EQUILIBRIUM STATES FOR FACTOR MAPS BETWEEN SUBSHIFTS

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1 EQUILIBRIUM STATES FOR FACTOR MAPS BETWEEN SUBSHIFTS DE-JUN FENG Abstract. Let π : X Y be a factor map, where (X, σ X and (Y, σ Y are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a = (a, a 2 R 2 with a > 0 and a 2 0. Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance. We show that there is a unique shift invariant measure µ on X that maximizes f dµ + a h µ (σ X + a 2 h µ π (σ Y. In particular, taking f 0 we see that there is a unique invariant measure µ on X that maximizes the weighted entropy a h µ (σ X + a 2 h µ π (σ Y, which answers an open question raised by Gatzouras and Peres in [6]. An extension is given to high dimensional cases. As an application, we show that for each compact invariant set K on the k-torus under a diagonal endomorphism, if the symbolic coding of K satisfies weak specification, then there is a unique invariant measure µ supported on K so that dim H µ = dim H K.. Introduction In this paper, we study the thermodynamic formalism on subshifts and give an application in non-conformal dynamical systems. Let k 2 be an integer. Assume that (X i, σ Xi, i =,..., k, are one-sided (or two-sided subshifts over finite alphabets, and X i+ is a factor of X i with a factor map π i : X i X i+ for i =,..., k. (See Sect. 2 for the definitions. For convenience, we use π 0 to denote the identity map on X. Define τ i : X X i+ by τ i = π i π i π 0 for i = 0,,..., k. Let M(X i, σ Xi denote the set of all σ Xi -invariant Borel probability measures on X i, endowed with the weak-star topology. Fix a = (a, a 2,..., a k R k with a > 0 and a i 0 for i 2. For µ M(X, σ X, we call k h a µ(σ X := a i h µ τ (σ X i i i= the a-weighted measure-theoretic entropy of µ with respect to σ X, or simply, the a-weighted entropy of µ, where h µ τ i (σ X i denotes the measure-theoretic entropy of µ τ i with respect to σ X i. For a real continuous function f on X, we say that Key words and phrases. Equilibrium states, factor maps, topological pressures, invariant measures, entropies, Hausdorff dimension Mathematics Subject Classification: Primary 37D35, Secondary 37B0, 37A35, 28A78.

2 µ M(X, σ X is an a-weighted equilibrium state of f for the factor maps π i s, or simply, a-weighted equilibrium state of f if { } (. f dµ + h a µ(σ X = sup f dη + h a η(σ X : η M(X, σ X. The value in the right hand side of (. is called the a-weighted topological pressure of f and is denoted by P a (σ X, f. The existence of at least one a-weighted equilibrium state of f follows from the upper semi-continuity of the entropy functions h ( (σ Xi. The notions of weighted topological pressure and weighted equilibrium state were recently introduced by Barral and the author in [], mainly motivated from the study of the multifractal analysis on self-affine sponges. When a = (, 0,..., 0, the a-weighted topological pressure and a-weighted equilibrium states are reduced back to the classical topological pressure and equilibrium states (cf. [32, 35, 29]. The main objective of this paper is to study the dynamical property of general weighted equilibrium states. We want to give conditions on f and X i s to guarantee a unique a-weighted equilibrium state. This study is mainly motivated from the following question raised by Gatzouras and Peres in [6, Problem 3]: Question.. Let π : X Y be a factor map between subshifts X and Y, where X is an irreducible subshift of finite type. Let α > 0. Is there a unique invariant measure µ on X maximizing the weighted entropy h µ (σ X + αh µ π (σ Y? Question. is closely related to dimension theory of certain non-conformal dynamical systems [6] (we will address it a little later. It still remains open except some partial results (see the remarks after Theorem.3. One of the difficulties is due to the complex structure of the fibres π (ν := {η M(X, σ X : η π = ν} for invariant measures ν on Y. For instance, there may exist different elements µ π (ν such that h µ (σ X = sup{h η (σ X : η π (ν} [30]. The reader is referred to [7] for some related open questions about the structure of π (ν. Let us return back to our general issue. We say that the subshift X satisfies weak specification if there exists p N such that, for any two words I and J that are legal in X (i.e., may be extended to sequences in X, there is a word K of length not exceeding p such that the word IKJ is legal in X. Similarly, say that X satisfies specification if there exists p N such that, for any two words I and J that are legal in X, there is a word K of length p such that the word IKJ is legal in X. For more details about the definitions, see Sect. 2. 2

3 Let C(X denote the collection of real continuous functions on X. For f C(X and n let n (.2 S n f(x = f(σx i x, x X. i=0 Let V (σ X denote the set of f C(X such that there exists c > 0 such that (.3 S n f(x S n f(y c whenever x i = y i for all 0 < i n, where x = (x i i= and y = (y i i=. Endow X with the usual metric (see Sect. 2. Clearly V (σ X contains all Hölder continuous functions on X. The main result of the paper is the following. Theorem.2. Assume that X satisfies weak specification. Then for any f V (σ X and a = (a, a 2,..., a k R k with a > 0 and a i 0 for i 2, f has a unique a-weighted equilibrium state µ. The measure µ is ergodic and, there exist p N and c > 0 such that lim inf n p i=0 µ ( A σ n i X (B cµ(aµ(b, Borel sets A, B X. Furthermore, if X satisfies specification, then there exists c > 0 such that (.4 lim inf n µ ( A σ n X (B cµ(aµ(b, Borel sets A, B X. The measure µ in Theorem.2 can be constructed as the limit of a sequence of discrete measures in the weak-star topology (see Remark 7.4. Taking f = 0 in Theorem.2 yields, whenever X satisfies weak specification, there is a unique invariant measure µ on X maximizing the a-weighted entropy. This yields a confirmative answer to Question., because each irreducible subshift of finite type satisfies weak specification (cf. Sect. 2. We remark that Theorem.2 is a natural extension of Bowen s result [5] about the classical equilibrium states. Restricted to the subshift case, Bowen [5] proved whenever X satisfies specification and f V (σ X, there is a unique invariant measure µ on X such that { } f dµ + h µ (σ X = sup f dη + h η (σ X : η M(X, σ X, and furthermore, µ satisfies (.4. This result corresponds to the special case a = (, 0,..., 0 in Theorem.2. 3

4 In the literature, an invariant measure µ satisfying (.4 is called partially mixing with respect to σ X. Recall that an invariant measure µ is called weakly mixing if n ( lim µ A σ i n X n (B µ(aµ(b = 0, Borel sets A, B X, i=0 and µ is called mixing if lim µ ( A σ n n X (B = µ(aµ(b, Borel sets A, B X. It is known that mixing implies partial mixing, and partial mixing implies weak mixing; these three properties are essentially different (cf. [5] and references therein. Theorem.2 has an interesting application in characterizing invariant measures of maximal Hausdorff dimension for certain non-conformal dynamical systems. Let T be the endmorphism on the k-dimensional torus T k = R k /Z k represented by an integral diagonal matrix Λ = diag(m, m 2,..., m k, where 2 m... m k. That is, T u = Λu ( mod for u T k. Let A denote the Cartesian product k {0,,..., m i } let R : A N T k be the canonical coding map given by R(x = Λ n x i, x = (x i i= A N, i= n= where each element in A is viewed as a k-dimensional column vector. For any nonempty D A, the set R(D N is called a self-affine Sierpinski sponge. Whenever k = 2, McMullen [26] and Bedford [4] determined the explicit value of the Hausdorff dimension of R(D N, and showed that there exists a Bernoulli product measure µ on D N such that dim H µ R = dim H R(D N. Kenyon and Peres [9] extended this result to the general case k 2, and moreover, they proved for each compact T - invariant set K T k, there is an ergodic σ-invariant µ on A N so that µ(r (K = and dim H µ R = dim H K. Furthermore, Kenyon and Peres [9] proved the uniqueness of µ M(D N, σ satisfying dim H µ R = dim H R(D N, by setting up the following formula for any ergodic η M(A N, σ: (.5 dim H η R = k h η (σ + log m k i= ( log m k i log m k i+ h η τ (σ i, i where τ i denotes the one-block map from A N to A N i, with A i = k i j= {0,,..., m j }, so that each element in A (viewed as a k-dimensional vector is projected into its first (k i coordinates; and σ i denotes the left shift on A N i. Formula (.5 is an 4

