sequences was first introduced by Falconer [5] while studying mixing repellers. They are a particular case of the nonadditive thermodynamic formalism of Barreira [1].
is almost-additive with c 1 (n) = c 1 < 0 and c 2 = 0. To show this we follow the argument in Feng and Lau [7]. Since ||·|| is a norm, we have that ||AB|| ≤ ||A||·||B||. Thus For the opposite inequality, Feng and Lau [7] show that there is a constant C > 0 such that We can apply Definition 2 to obtain the topological pressure Applying Theorem 3 gives the variational principle: The above pressure and variational principle are those given by Feng and Lau [7], [6]. Applying Theorem 6 gives the existence of a unique Gibbs and equilibrium measure, which recovers the results of Feng and Lau. As mentioned in the introduction, the thermodynamic formalism of the sequence ϕ n (x) = log ||M (x) . . . M (σ n−1 (x))|| was studied by Feng and Lau, [7] [6], in the context of multifractal analysis for iterated function systems with overlaps.
3. Properties of Almost-Additive Sequences of Functions. We first note that if c 2 (n) = 0 for every n, then Φ is a sub-additive sequence. If in addition, c 1 (n) = 0 for all n, then the almost-additive sequence of functions is additive. Lemma 1. Let Φ be an almost-additive sequence of functions. Then the limit lim n→∞ ϕ n /n exists almost everywhere; it is possibly −∞ or ∞. Proof. The sequence (ϕ n + C 2 )/n is sub-additive since for every m, n we have By the subadditive ergodic theorem (see [12]) lim n→∞ (ϕ n + C 2 )/n exists almost everywhere. As we have the result is shown.
The following two lemmas are easily shown from Definition 1 (c).

Lemma 2.
Let Φ be an almost-additive sequence of functions. Then ϕ n dµ is an almost-additive sequence of numbers, i.e. for a n = ϕ n dµ we have that c 1 (n + m) + a n + a m ≤ a n+m ≤ a n + a m + c 2 (n + m) Lemma 3. Let Φ be an almost-additive sequence of functions. Then for a fixed m ≥ 0, the sequence is additive.

Topological Pressure.
We begin with the definition of the nonadditive topological pressure, see [1] or [9]. For every Z ⊂ X define where the infimum is taken over all Γ ⊂ ∪ k≥n W k (U) that cover Z. Similarly, define where the infimum are each taken over all Γ ⊂ W n (U) that cover Z. In equations Theorem 8 (Barreira [1]). Let Φ be a sequence of functions satisfying (1). The following limits exists Definition 5. The topological pressure, on any set Z ⊂ X, of a sequence Φ satisfying (1) is given by P Z (Φ). The values CP Z (Φ) and CP Z (Φ) are the lower and upper capacity pressure of Φ on Z, respectively.
The following theorem shows that Definition 2 is the topological pressure for an almost-additive sequence satisfying (1) on a compact, f -invariant set.
Theorem 9. Let Φ be an almost-additive sequence of functions satisfying (1) and Proof. (a) Given sets Γ m ⊂ W m (U) and Γ n ⊂ W n (U), define the set Since Z is f -invariant, if the two collections Γ m and Γ n cover Z, then their concatenation Γ m,n also covers Z. As Φ is almost-additive we have that for every U V ∈ Γ m,n . Thus we have that As U is a finite cover we have that Almost-additivity implies that the limit as n goes to infinity of log Z n (Φ, U)/n exists and is finite. Thus, as a result of Theorem 2.2 in [9] and Theorem 8, we have that (b) For an almost-additive sequence of functions, we have that ϕ n < ϕ n+1 + K, where K = sup x∈X |ϕ 1 (x)| + max{|C 1 |, |C 2 |} + 1. Let Γ ⊂ ∪ n≥1 W n (U) be a cover of Z. As Z is compact, we can assume that Γ is finite. Thus there exists an M such that Γ ⊂ ∪ n≤M W n (U).
