THE THERMODYNAMIC FORMALISM FOR SUB-ADDITIVE POTENTIALS. Yong-Luo Cao. De-Jun Feng. Wen Huang. (Communicated by )

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1 Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X Website: pp. X XX THE THERMODYNAMIC FORMALISM FOR SUB-ADDITIVE POTENTIALS Yong-Luo Cao Department of Mathematics, Suzhou University Suzhou, 25006, Jiang Su, P. R. China De-Jun Feng Department of Mathematics, The Chinese University of Hong Kong Shatin, Hong Kong and Department of Mathematical Sciences, Tsinghua University Beijing, 00084, P. R. China Wen Huang Department of Mathematics, University of Science and Technology of China Hefei, , Anhui, P. R. China (Communicated by Abstract. The topological pressure is defined for sub-additive potentials via separated sets and open covers in general compact dynamical systems. A variational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the topological pressure are discussed. The result will have some potential applications in the multifractal analysis of iterated function systems with overlaps, the distribution of Lyapunov exponents and the dimension theory in dynamical systems.. Introduction and main results. The well-known notion of topological pressure for additive potentials was first introduced by Ruelle [25] in 973 for expansive maps acting on compact metric spaces. In the same paper he formulated a variational principle for the topological pressure. Later Walters [28] generalized these results to general continuous maps on compact metric spaces. The variational principle formulated by Walters can be stated precisely as follows: Let T : X X be a continuous transformation on a compact metric space (X, d and φ : X R an arbitrary continuous function. Let P (φ, T denote the topological pressure of φ (see [29]. Then { P (φ, T = sup h µ (T + } φ dµ : µ M(X, T, ( where M(X, T denotes the space of all T -invariant Borel probability measures on X endowed with the weak* topology, and h µ (T denotes the measure-theoretical entropy of µ. A T -invariant Borel probability measure µ such that P (φ, T = h µ (T + φ dµ, if it exists, is called an equilibrium state for φ. The theory about 2000 Mathematics Subject Classification. Primary: 37D35; Secondary: 34D20. Key words and phrases. Thermodynamical formalism, Variational principle, Topological pressures, Lyapunov exponents, Entropies, Products of matrices.

2 2 CAO, FENG AND HUANG the topological pressure, variational principle and equilibrium states plays a fundamental role in statistical mechanics, ergodic theory and dynamical systems (see, e.g., [4, 8, 26, 29]. Since the works of Bowen [5] and Ruelle [27], the topological pressure has also become a basic tool in the dimension theory in dynamical systems (see, e.g., [23, 30]. In 984, Pesin and Pitskel [24] defined the topological pressure of additive potentials for non-compact subsets of compact metric spaces and proved the variational principle under some supplementary conditions. Their work extended Bowen s results in [3] on topological entropy for non-compact sets. In this paper, we generalize Ruelle and Walters s results to sub-additive potentials in general compact dynamical systems. We define the topological pressure for subadditive potentials F = {log f n } n= and set up a variational principle between the topological pressure, measure-theoretical entropies and Lyapunov exponents. It is now log f n which plays the role of the classical potential. More precise, log f n plays the role of φ + φ T φ T n. To formulate our results, let T : X X be a continuous transformation on a compact metric space (X, d. A sequence F = {log f n } n= of functions on X is called sub-additive if each f n is a continuous non-negative valued function on X such that 0 f n+m (x f n (xf m (T n x, x X, m, n N. (2 We first define the topological pressure of F with respect to T. As usual for any n N and ɛ > 0, a set E X is said to be an (n, ɛ-separated subset of X with respect to T if max 0 i n d(t i x, T i y > ɛ for any two different points x, y E. Define { } P n (T, F, ɛ = sup f n (x : E is an (n, ɛ-separated subset of X. x E It is clear that P n (T, F, ɛ is a decreasing function of ɛ. Define P (T, F, ɛ = lim sup n n log P n(t, F, ɛ. P (T, F, ɛ is also a decreasing function of ɛ. Set P (T, F = lim ɛ 0 P (T, F, ɛ and we call it the topological pressure of F with respect to T. By the definition, P (T, F takes a value in [, +. For a T -invariant Borel probability measure µ, denote F (µ = lim n n log f n dµ. The existence of the above limit follows from a sub-additive argument. We call F (µ the Lyapunov exponent of F with respect to µ. It also takes a value in [, +. As a main result, we obtain the following variational principle. Theorem.. Let F = {log f n } n= be a sub-additive potential on a compact dynamical system (X, T. Then, if F (µ = for all µ M(X, T, P (T, F = sup {h µ (T + F (µ : µ M(X, T, F (µ }, otherwise. In the above theorem we adopt the rigorous expression just in order to avoid the situation that h µ (T = + and F (µ = take place simultaneously. If there is no such confusion, for instance the topological entropy of T is assumed to be finite,