5 analogue of that for the Hausdorff dimension of C +α hyperbolic measures along an unstable (respectively, a stable manifold established by Ledrappier and Young [24]. As Gatzouras and Peres pointed out in [6], the uniqueness has not been known for more general invariant subsets K, even if K = R(X, where X A N is a general irreducible subshift of finite type. However, as a direct application of (.5 and Theorem.2, we have the following answer. Theorem.3. Let K = R(X, where X A N is a subshift satisfying weak specification. Then there is a unique T -invariant measure µ on K such that dim H µ = dim H K. Now we give some historic remarks about the study of Question.. Assume that π is a factor map between subshifts X and Y, where X is an irreducible subshift of finite type. Recall that a compensation function for π is a continuous function F : X R such that sup ν M(Y,σ Y ( φ dν + h ν (σ Y ( = sup µ M(X,σ X (φ π + F dµ + h µ (σ X for all φ C(Y. Compensation functions were introduced in [8] and studied systematically in [36]. Shin [33] showed that if there exists a compensation function of α the form f π, with f C(Y, and if f π has a unique equilibrium state, then +α there is a unique measure µ maximizing the weighted entropy h µ (σ X +αh µ π (σ Y. However, there exist factor maps between irreducible subshifts of finite type for which there are no such compensation functions [34]. Later, Petersen, Quas and Shin [30] proved that for each ergodic measure ν on Y, the number of ergodic measures µ of maximal entropy in the fibre π (ν is uniformly bounded; in particular, if π is a one-block map and there is a symbol b in the alphabet of Y such that the pre-image of b is a singleton (in this case, π : X Y is said to have a singleton clump, then there is a unique measures µ of maximal entropy in the fibre π (ν for each ergodic measure ν on Y. Recently, Yayama [37, 38] showed the uniqueness of invariant measures of maximal weighted entropy if π : X Y has a singleton clump. The uniqueness is further proved by Olivier [27] and Yayama [38] under an assumption that the projection of the Parry measure on X has certain Gibbs property (however the assumption only fulfils in some special cases. We remark that a special case of Theorem.2 was studied in []. It was proved in [] that, whenever π i : X i X i+ (i =,..., k are one-block factor maps between one-sided full shifts (X i, σ Xi, each f V (σ X has a unique a- weighted equilibrium state, which is Gibbs and mixing. This result has an interesting application in the multifractal analysis []. See [2, 3, 20, 28] for related results. The approach given in [] is based on the (relativized thermodynamic formalism of 5

6 almost additive potentials, which depends strongly upon the simple fibre structure in this special setting. For the proof of Theorem.2 in the general case, due to the complexity of fibre structures, it seems rather intractable to use classical thermodynamic formalism as in [5, 0] or take an approach as in []. In this paper, we manage to prove Theorem.2 by showing the uniqueness of equilibrium states and conditional equilibrium states for certain sub-additive potentials. A crucial step in our approach is to prove, for certain functions f defined on A (the set of finite words over A, there exists an ergodic invariant measures µ on the full shift space A N and c > 0, so that µ(i cf(i for I A (see Proposition 4.3. After we completed the first version of this paper, François Ledrappier informed us the following result which improves Theorem.2. Theorem.4. When X satisfies specification, the unique a-weighted equilibrium state µ in Theorem.2 is in fact a K-system; in particular, it is mixing. This result can be proved by using a similar argument as in [22, Proposition 4]. To see it, consider on X X the function (x, x F (x, x = f(x + f(x. By Theorem.2, F has a unique a-weighted equilibrium state. Let P µ be the Pinsker σ-algebra of (X, µ (see [35] and let η be relatively independent product of µ by µ over P µ, i.e., η(a B = E(χ A P µ E(χ B P µ dµ, Borel sets A, B X, where E( denotes the conditional expectation, and χ A, χ B the indicator functions of A, B respectively. Then both µ µ and η are a-weighted equilibrium states of F. So they coincide and thus the σ-algebra P µ has to be trivial. Therefore µ is a K-system. The paper is organized as follows: In Sect. 2, we give some basic notation and definitions about subshifts. In Sect. 3, we present and prove some variational principles about certain sub-additive potentials. In Sect. 4, we prove Proposition 4.3. In Sect. 5, we prove the uniqueness of equilibrium states for certain sub-additive potentials. In Sect. 6, we prove the uniqueness of weighted equilibrium states for certain sub-additive potentials in the case k = 2. The extension to the general case k 2 is given in Sect. 7, together with the proof of Theorem Preliminaries about subshifts In this section, we introduce some basic notation and definitions about subshifts. The reader is referred to [25] for the background and more details. 6

7 2.. One-sided subshifts over finite alphabets. Let A be a finite set of symbols which we will call the alphabet. Let A = k=0 denote the set of all finite words with letters from A, including the empty word ε. Let A k A N = {(x i i= : x i A for i } denote the collection of infinite sequences with entries from A. Then A N is a compact metric space endowed with the metric d(x, y = 2 inf{k: x k y k }, x = (x i i=, y = (y i i=. For any n N and I A n, we write (2. [I] = {(x i i= A N : x... x n = I} and call it an n-th cylinder set in A N. In this paper, a topological dynamical system is a continuous self map of a compact metrizable space. The shift transformation σ : A N A N is defined by (σx i = x i+ for all i N. The pair (A N, σ forms a topological dynamical system which is called the one-sided full shift over A. If X is a compact σ-invariant subset of A N, that is, σ(x X, then the topological dynamical system (X, σ is called a one-sided subshift over A, or simply, a subshift. Sometimes, we denote a subshift (X, σ by X, or (X, σ X. A subshift X over A is called a subshift of finite type if, there exists a matrix A = (A(α, β α,β A with entries 0 or such that X = { (x i i= A N : A(x i, x i+ = for all i N }. If A is irreducible (in the sense that, for any α, β A, there exists n > 0 such that A n (α, β > 0, X is called an irreducible subshift of finite type. Moreover if A is primitive (in the sense that, there exists n > 0 such that A n (α, β > 0 for all α, β A, X is called a mixing subshift of finite type. The language L(X of a subshift X is the set of all finite words (including the empty word ε that occur as consecutive strings x... x n in the sequences x = (x i i= which comprise X. That is, L(X = {I A : I = x... x n for some x = (x i i= X and n } {ε}. Denote I the length of a word I. For n 0, denote L n (X = {I L(X : I = n}. 7