Set Γ n = {U 1 . . . U n : U i ∈ Γ} for each n ≥ 1. As Z is f -invariant and Γ covers Z, for all n the collection Γ n covers Z. Since Φ is almost-additive, If α = P Z (Φ, U) then there exists an m ≥ 1 and a cover Γ ⊂ ∪ n≥m W n (U) of Z such that (exp C 2 )N (Γ) < 1. Set Γ ∞ = {U : U ∈ Γ n for some n}. Then As Γ covers Z, for any N ≥ 1 and x ∈ Z there exists U ∈ Γ ∞ such that x ∈ X(U ) and N ≤ m(U ) ≤ N + M . Let Γ * ⊂ W N (U) be the collection of strings U * that consist of the first N elements of some string U ∈ Γ ∞ . We have that Thus M (Z, α, Φ, U) < ∞, which implies that α > CP Z (Φ, U). Thus P Z (Φ, U) ≥ CP Z (Φ, U). Then using part (a) above, the result is shown.
Let σ : Σ A → Σ A be the shift map on one-sided sequences of p symbols, with adjacency matrix A. Write U n for the open cover of Σ p formed by the n-cylinder sets. We note that diam(U n ) goes to zero as n goes to infinity.
Consider any set Z ⊂ Σ p . There is a unique Γ ⊂ W n (U l ) covering Z. For each U ∈ W n (U l ) the set X(U ) is an (n + l − 1)-cylinder, and X(U ) ∩ X(V ) = ∅ if U = V . Thus for each l ≥ 1 we have that For an almost-additive sequence satisfying (1) this implies that An almost-additive sequence satisfying (1) on a subshift of finite type the topological pressure can be computed without the limit as the diameter of the cover goes to zero. Also, we note that it is possible to prove directly that exists for an almost-additive sequence of functions on a compact f -invariant set Z. This direct proof shows that the condition (1) is required for the limit as the diameter of the cover goes to zero. Thus for an almost-additive sequence on a subshift of finite type, we define the topological pressure without this condition, as in Definition 2.

Variational Principle.
For the rest of the paper, we will be considering Z a compact f -invariant subset of X; thus without loss of generality we assume that X = Z, and write P (Φ) = P Z (Φ). We first show that for Φ an almost-additive sequence satisfying condition (1) We note that condition (2) is not needed for the proof of this inequality. The proof of the inequality follows the line of argument in Bowen [3] (see also [9]). We only indicate necessary modifications for the almost-additive case. Let Γ ∈ W n (U) be a cover of X.
By almost-additivity and Equality (14), which implies that The desired result follows from Lemma 5.
Lemma 8 (Bowen [3]). Let real numbers a 1 , . . . a n be given. Then the quantity has maximum value log n i=1 exp a i for (p 1 , . . . p n ) with p i ≥ 0 and p 1 +· · ·+p n = 1, and the maximum is assumed only at p j = e ai ( n i=0 e ai ) −1 . Set M (X) to be the set of all Borel probability measures on X. We set H µ (C) as the entropy of µ with respect to the partition C, and H µ (C|D) as the conditional entropy of the partition C with respect to the partition D.
Lemma 9 (Bowen [3]). Let X be a compact metric space, µ ∈ M (X), > 0, and C a finite Borel partition. There is a δ > 0 so that H µ (C|D) < whenever D is a partition with diameter less than δ.
. Now, using properties of entropy (see for example Katok and Hasselblatt [8]), and Lemma 8 we have For each , one has by inequalities (15) and (16) h µ (f, D) + lim By letting m → ∞ and then diameter U → 0, we obtain the desired inequality.
Proof of Inequality (12). Let C be a Borel partition and > 0. By Lemma 9 we can find a finite open cover A of X so that H µ (C|D) ≤ whenever D is a partition every member of which is contained in some member of A. Fix n > 0. Let C n = C ∨ · · · ∨ f −n+1 C, and D(n) be as in Lemma 10. Then by Lemma 11 we have n (P f n (S n Φ) + log n|A|) + 1 n H µ (C n |D(n)).
We have H µ (C n |D(n)) ≤ n−1 k=0 H µ (f −k D|D(n)). Since D(n) refines f −k A for each k, one has H µ (f −k C|D(n)) ≤ (since µ is f -invariant, f −k A bears the same relation to f −k C as A to C). Hence, using Lemma 6, Letting n → ∞ and then → 0 shows the desired inequality.