8 Let p N. A subshift X is said to satisfy p-specification if for any I, J L(X, there exists K L p (X such that IKJ L(X. We say that X satisfies specification if it satisfies p-specification for some p N. Similarly, X is said to satisfy weak p- specification if for any I, J L(X, there exists K p i=0 L i(x such that IKJ L(X; and X is said to satisfy weak specification if it satisfies weak p-specification for some p N. It is easy to see that an irreducible subshift of finite type satisfies weak specification, whilst a mixing subshift of finite type satisfies specification. Let (X, σ X and (Y, σ Y be two subshifts over finite alphabets A and A, respectively. We say that Y is a factor of X if, there is a continuous surjective map π : X Y such that πt = Sπ. Here π is called a factor map. Furthermore π is called a -block map if there exists a map π : A A such that π(x = (π(x i i=, x = (x i i= X. It is well known (see, e.g. [25, Proposition.5.2] that each factor map π : X Y between two subshifts X and Y, will become a -block factor map if we enlarge the alphabet A and recode X through a so-called higher block representation of X. Whenever π : X Y is -block, we write πi = π(x... π(x n for I = x... x n L n (X; clearly πi L n (Y Two-sided subshifts over finite alphabets. For a finite alphabet A, let A Z = {x = (x i i Z : x i A for all i Z} denote the collection of all bi-infinite sequences of symbols from A. Similarly, A Z is a compact metric space endowed with the metric d(x, y = 2 inf{ k : x k y k }, x = (x i i Z, y = (y i i Z. The shift map σ : A Z A Z is defined by (σx i = x i+ for x = (x i i Z. The topological dynamical system (A Z, σ is called the two-sided full shift over A. If X A Z is compact and σ(x = X, the topological dynamical system (X, σ is called a two-sided subshift over A. The definitions of L(X, (weak specification and factor maps for two-sided subshifts can be given in a way similar to the one-sided case Some notation. For two families of real numbers {a i } i I and {b i } i I, we write a i b i a i b i if there is c > 0 such that b c i a i cb i for i I; if there is c > 0 such that a i cb i for i I; a i b i if there is c > 0 such that a i cb i for i I; a i = b i + O( if there is c > 0 such that a i b i c for i I; a i b i + O( if there is c > 0 such that a i b i c for i I; a i b i + O( if there is c > 0 such that a i b i c for i I. 8

9 3. Variational principles for sub-additive potentials In this section we present and prove some variational principles for certain subadditive potentials. This is the starting point in our work. First we give some notation and definitions. Let (X, σ X be a one-sided subshift over a finite alphabet A. We use M(X to denote the set of all Borel probability measures on X. Endow M(X with the weak-star topology. Let M(X, σ X denote the set of all σ X -invariant Borel probability measures on X. The sets M(X and M(X, σ X are non-empty, convex and compact (cf. [35]. Let L(X denote the language of X (cf. Sect 2. For convenience, for µ M(X and I L(X, we would like to write µ(i := µ([i] X, where [I] denotes the n-th cylinder in A N defined as in (2.. For µ M(X, σ X, the measure theoretic entropy of µ with respect to σ X defined as (3. h µ (σ X := lim µ(i log µ(i. n n The above limit exists since the sequence (a n n=, where a n = µ(i log µ(i, satisfies a n+m a n + a m for n, m N. It follows that (3.2 h µ (σ X = inf µ(i log µ(i. n N n The function µ h µ (σ X is affine and upper semi-continuous on M(X, σ X (cf. [35]. A sequence Φ = (log φ n n= of functions on a subshift X is called a sub-additive potential on X, if each φ n is a non-negative continuous function on X and there exists c > 0 such that (3.3 φ n+m (x cφ n (xφ m (σ n Xx, x X, n, m N. For convenience, we denote by C sa (X, σ X the collection of sub-additive potentials on X. For Φ = (log φ n n= C sa (X, σ X, define Φ : M(X, σ X R { } by (3.4 Φ (µ = lim log φ n (xdµ(x. n n The limit in (3.4 exists by the sub-additivity of log(cφ n dµ. 9 is

10 Remark 3.. One observes that for f C(X, if Φ = (log φ n n= is given by φ n (x = exp(s n f(x, then Φ C sa (X, σ X and Φ (µ = f dµ for each µ M(X, σ X. By the sub-additivity (3.3, we have the following simple lemma (cf. Proposition 3. in [3]. Lemma 3.2. (i Φ is affine and upper semi-continuous on M(X, σ X. (ii There is a constant C R such that log φ n (xdµ(x nφ (µ C for n N and µ M(X, σ X. Definition 3.3. For Φ C sa (X, σ X, µ M(X, σ X is called an equilibrium state of Φ if Φ (µ + h µ (σ X = sup{φ (η + h η (σ X : η M(X, σ X }. Let I(Φ denote the collection of all equilibrium states of Φ. Definition 3.4. A function φ : L(X [0, is said to be sub-multiplicative if, φ(ε = and there exists a constant c > 0 such that φ(ij cφ(iφ(j for any IJ L(X. Furthermore, say Φ = (log φ n n= C sa (X, σ X is generated by φ if φ n (x = φ(x... x n, x = (x i i= X. Proposition 3.5. Assume that Φ = (log φ n n= C sa (X, σ X is generated by a sub-multiplicative function φ : L(X [0,. Then (i sup{φ (µ + h µ (σ X : µ M(X, σ X } = lim n log u n n, where u n is given by u n = φ(i. (ii I(Φ is a non-empty compact convex subset of M(X, σ X. Furthermore each extreme point of I(Φ is an ergodic measure. We remark that Proposition 3.5(i is a special case of Theorem. in [9] on the variational principle for general sub-additive potentials. It was also obtained in [8] for the case that φ > 0. Proposition 3.5(ii actually holds for any Φ C sa (X, σ X, by the affinity and upper semi-continuity of Φ ( and h ( (σ X on M(X, σ X (see the proof of Proposition 3.7(ii for details. Now let (X, σ X and (Y, σ Y be one-sided subshifts over A, A, respectively. Assume that Y is a factor of X with a -block factor map π : X Y. Definition 3.6. For ν M(Y, σ Y, µ M(X, σ X is called a conditional equilibrium state of Φ with respect to ν if, µ π = ν and Φ (µ + h µ (σ X = sup{φ (η + h η (σ X : η M(X, σ X, η π = ν}. 0

11 Let I ν (Φ denote the collection of all conditional equilibrium states of Φ with respect to ν. The following result is a relativized version of Proposition 3.5. Proposition 3.7. Let Φ = (log φ n n= C sa (X, σ X be generated by a sub-multiplicative function φ : L(X [0,. Let ν M(Y, σ Y. Then (i sup{φ (µ+h µ (σ X h ν (σ Y : µ M(X, σ X, µ π = ν} = Ψ (ν, where Ψ = (log ψ n n= C sa (Y, σ Y is generated by a sub-multiplicative function ψ : L(Y [0,, which satisfies (3.5 ψ(j = φ(i, J L(Y. I L(X: πi=j (ii I ν (Φ is a non-empty compact convex subset of M(X, σ X. Furthermore, if ν is ergodic, then each extreme point of I ν (Φ is an ergodic measure on X. We remark that Proposition 3.5 can be obtained from Proposition 3.7 by considering the special case that Y is a singleton (correspondingly, A consists of one symbol. To prove Proposition 3.7, we need the following lemmas. Lemma 3.8 ([6], p. 34. Suppose 0 p,..., p m, s = p + + p m and a,..., a m 0. Then m p i (log a i log p i s log(a + + a m s log s. i= Lemma 3.9 ([9], Lemma 2.3. Let Φ = (log φ n n= C sa (X, σ X. Suppose (η n n= is a sequence in M(X. We form the new sequence (µ n n= by µ n = n n i=0 η n σ i X. Assume that µ ni converges to µ in M(X for some subsequence (n i of natural numbers. Then µ M(X, σ X, and moreover lim sup log φ ni (x dη ni (x Φ (µ. i n i Lemma 3.0 ([9], Lemma 2.4. Denote k = #A. Then for any ξ M(X, and positive integers n, l with n 2l, we have where ξ n = n n ξ(i log ξ(i l I L l (X n i=0 ξ σ i X. ξ n (I log ξ n (I 2l n log k,

12 Proof of Proposition 3.7. Fix ν M(Y, σ Y. For any µ M(X, σ X with µ π = ν, and n N, we have µ(i log φ(i µ(i log µ(i = J L n(y : πi=j J L n(y µ(i log φ(i µ(i log µ(i ν(j log ψ(j ν(j log ν(j (by Lemma 3.8. Dividing both sides by n and letting n, we obtain Φ (µ + h µ (σ X h ν (σ Y Ψ (ν. Thus to complete the proof of (i, it suffices to show that there exists µ with µ π = ν, such that Φ (µ+h µ (σ X h ν (σ Y Ψ (ν. For this purpose, construct a sequence (η n n= in M(X such that η n (I = ν(πiφ(i, I L n (X, ψ(πi where we take the convention 0 = 0. Clearly, η 0 n π (J = ν(j for all J L n (Y. Set µ n = n n i=0 η n σ i X. Assume that µ n i converges to µ in M(X for some subsequence (n i of natural numbers. By Lemma 3.9, µ M(X, σ X and (3.6 Φ (µ lim sup log φ ni (x dη ni (x = lim sup η ni (I log φ(i. i i n i n i I L ni (X We first show that µ π = ν. Let J L(Y. Denote l = J. For n > l and 0 i n l, we have η n σ i X π (J = η n π σ i Y (J = = J L i (Y, J 2 L n i l (Y : J JJ 2 L n(y J L i (Y, J 2 L n i l (Y : J JJ 2 L n(y η n π (J JJ 2 ν(j JJ 2 = ν(j. It follows that µ n π (J = n n i=0 η n σ i X π (J ν(j, as n. Therefore µ π (J = ν(j. Since J L(Y is arbitrary, we have µ π = ν. We next show that (3.7 Φ (µ + h µ (σ X h ν (σ Y Ψ (ν. Fix l N. By Lemma 3.0, we have for n 2l, η n (I log η n (I µ n (I log µ n (I 2l log k, n l n I L l (X 2

13 where k := #A. Since µ ni µ as i, we obtain lim inf η ni (I log η ni (I i l Taking l yields n i I L ni (X (3.8 lim inf i Observe that η n (I log φ(i = n i I L ni (X = I L l (X µ(i log µ(i. η ni (I log η ni (I h µ (σ X. η n (I log η n(iψ(πi ν(πi η n (I log η n (I + J L n(y ν(j(log ψ(j log ν(j. This together with (3.8 yields lim inf η ni (I log φ(i h µ (σ X + Ψ (ν + h ν (σ Y. i n i I L ni (X Applying (3.6, we have Φ (µ h µ (σ X + Ψ (ν + h ν (σ Y. This proves (3.7. Hence the proof of (i is complete. Now we show (ii. By the above proof, we see that I ν (Φ. The convexity of I ν (Φ follows directly from the affinity of Φ ( and h ( (σ X on M(X, σ X. Furthermore, the compactness of I ν (Φ follows from the upper semi-continuity of Φ ( and h ( (σ X on M(X, σ X. Next, assume that ν is ergodic and let µ be an extreme point of I ν (Φ. We are going to show that µ is ergodic. Assume it is not true, that is, there exist µ, µ 2 M(X, σ X with µ µ 2, and α, α 2 (0, with α + α 2 =, such that µ = 2 i= α iµ i. Then ν = µ π = 2 i= α iµ i π. Since µ i π M(Y, σ Y for i =, 2 and ν is ergodic, we have µ π = µ 2 π = ν. Note that 2 Ψ (ν = Φ (µ + h µ (σ X h ν (σ Y = α i (Φ (µ i + h µi (σ X h ν (σ Y and Φ (µ i + h µi (σ X h ν (σ Y Ψ (ν by (i. Hence we have i= Φ (µ i + h µi (σ X h ν (σ Y = Ψ (ν, i =, 2. That is, µ i I ν (Φ for i =, 2. However µ = 2 i= α iµ i. It contradicts the assumption that µ is an extreme point of I ν (Φ. This finishes the proof of the proposition. 3

14 Definition 3.. Let a = (a, a 2 R 2 with a > 0 and a 2 0. For Φ C sa (X, σ X, µ M(X, σ X is called an a-weighted equilibrium state of Φ for the factor map π, or simply, a-weighted equilibrium state of Φ, if (3.9 Φ (µ + a h µ (σ X + a 2 h µ π (σ Y = sup{φ (η + a h η (σ X + a 2 h η π (σ Y : η M(X, σ X }. We use I(Φ, a to denote the collection of all a-weighted equilibrium states of Φ. The value in the right hand side of (3.9 is called the a-weighted topological pressure of Φ and is denoted by P a (σ X, Φ. As a corollary of Propositions 3.5 and 3.7, we have Corollary 3.2. Let a = (a, a 2 R 2 with a > 0 and a 2 0. Let Φ = (log φ n n= C sa (X, σ X be generated by a sub-multiplicative function φ : L(X [0,. Define φ (2 : L(Y [0, by ( φ (2 (J = : πi=j φ(i a a for J Ln (Y, n N. Let Φ (2 = (log ψ n n= C sa (Y, σ Y be generated by φ (2. Then (i µ I(Φ, a if and only if µ π I( a +a 2 Φ (2 and µ I µ π ( a Φ, where a +a 2 Φ (2 := (log(ψ /(a +a 2 n=. n n= and a Φ := (log(φ /a n (ii Furthermore, I(Φ, a is a non-empty compact convex set, and each extreme point of I(Φ, a is ergodic. (iii I(Φ, a is a singleton if and only if I( a +a 2 Φ (2 is a singleton {ν} and, I ν ( a Φ contains a unique ergodic measure. Proof. Note that for µ M(X, σ X, Φ (µ + a h µ (σ X + a 2 h µ π (σ Y By Proposition 3.7, = Φ (µ + a (h µ (σ X h µ π (σ Y + (a + a 2 h µ π (σ Y. sup { Φ (η + a (h η (σ X h η π (σ Y : η M(X, σ X, η π = µ π } = Φ (2 (µ π. Hence µ I(Φ, a if and only if that Φ (µ + a (h µ (σ X h µ π (σ Y = Φ (2 (µ π and { Φ (2 (µ π + (a + a 2 h µ π (σ Y = sup Φ (2 (ν + (a + a 2 h ν (σ Y } ν M(Y,σ Y hold simultaneously. That is, µ I(Φ, a if and only if µ I µ π ( a Φ and µ π I( a +a 2 Φ (2. This proves (i. The proof of (ii is essentially identical to that of Proposition 3.7(ii. Part (iii follows from (i and (ii. 4

15 Remark 3.3. Proposition 3.7 was proved in [] in the special case that π : X Y is a one-block factor map between full shifts. Independently, Proposition 3.7 and Corollary 3.2 were set up in [38] for the special case that φ and X is an irreducible subshift of finite type, by a direct combination of [23, Theorem 2.] and [3, Corollary]. 4. Ergodic invariant measures associated with certain functions on A Let A be a finite alphabet and let A = n=0 An. We define two collections of functions over A. Definition 4.. Let p N. Define Ω w (A, p to be the collection of functions f : A [0, ] such that there exists 0 < c so that (H I A f(i = for any n 0. n (H2 For any I, J A, there exists K p i=0 Ai such that f(ikj cf(if(j. (H3 For each I A, there exist i, j A such that f(ii cf(i, f(ij cf(i. Definition 4.2. Let p N. Let Ω(A, p denote the collection of functions g : A [0, ] such that there exists 0 < c so that (A I A n g(i = for any n 0. (A2 For any I, J A, there exists K A p such that g(ikj cg(ig(j. For f Ω w (A, p Ω(A, p, define a map f : A [0, by (4. f (I = sup f m,n (I, I A, m,n 0 where f m,n (I := I A m I 2 A n f(i II 2. Clearly, f(i = f 0,0 (I f (I for any I A. The main result in this section is the following proposition, which plays a key role in our proof of Theorem.2. Proposition 4.3. Let f Ω w (A, p Ω(A, p and f be defined as in (4.. Let (η n n= be a sequence of Borel probability measures on A N satisfying η n (I = f(i, I A n. We form the new sequence (µ n n= by µ n = n n i=0 η n σ n. Assume that µ ni converges to µ for some subsequence (n i of natural numbers. Then µ M(A N, σ and it satisfies the following properties: (i There is a constant C > 0 such that C f (I µ(i f (I for all I A. 5