Next we show that for an almost-additive sequence Φ satisfying conditions (1) and (2) The following lemmas contain several steps needed in the proof the inequality.
Lemma 12 (Bowen [3]). Fix a finite set E and h ≥ 0. Let Then lim k→∞ 1 k log |R(k, h)| ≤ h. Let δ x be the unit measure on x ∈ X, and δ x,n = 1 n (δ x + · · · + δ f n−1 x ). Define the set V (x) = {µ ∈ M (X) : δ x,n k → µ for some n k → ∞}. As X is a compact metric space V (x) is not empty. Also, every measure in Lemma 13. Let x ∈ X, µ ∈ V (x). Fix δ > 0, and > 0. Then there exists m, N ∈ N for which one can find a string U ∈ W N (U) satisfying the following a.
x ∈ X(U ) b. U contains a substring of length km ≥ N − m which, when viewed as a = a 0 , . . . , Proof. Parts (a) and (b) can be found in Bowen [3]. Let N be large enough so that As (2) is satisfied we have for N large enough that B(N ) < , and the result is shown.
Proof of Inequality (17). Let U be a finite open cover of X and > 0. Cover X with countably many sets X m , where each X m is the set of points for which Lemma 13 holds with this m and some µ ∈ V (X). Choose an -dense set u 1 , . . . , u r in the closed set Cover each X m with the sets Y m (u i ), where each Y m (u i ) is the set of x ∈ X m for which Lemma 13 holds for this m and some µ ∈ V (x) with lim n→∞ (ϕ n dµ)/n ∈ [u i − , u i + ]. Next we cover each Y m (u) by Γ m,u and show that we can make as small as desired. Taking the unions of such Γ m,u we obtain a Γ covering X with (exp C 2 )Z(Γ, λ) < 1.
Let S(N ) be the number of all possible strings U satisfying Lemma 13 for some x ∈ Y m (u). By Lemma 13 part (2), and Lemma 12 for all sufficiently large N.
For every integer N 0 , the strings satisfying Lemma 13, for some x ∈ Y m (u) and N ≥ N 0 , cover the set Y m (u). Let Γ m,u be the collection of U showing up in the present situation for some sufficiently large N > N 0 fixed. One can show any Y m (u) can be covered by Γ m,u ⊂ m≥0 W m (U) with (exp c 2 )Z(Γ m,u , λ) as small as desired. Taking the unions of such Γ m,u we obtain a Γ covering X with (exp c 2 )Z(Γ, λ) < 1. By Lemma 7, λ ≤ exp(−P (Φ, U)), meaning The result is shown by letting → 0 and diam(U) → 0.
To prove Theorem 4, we demonstrate a new proof of Lemma 13 part (c) without condition (2).
Proof of Lemma 13 part(c). Let x ∈ V (x) and > 0. For each n there exists a sequence of integers m k = m k (n) such that δ x,m k (ϕ n ) − ϕ n dµ < for every k ∈ N. Since the sequence Φ is almost-additive, for each n ∈ N we have ϕ n • f n j + Cm. ϕ n (f nj ) < C n .
The proof of Theorem 4 now procedes as in the proof of Theorem 2.
Proof of Theorem 5. We show that the function µ → h µ (f ) + lim n→∞ 1 n ϕ n dµ is upper semi-continuous. Then use the fact that upper semi-continuous functions achieve their supremum on a compact set.
The map µ → h µ (f ) is known to be upper semi-continuous for an expansive homeomorphism, see for example Bowen [3].
As a consequence of almost-additivity for the sequence of numbers ϕ n dµ we have 1 m ϕ m dµ + C 1 ≤ lim As the system is mixing we have that there is an integer p > 0 such that A p > 0. This implies that for any cylinders C n ⊂ Σ A and J l ⊂ Σ A , there exists a p-cylinder K p ⊂ Σ A such that C n K p J l ⊂ Σ A .
We use the notation A ≈ B when there exists D 1 , D 2 so that D 1 A ≤ B ≤ D 2 A. The following lemma can be shown through a series of computations that follow the work of Feng and Lau [7] [6].

Lemma 14.