16 (ii There is a constant C 2 > 0 such that p lim inf µ(a σ n i (B C 2 µ(aµ(b n i=0 for any Borel sets A, B A N. (iii µ is ergodic. (iv µ is the unique ergodic measure on A N such that µ(i C 3 f(i for all I A and some constant C 3 > 0. (v n n i=0 η n σ n converges to µ in the weak-star topology, as n. Furthermore if f Ω(A, p, we have (vi There is a constant C 4 > 0 such that lim inf n for any Borel sets A, B A N. µ(a σ n (B C 4 µ(aµ(b To prove the above proposition, we need several lemmas. Lemma 4.4. Let f Ω w (A, p Ω(A, p. Then there is a constant C > 0, which depends on f, such that (i f m,n (I Cf m,n(i for any I A, m m + p and n n + p. (ii For each I A, there exists an integer N = N(I such that f m,n (I (C/2f (I, m, n N. Proof. To show (i, we first assume f Ω w (A, p. Let c be the constant associated with f in Definition 4.. Fix I A and m, n, m, n N {0} such that m m + p and n n + p. By (H2, for given I A m, I 2 A n, I 3 A m m p and I 4 A n n p, there exist K, K 2 p i=0 Ai so that f(i 3 K I II 2 K 2 I 4 c 2 f(i 3 f(i II 2 f(i 4. Furthermore by (H3, there exist K 3, K 4 p i=0 Ai so that K + K 3 = p, K 2 + K 4 = p and (4.2 f(k 3 I 3 K I II 2 K 2 I 4 K 4 c 2p f(i 3 K I II 2 K 2 I 4 c 2p+2 f(i 3 f(i II 2 f(i 4. Summing over I A m, I 2 A n, I 3 A m m p and I 4 A n n p, and using (H, we obtain f m,n (I M c2p+2 f m,n (I, where M denotes the number of different tuples (J, J 2, J 3, J 4 (A 4 with J + J 3 = p and J 2 + J 4 = p. 6

17 Now assume f Ω(A, p. Instead of (4.2, by (A2, we can find K, K 2 A p such that Summing over I, I 2, I 3, I 4 yields f(i 3 K I II 2 K 2 I 4 c 2 f(i 3 f(i II 2 f(i 4. f m,n (I c2 f m,n (I. This proves (i by taking C = min{c 2, M c2p+2 } = M c2p+2. To show (ii, note that f (I = sup m,n 0 f m,n (I. Hence we can pick m 0, n 0 such that f m0,n 0 (I f (I/2. Let N = m 0 + n 0 + p. Then by (i, for any m, n N, we have This finishes the proof of the lemma. f m,n (I Cf m0,n 0 (I C 2 f (I. Lemma 4.5. Let f Ω w (A, p Ω(A, p. Then there exists a constant C > 0 such that for any I, J A, there exists an integer N = N(I, J such that p f (IKJ C f (If (J, n N. i=0 K A n+i In particular, if f Ω(A, p, then the above inequality can be strengthened as K A n f (IKJ C f (If (J, n N. Proof. First assume f Ω w (A, p. Let C be the constant associated with f in Lemma 4.4. Fix I, J A. By Lemma 4.4(ii, there exists k N such that for m, m 2, m 3, m 4 k, f m,m 2 (I C 2 f (I, f m3,m 4 (J C 2 f (J. Take N = 2k. Let n N. Then we have f k,n k (I C 2 f (I, f k,k (J C 2 f (J. By (H2, for any I A k, I 2 A n k, J, J 2 A k, we have p (4.3 f(i II 2 UJ JJ 2 cf(i II 2 f(j JJ 2. i=0 U A i Summing over I, I 2, J, J 2 yields p f k,k (IKJ cf k,n k (If k,k (J. i=0 K A n+i 7

18 Hence, we have p i=0 K A n+i f (IKJ cf k,n k (If k,k (J c(c/2 2 f (If (J. Next assume f Ω(A, p. By (A2, instead of (4.3, we have U A p f(i II 2 UJ JJ 2 cf(i II 2 f(j JJ 2 for any I A k, I 2 A n k, J, J 2 A k. Summing over I, I 2, J, J 2 we obtain K A n+p f k,k (IKJ cf k,n k (If k,k (J c(c/2 2 f (If (J. Hence K A f (IKJ c(c/2 2 f (If (J. n+p lemma. This finishes the proof of the Proof of Proposition 4.3. By [35, Theorem 6.9], µ is σ-invariant. Fix I A. Let m = I. For n > m, we have ( µ n (I = n m n η n σ i (I + η n σ j (I n = n i=0 ( n m i=0 f i,n m i (I + j=n m+ n j=n m+ Applying Lemma 4.4(ii to the above equality yields C 2 f (I lim inf n η n σ j (I µ n(i lim sup µ n (I f (I, n where C > 0 is a constant independent of I. Hence This proves (i by taking C = C/2. (C/2f (I µ(i f (I. By (i and Lemma 4.5, we have p lim inf µ([i] σ n i ([J] C lim inf (4.4 n n i=0 C C f (If (J C C µ(iµ(j p i=0 K A n+i f (IKJ for some constant C > 0 and all I, J A. Take C 2 = C C. Since {[I] : I A } generates the Borel σ-algebra of A N, (ii follows from (4.4 by a standard argument. As a consequence of (ii, for any Borel sets A, B A N with µ(a > 0 and µ(b > 0, there exists n such that µ(a σ n (B > 0. This implies that µ is ergodic (cf. [35, Theorem.5]. This proves (iii. 8.

19 To prove (iv, assume that η is an ergodic measure on A N so that there exists C 3 > 0 such that η(i C 3 f(i, I A. Then for any I A and m, n N, η(i = I A m I 2 A n η(i II 2 C 3 I A m I 2 A n f(i II 2 = C 3 f m,n (I. Hence η(i C 3 f (I C 3 µ(i. It implies that µ is absolutely continuous with respect to η. Since any two different ergodic measures on A N are singular to each other (cf. [35, Theorem 6.0(iv], we have η = µ. This proves (iv. Notice that (v follows directly from (i, (iii and (iv. Now assume that f Ω(A, p. Instead of (4.4, by (i and Lemma 4.5 we have lim inf µ([i] n σ n ([J] C lim inf f (IKJ n K A n C C f (If (J C C µ(iµ(j = C 2 µ(iµ(j, from which (vi follows. This finishes the proof of Proposition Equilibrium states for certain sub-additive potentials In this section, we show the uniqueness of equilibrium states for certain subadditive potentials on one-sided subshifts. Let (X, σ X be a subshift over a finite alphabet A. Let p N. We use D w (X, p to denote the collection of functions φ : L(X [0, such that φ(w > 0 for at least one W L(X\{ε}, and there exists 0 < c so that ( φ(ij c φ(iφ(j for any IJ L(X. (2 For any I, J L(X, there exists K p i=0 L i(x such that IKJ L(X and φ(ikj cφ(iφ(j. Furthermore, we use D(X, p to denote the collection of functions φ : L(X [0, such that φ(w > 0 for at least one W L(X\{ε}, and there exists 0 < c so that φ satisfies the above condition (, and (2 For any I, J L(X, there exists K L p (X such that IKJ L(X and φ(ikj cφ(iφ(j. Remark 5.. (i D(X, p D w (X, p. (ii D w (X, p if and only if X satisfies weak p-specification. The necessity is obvious. For the sufficiency, if X satisfies weak p-specification, then the constant function φ on L(X is an element in D w (X, p. Similarly, D(X, p if and only if X satisfies p-specification. 9

20 Lemma 5.2. Suppose φ D w (X, p. Then the following two properties hold: (i There exists a constant γ > 0 such that for each I L(X, there exist i, j A such that φ(ii γφ(i and φ(ij γφ(i. (ii Let u n = J L n(x φ(j. Then the limit u = lim n (/n log u n exists and u n exp(nu. Proof. Let φ D w (X, p with the corresponding constant c (0, ]. For (i, we only prove there exists a constant γ > 0 such that for each I L(X, there exists j A such that φ(ij γφ(i. The other statement (there exists i A so that φ(ii γφ(i follows by an identical argument. Fix a word W L(X\{ε} such that φ(w > 0. Let I L(X so that φ(i > 0. Then there exists K p i=0 L i(x such that φ(ikw cφ(iφ(w. Write KW = ju, where j is the first letter in the word KW. Then φ(ijφ(u cφ(iju = cφ(ikw c 2 φ(iφ(w. Hence φ(u > 0 and φ(ij c 2 φ(iφ(w /φ(u. Since there are only finite possible U (for U W + p, φ(ij/φ(i γ for some constat γ > 0. To see (ii, we have (5. u n+m =, J L m(x: IJ L n+m (X, J L m(x φ(ij c φ(iφ(j = c u n u m and (5.2 p u n+m+k = k=0, J L m(x, J L m(x K p i=0 L i(x: IKJ L(X cφ(iφ(j = cu n u m. φ(ikj On the other hand, u n+ = φ(ij γ φ(i = γu n, j A: Ij L n+ (X and u n+ c u u n by (5.. Hence u n+ u n. This together with (5. and (5.2 yields u n+m u n u m, from which (ii follows. Note that we have introduced Ω w (A, p and Ω(A, p in Sect. 4. consequence of Lemma 5.2, we have 20 As a direct

21 Lemma 5.3. Let φ D w (X, p. Define f : A [0, ] by φ(i (5.3 f(i = J L φ(j if I L n (X, n 0, n(x 0 if I A \L(X, Then f Ω w (A, p, and f(ij f(if(j for I, J A. Moreover if φ D(X, p, then f Ω(A, p. Lemma 5.4. Let η, µ M(X, σ X. Assume that η is not absolutely continuous with respect to µ. Then lim η(i log µ(i η(i log η(i =. n Proof. We take a slight modification of the proof of Theorem.22 in [6]. Since η is not absolutely continuous with respect to µ, there exists c (0, such that for any 0 < ɛ < c/2, there exists a Borel set A X so that η(a > c and µ(a < ɛ. Applying [6, Lemma.23], we see that for each sufficiently large n, there exists F n L n (X so that µ(a A n + η(a A n < ɛ with A n := [I] X, I F n which implies η(a n > c ɛ > c/2 and µ(a n < 2ɛ. Using Lemma 3.8, we obtain η(i log µ(i η(i log η(i η(a n log µ(a n + sup s log(/s 0 s (5.4 I F n (c/2 log(2ɛ + log 2 and (5.5 \F n η(i log µ(i η(i log η(i η(x\a n log µ(x\a n + sup s log(/s log 2. 0 s Combining (5.4 and (5.5 yields η(i log µ(i η(i log η(i (c/2 log(2ɛ + 2 log 2, from which the lemma follows. The main result in this section is the following 2

22 Theorem 5.5. Let φ D w (X, p. Let Φ = (log φ n n= C sa (X, σ X be generated by φ, i.e. φ n (x = φ(x... x n for x = (x i i= X. Then Φ has a unique equilibrium state µ. The measure µ is ergodic and has the following Gibbs property (5.6 µ(i φ(i J L n(x φ(j exp( np φ(i, I L n(x, n N, where P = lim n log n J L n(x φ(j. Furthermore, we have the following estimates: µ(i log φ(i = nφ (µ + O(, µ(i log µ(i = nh µ (σ X + O(. Proof. Define f : A [0, ] as in (5.3. By Lemma 5.3, f Ω w (A, p and f satisfies f(ij f(if(j for I, J A. Let f : A [0, be defined as f (I = sup f(i II 2, I A. n,m 0 I A n I 2 A m Since f(ij f(if(j for I, J A, we have f (I f(i. Hence by Proposition 4.3, there exists an ergodic measure µ on A N such that µ(i f(i, I A. Since f(i = 0 for I A \L(X, µ is supported on X. By Lemma 5.2(ii, φ(i exp(np, hence we have µ(i f(i exp( np φ(i, I L n (X, n N. Let η be an ergodic equilibrium state of Φ. By Proposition 3.5(i, Φ (η+h η (σ X = P. By Lemma 3.2 and (3.2, we have (5.7 η(i log φ(i nφ (η + O(, η(i log η(i nh η (σ X + O(. Thus we have (5.8 O( ( η(i log φ(i η(i log η(i np ( η(i log µ(i η(i log η(i + O(. That is, η(i log µ(i η(i log η(i O(. By Lemma 5.4, η is absolutely continuous with respect to µ. Since both µ and η are ergodic, we have η = µ (cf. [35, Theorem 6.0(iv]. This implies that µ is the unique ergodic equilibrium state of Φ. By Proposition 3.5(ii, µ is the unique equilibrium state of Φ. 22

23 Since η = µ, by (5.8, we have ( η(i log φ(i η(i log η(i nφ (η nh η (σ X = This together with (5.7 yields the estimates: η(i log φ(i = nφ (η + O(, This completes the proof of Theorem 5.5. ( η(i log φ(i η(i log η(i np O(. η(i log η(i = nh η (σ X + O(. Remark 5.6. The introduction of D w (X, p and D(X, p was inspired by the work [4]. Indeed, Theorem 5.5 was first setup in [4] for a class of φ D w (X, p, where X is an irreducible subshift of finite type and, φ is given by the norm of products of non-negative matrices satisfying an irreducibility condition (see [4, Theorem 3.2], [, Theorem 3.]. Although the approach in [4] can be adapted to prove (5.6 under our general settings, we like to provide the above short proof using Proposition 4.3. Independently, Theorem 5.5 was set up in [38] in the special case that X is a mixing subshift of finite type, and φ a certain element in D(X, p, through an approach similar to [4]. In the end of this section, we give the following easy-checked, but important fact. Lemma 5.7. Let (X, σ X, (Y, σ Y be one-sided subshifts over finite alphabets A, A, respectively. Assume that Y is a factor of X with a one-block factor map π : X Y. Let p N and a > 0. For φ D w (X, p, define φ a : L(X [0, and ψ : L(Y [0, by φ a (I = φ(i a for I L(X, ψ(j = φ(i for J L(Y. I L(X: πi=j Then φ a D w (X, p and ψ D w (Y, p. Furthermore if φ D(X, p, then φ a D(X, p and ψ D(Y, p. Proof. Let φ D w (X, p with the corresponding constant c (0; ]. Clearly φ a D w (X, p. Here we show ψ D w (Y, p. Observe that for J J 2 L(Y, ψ(j J 2 = φ(i I 2 I I 2 L(X: πi =J, πi 2 =J 2 c φ(i φ(i 2 I I 2 L(X: πi =J, πi 2 =J 2 c φ(i φ(i 2 = c ψ(j ψ(j 2. I 2 L(X: πi 2 =J 2 I L(X: πi =J 23