Cn∈Σ A sup x∈Cn exp ϕ n (x) ≈ exp(nP (Φ)). For each integer n > 0 let B n be the σ-algebra generated by the cylinders C n ⊂ Σ A . We define a sequence of probability measures {ν n,Φ } on B n by ν n,Φ (C n ) = sup x∈Cn exp ϕ n (x) There exists a subsequence {ν n k ,Φ } that converges in the weak-star topology to a probability measure ν Φ .
Proof. Let µ Φ be a limit point of a subsequence of in the weak-star topology. By definition µ Φ is a σ-invariant probability measure on Σ A . For each C n ⊂ Σ A and l > p we have that σ −l (C n ) is the union of all cylinders D l C n ⊂ Σ A . Thus we can show for every l > p that Sum over all C n ⊂ Σ A , and divide by m. Taking the limit as m goes to infinity yields that µ Φ satisfies equation (5). Given any C n and D l ⊂ Σ A and any i > n + 2p we have As any Borel set can be approximated within by a finite disjoint union of cylinder sets, the above inequality holds for all Borel sets. Thus for any µ Φ positive measure Borel sets A, B ⊂ Σ there exists n > 0 such that µ Φ (A ∩ σ −n (B)) > 0. Hence µ Φ is ergodic. Any two distinct ergodic measures must be singular, but property (5) shows that they must be absolutely continuous to each other. Thus the measure µ Φ is unique.
Lemma 17. The Gibbs measure µ Φ is an equilibrium measure for Φ.
Proof. Since µ Φ satisfies equation (5), for each n ∈ N, C n ⊂ Σ A , and x ∈ C n we have log Integrate over x ∈ C n , sum over all C n ⊂ Σ A , and divide by n to get log C 1 n ≤ P (Φ) + 1 n Letting n go to infinity and combining with Inequality (12) yields the result.
During the proof of the above lemma we also showed the variational principle for subshifts of finite type without requiring (2), i.e. Theorem 3.
Lemma 18. The Gibbs measure µ Φ is the unique equilibrium measure.
To prove Lemma 18 we use the following two lemmas.
Lemma 19 (Bowen [3]). Let X be a compact metric space, µ ∈ M (X), and D = {D 1 , . . . , D n } a Borel parition of X. Suppose {C m } ∞ m=1 is a sequence of partitions so that diam(C m ) = max C∈Cm diam(C) → 0 as m → ∞. Then there is a sequence Lemma 20 (Bowen [3]). Suppose 0 ≤ p i , . . . , p m ≤ 1, s = p 1 + · · · + p m ≤ 1 and a 1 , . . . , a m ∈ R. Then ] E m is a union of members of σ −m+1 U ∨ · · · ∨ U. For every there exists an m large enough so that As ϕ m is continuous there exists a d so that for each B one can find Applying Lemma 20 gives where K = sup 0≤s≤1 (−s log s). Using the fact that µ Φ is a Gibbs measure, we have that ≤ log A −1 1 + ν(F m ) log µ Φ (F m ) + ν(X \ F m ) log µ(X \ F m ).
Proof of Theorem 7. Let R be a Markov partition for Ω of diameter less or equal , A the transition matrix for R, and π : Σ A → Ω. As in Bowen [3], π is one-to-one except on a set of measure zero.
First we assume that f |Ω is mixing. Then σ |Σ A is mixing and there exists an equilibrium measure µ Φ•π . Let µ Φ = π * µ Φ•π , meaning µ Φ (E) = µ Φ•π (π −1 E). Then µ Φ is f -invariant. The measure spaces (Σ A , σ, µ Φ•π ) and (X, f, µ Φ ) are conjugate, since π is one-to-one except on a set of µ φ • π measure zero. In particular, h µΦ (f ) = h µΦ•π (σ) and so by Lemma 21 Hence P σ (Φ • π) = P f (Φ) and µ Φ is an equilibrium state for Φ. If f |Ω is not mixing, then use the spectral decomposition to show that there exists a bijection between M (Ω, f ) and M (X 1 , f m ). Thus we have that  For Φ Hölder on Ω, S m Φ will be Hölder on X 1 and therefore we are done since X 1 is a mixing basic set of f m .
During the proof of the above lemma we also showed the variational principle for Axiom A diffeomorphisms without requiring (2), i.e. Corollary 2.