24 Furthermore for any J, J 2 L(Y, ψ(j W J 2 W p i=0 L i(y : J W J 2 L(Y = I L(X: πi =J I 2 L(X: πi 2 =J 2 I L(X: πi =J K p i=0 L i(x: I KI 2 L(X φ(i KI 2 I 2 L(X: πi 2 =J 2 cφ(i φ(i 2 = cψ(j ψ(j 2. Therefore there exists W p i=0 L i(y, such that J W J 2 L(Y, and ψ(j W J 2 c ψ(j L ψ(j 2, where L denotes the cardinality of p i=0 L i(y. Hence ψ D w (Y, p. A similar argument shows that ψ D(Y, p whenever φ D(X, p. 6. Uniqueness of weighted equilibrium states: k = 2 Assume that (X, σ X is a one-sided subshift over a finite alphabet A. Let (Y, σ Y be a one-sided subshift factor of X with a one-block factor map π : X Y. Let a = (a, a 2 R 2 so that a > 0 and a 2 0. Assume that D w (X, p for some p N, equivalently, X satisfies weak p-specification. Let φ D w (X, p. Define φ (2 : L(Y [0, by ( (6. φ (2 (J = : πi=j φ(i a a for J Ln (Y, n N. Furthermore, define φ (3 : N [0, by (6.2 φ (3 (n = φ (2 a (J +a 2, n N. J L n(y The main result of this section is the following. Theorem 6.. Let φ D w (X, p. Let Φ = (log φ n n= C sa (X, σ X be generated by by φ, i.e. φ n (x = φ(x x n for x = (x i i= X. Then Φ has a unique a-weighted equilibrium state µ. Furthermore, µ is ergodic and has the following properties: (i µ(i φ (I φ(i for I L(X, where φ, φ : L(X [0, are defined by (6.3 φ(i = φ(i a and φ (I = sup φ (2 (πi a φ(2 (πi φ (3 (n a +a 2 m,n 0 I L m(x, I 2 L n(x: I II 2 L(X where in (6.3 we take the convention 0/0 = 0. 24, I L n (X, n N φ(i II 2, I L(X,

25 (ii lim inf p n i=0 µ ( A σ n i X (B µ(aµ(b for Borel sets A, B X. (iii We have the estimates: µ(i log µ(i = µ(i log φ(i + O( = nh µ (σ X + O(, µ(i log φ(i = nφ (µ + O(. Moreover, if φ D(X, p, then instead of (ii we have (iv lim inf n µ ( A σ n X (B µ(aµ(b for Borel sets A, B X. Proof. By (6.3, we have where θ(i is given by φ(i = φ(i a θ(i, I L n(x, n N, θ(i = φ (3 (nφ (2 (πi a a +a 2, I L n (X, n N. We claim that φ and θ satisfy the following properties: (a φ(i = for each n N. (b For any I L(X, if φ(i > 0 then θ(i > 0. (c θ(i I 2 θ(i θ(i 2 for I I 2 L(X. Property (a follows immediately from the definition of φ. To see (b, one observes that if φ(i > 0 for some I L n (X, then so are φ (2 (πi and φ (3 (n, hence θ(i > 0. To see (c, by Lemma 5.7, φ (2 D w (Y, p and thus φ (2 (π(i I 2 φ (2 (πi φ (2 (πi 2, I I 2 L(X. Furthermore by Lemma 5.2, φ (3 (n + m φ (3 (nφ (3 (m. Hence (c follows. Extend φ, φ : A [0, by setting φ(i = φ (I = 0 for I A \L(X. By (a, (b, (c and Lemma 5.2(i, we see that φ Ω w (A, p. Hence by Proposition 4.3, there exists an ergodic measure µ M(A N, σ such that (6.4 µ(i φ (I φ(i, I A N. Moreover, µ satisfies p (6.5 lim inf µ ( A σ n i (B µ(aµ(b for Borel sets A, B A N. n i=0 By (6.4, µ is supported on X and µ M(X, σ X. Let Φ (2 = (log φ (2 n n= C sa (Y, σ Y be generated by φ (2, i.e. φ (2 n (y = φ (2 (y y n for y = (y i i= Y. 25

26 Define ψ : L(Y [0, by a +a 2 ψ(j = φ(2 (J φ (3 (n, J L n (Y, n N. By the definitions of φ and ψ, we have (6.6 φ(i = φ(i a φ (2 (πi a ψ(πi, I L(X. Since φ (2 D w (Y, p, by Lemma 5.7, (φ (2 /(a +a 2 D w (Y, p. Hence by Theorem 5.5, a +a 2 Φ (2 has a unique equilibrium state ν M(Y, σ Y and ν satisfies the properties (6.7 ν(j log ν(j = ν(j log ψ(j + O( = nh ν (σ Y + O(, J L n(y J L n(y and (6.8 J L n(y ν(j log φ (2 (J = nφ (2 (ν + O(. Assume that η is an ergodic a-equilibrium state of Φ. By Corollary 3.2(i, η π = ν and η is a conditional equilibrium state of a Φ with respect to ν, that is, (6.9 Φ (η + h η (σ X h ν (σ Y = Φ (2 (ν. a a By (6.7 and (6.8, we have (6.0 nh ν (σ Y + n a Φ (2 (ν = J L n(y ν(j log ψ(j φ (2 (J a + O(. By Lemma 3.2(ii and (3.2, we have (6. η(i log φ(i nφ (η + O(, η(i log η(i nh η (σ X. 26

27 Combining (6.9, (6.0 and (6., we obtain O( (η(i log(φ(i a η(i log η(i n Φ (η nh η (σ X a (6.2 = = = = (η(i log(φ(i a η(i log η(i (η(i log(φ(i a η(i log η(i + J L n(y ν(j log (η(i log φ(i a ψ(πi φ (2 (πi a ( η(i log φ(i η(i log η(i ψ(j φ (2 (J a nh ν (σ X2 n Φ (2 (ν a + O( η(i log η(i + O( + O( (by (6.6. That is, (6.3 ( η(i log φ(i η(i log η(i O(. Combining (6.3 and (6.4 yields ( η(i log µ(i η(i log η(i (6.4 ( η(i log φ(i η(i log η(i + O( O(. By (6.4 and Lemma 5.4, η is absolutely continuous with respect to µ. Since both µ and η are ergodic, we have η = µ (cf. [35, Theorem 6.0(iv]. This implies that µ is the unique ergodic a-weighted equilibrium state of Φ. By Corollary 3.2(iii, µ is the unique a-weighted equilibrium state of Φ. Now parts (i, (ii of the theorem follow from (6.4-(6.5. To show (iii, due to η = µ, the left hand side of (6.4 equals 0. Hence by (6.4, ( (6.5 η(i log φ(i η(i log η(i = O(. Combining (6.5 and (6.2 yields (6.6 (η(i log(φ(i a η(i log η(i n Φ (η nh η (σ X = O(. a 27

28 However (6.6 and (6. imply (6.7 η(i log φ(i = nφ (η + O(, η(i log η(i = nh η (σ X + O(. Now part (iii follows from (6.7 and (6.5. To see (iv, note that whenever φ D(X, p, we have φ Ω(A, p, following from (a-(c. Now (iv follows from Proposition 4.3(vi. This finishes the proof of the theorem. 7. Uniqueness of weighted equilibrium states: k 2 Let k 2 be an integer. Assume that (X i, σ Xi (i =,..., k are one-sided subshifts over finite alphabets so that X i+ is a factor of X i with a one-block factor map π i : X i X i+ for i =,..., k. For convenience, we use π 0 to denote the identity map on X. Define τ i : X X i+ by τ i = π i π i π 0 for i = 0,,..., k. Let a = (a,..., a k R k so that a > 0 and a i 0 for i >. Let φ D w (X, p. Set φ ( = φ and define φ (i : L(X i [0, (i = 2,..., k recursively by ( φ (i (J = φ (i a + +a i a (I + +a i I L n(x i : π i I=J for n N, J L n (X i. Furthermore, define φ (k+ : N [0, by φ (k+ (n = φ (k a (I + +a k. I L n(x k Definition 7.. Let Φ = (log φ n n= C sa (X, σ X be generated by φ. Say that µ M(X, σ X is an a-weighted equilibrium state of Φ if Φ (µ + h a µ(σ X = sup { Φ (η + h a η(σ X : η M(X, σ X }, where h a µ(σ X := k i= a ih µ τ i (σ X i. Let I(Φ, a be the collection of all a-weighted equilibrium states of Φ. Let Φ (2 C sa (X 2, σ X2 be generated by φ (2. By a proof essentially identical to that of Corollary 3.2, we have Lemma 7.2. (i I(Φ, a is a non-empty compact convex subset of M(X, σ X. Each extreme point of I(Φ, a is ergodic. (ii µ I(Φ, a if and only if µ I µ π ( a Φ together with µ π I(Φ (2, b, where b = (a + a 2, a 3,..., a k R k. (iii I(Φ, a is a singleton if and only if I(Φ (2, b is a singleton {ν} and, I ν ( a Φ contains a unique ergodic measure. 28

29 As the high dimensional version of Theorem 6., we have Theorem 7.3. Let φ D w (X, p. Let Φ = (log φ n n= C sa (X, σ X be generated by φ. Then Φ has a unique a-weighted equilibrium state µ. Furthermore, µ is ergodic and has the following properties: (i µ(i φ (I φ(i for I L(X, where φ, φ : L(X [0, are defined respectively by ( k φ (7. φ(i (i a (τ i I + +a i = φ (i+ φ(k a (τ k I + +a k a (τ i I + +a i φ (k+ (n i= for I L n (X, n N, and φ (I = sup m,n 0 I L m(x, I 2 L n(x : I II 2 L(X φ(i II 2, I L(X. (ii lim inf p n i=0 µ ( A σ n i X (B µ(aµ(b for Borel sets A, B X. (iii We have the estimates: µ(i log µ(i = µ(i log φ(i + O( = nh µ (σ X + O(, I L n(x I L n(x I L n(x µ(i log φ(i = nφ (µ + O(. Moreover, if φ D(X, p, then instead of (ii we have (iv lim inf n µ ( A σ n X (B µ(aµ(b for Borel sets A, B X. Proof. We prove the theorem by induction on the dimension k. By Theorem 6., Theorem 7.3 is true when the dimension equals 2. Now assume that the theorem is true when the dimension equals k. In the following we prove that the theorem is also true when the dimension equals k. By (7., we have where θ(i is given by θ(i = φ (k+ (n φ(i = φ(i a θ(i, I L n(x, n N, k φ (i (τ i I i=2 a + +a i a + +a i, I Ln (X, n N. By Lemma 5.7 and Lemma 5.2(ii, we have φ (i D w (X i, p for i = 2,..., k, and φ (k+ (n + m φ (k+ (nφ (k+ (m. Similar to the proof of Theorem 6., we can show that φ and θ satisfy the following properties: (a I L n(x φ(i = for each n N. 29

30 (b For any I L(X, if φ(i > 0 then θ(i > 0. (c θ(i I 2 θ(i θ(i 2 for I I 2 L(X. Extend φ, φ : A [0, by setting φ(i = φ (I = 0 for I A \L(X. By (a, (b, (c and Lemma 5.2(i, we see that φ Ω w (A, p. Hence by Proposition 4.3, there exists an ergodic measure µ M(A N, σ such that (7.2 µ(i φ (I φ(i, I A N. Moreover, µ satisfies p lim inf µ ( A σ n i (B µ(aµ(b for Borel sets A, B A N. n i=0 By (7.2, µ is supported on X and µ M(X, σ X. Let Φ (2 = (log φ (2 n n= C sa (X 2, σ X2 be generated by φ (2, i.e. φ (2 n (x = φ (2 (x x n for x = (x i i= X 2. Let b = (a + a 2, a 3,..., a k R k. Define ψ : L(X 2 [0, by ( k φ ψ(j (i a (ξ i J + +a i = φ (i+ φ(k a (ξ k J + +a k, J L(X a (ξ i J + +a i φ (k+ 2, n N, (n i=2 where ξ := Id, and ξ i = π i π 2 for i 2. By the definitions of φ and ψ, we have (7.3 φ(i = φ ( (I a φ (2 (π I a ψ(π I, I L(X. Since φ (2 D w (X 2, p, by the assumption of the induction, Φ (2 has a unique b-weighted equilibrium state ν M(X 2, σ X2 and ν satisfies the properties (7.4 ν(j log ν(j = ν(j log ψ(j + O( = nh ν (σ X2 + O(, and (7.5 J L n(x 2 J L n(x 2 J L n(x 2 ν(j log φ (2 (J = nφ (2 (ν + O(. Assume that η is an ergodic a-equilibrium state of Φ. By Lemma 7.2, η π = ν and η is a conditional equilibrium state of a Φ with respect to ν, that is, (7.6 Φ (η + h η (σ X h ν (σ X2 = Φ (2 (ν. a a Using (7.2, (7.3, (7.4-(7.6, and taking a process the same as in the proof of Theorem 6., we prove Theorem 7.3 when the dimension equals k. 30

31 Remark 7.4. Let φ be defined as in (7., and let (η n be a sequence in M(X so that η n (I = φ(i for each I L n (X. Then by Proposition 4.3(v and the above proof, the measure µ in Theorem 7.3 satisfies n µ = lim η n σ i n X n. i=0 Proof of Theorem.2. We first consider the case that X i (i =,..., k are onesided subshifts. Recoding X k, X k,..., X recursively through their higher block representations (cf. Proposition.5.2 in [25], if necessary, we may assume that π i : X i X i+ (i =,..., k are all one-block factor maps. Recall that X satisfies weak specification. (Notice that this property is preserved by recoding via higher block representations. Let f V (σ X (see (.3 for the definition. Define φ : L(X [0, by φ(i = sup exp(s n f(x, I L n (X, n N, x X [I] where S n f is defined as in (.2. Since f V (σ X, it is direct to check that φ D w (X, p, where p is any integer so that X satisfies weak p-specification. Let Φ = (log φ n n= C sa (X, σ X be generated by φ. Again by f V (σ X, we have Φ (µ = f dµ for any µ M(X, σ X. It follows that µ is an a-weighted equilibrium state of f if and only if that, µ is an a-weighted equilibrium state of Φ. Now the theorem follows from Theorem 7.3. Next we consider the case that X i s are two-sided subshifts over finite alphabets A i s. Again we may assume that π i s are one-block factor maps. Define for i =,..., k, X + i := { (x j j= A N i : (y j j Z X i such that x j = y j for j }. Then (X + i, σ X + becomes a one-sided subshift for each i. Furthermore define Γ i : i X i X + i by (x j j Z (x j j N. Then for each i k, the mapping µ µ Γ i is a homeomorphism from M(X i, σ Xi to M(X + i, σ X + which preserves the measure i theoretic entropy. Now π i : X + i X + i+ becomes a one-block factor between onesided subshifts for i =,..., k. Let f V (σ X. Define φ : L(X + [0, by φ(i = sup x X : x...x n=i exp(s n f(x, I L n (X +, n N. Similarly, φ D w (X, p for some p N. Let Φ = (log φ n n= C sa (X +, σ X + be generated by φ. Due to f V (σ X, we have f dµ = Φ (µ Γ, µ M(X, σ X. 3