3 SUB-ADDITIVE POTENTIALS 3 then we can state the variational principle simply as: P (T, F = sup{h µ (T +F (µ : µ M(X, T }. Our original motivation for studying the above issue comes from the study of multifractal formalism for self-similar measures with overlaps (the reader is referred to [, 23] for some basic information about self-similar measures and multifractals. It is known that in many interesting cases a self-similar measure with overlaps can be locally represented as infinite products of a family of matrices (see, e.g., [2, 5, 9, 20]. Thus to understand the corresponding multifractal formalism for these measures one needs to study the distribution of Lyapunov exponents for products of matrices. A similar task also arises in the study of regularity of the solutions of refinement equations in wavelets (see [6, 7, 8]. For the above purpose, a thermodynamic formalism (pressure functions, Gibbs measures and the distribution of Lyapunov exponents for non-negative matrix-valued potentials was developed in [3, 6]. Moreover, a variational principle for products of non-negative matrices in symbolic spaces was set up by Feng in [4]. Namely, let M be a continuous nonnegative matrix-valued function defined on a mixing sub-shift space and denote F = {log f n } where f n (x = n M(T i x, and denotes the norm of matrices, then Theorem. was proved in [4] under this special setting. However due to Theorem., we have the following more general result about products of matrices: Corollary.2. Let M be a continuous function defined on (X, T taking values in the set of all d d (real or complex matrices. Let F = {log f n } where f n (x = n M(T i x, and denotes the norm of matrices, then the result of Theorem. holds. We point out that Falconer and Barreira had some earlier contributions in the study of thermodynamic formalism for sub-additive potentials. In [9], Falconer considered the thermodynamic formalism for sub-additive potentials on mixing repellers. He proved the variational principle for the topological pressure under some Lipschitz conditions and bounded distortion assumptions on the sub-additive potentials. More precise, he assumed that there exist constants M, a, b > 0 such that n log f n(x M, n log f n(x log f n (y a x y, x, y X, n N and log f n (x log f n (y b whenever x, y belong to the same n-cylinder of the mixing repeller X. In 996, Barreira [] extended the work of Pesin and Pitskel [24]. He defined the topological pressure for an arbitrary sequence of continuous functions on an arbitrary subset of compact metric space and proved the variational principle under a strong convergence assumption on the potentials. Corresponding to our setting, Barreira assumed that there exists a continuous function ψ : X R such that log f n+ log(f n T converges to ψ uniformly on X. We remark that the assumptions given by Falconer and Barreira are usually not satisfied by general sub-additive potentials, for instance, the one generated by the product of general matrices in Corollary.2. The major virtue of our result is that we don t need any additional assumptions on the compact continuous dynamical system (X, T nor any regularity assumption on F. Our definition of the topological pressure for sub-additive potentials in this section adopts an approach via separated sets, which is similar to the classical additive case (see Walters book [29]. It is proved to be equivalent to Falconer s definition in the mixing repeller case. Nevertheless our definition is different from

4 4 CAO, FENG AND HUANG that of Barreira through open covers. We don t know if in general these two definitions are equivalent. However, we do prove that they are equivalent in some situations (for instance, when the entropy map is upper semi-continuous and the topological entropy is bounded. In section 4, we provide another definition of the topological pressure via open covers, which is equivalent to the one given in this section. Our proof of Theorem. is a generalization of Misiurewicz s elegant proof of the classical variational principle [2]. The principal applications of Theorem. are closely related to the dimension estimates for a broad class of Cantor-like sets and dynamical repellers [, ], the multifractal structure of iterated function systems with overlaps [2, 20], the distribution of Lyapunov exponents of a dynamical system [9, 3]. After a first version of this paper was completed, we were informed that Barreira [2] and Mummert [22] independently considered the thermodynamic formalism of almost-additive potentials, where F = {log f n } is assumed to satisfy 0 < c f n (xf m (T n x f n+m (x cf n (xf m (T n x. They proved the variational principle under an additional tempered variation condition and gave some conditions for the existence and uniqueness of equilibrium states and Gibbs measures. We were also informed that Käenmäki [7] proved the variational principle on full shift symbolic spaces for a class of special sub-additive potentials F = {log f n } (where f n (x were assumed to depend upon only the first n coordinates of x using the upper semi-continuity of the entropy map on symbolic spaces, she also showed that for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set itself. The paper is arranged in the following manner: in section 2 we provide some useful lemmas, in section 3 we prove Theorem., and in section 4 we discuss the relations between different approaches in the definition of the topological pressure. 2. Some Lemmas. In this section, we give some lemmas which are needed in our proof of Theorem.. Let F = {log f n } n= be a sequence of functions defined on a compact metric space (X, d, and T : X X a continuous map. Assume that F is sub-additive. Let P (T, F be defined as in section. We begin with the following lemma. Lemma 2.. For any k N, we have ( P T k, F (k = kp (T, F, where T k := T } {{ T } and F (k := {log f kn } n=. k times Proof. Fix k N. Observe that if E is an (n, ɛ-separated subset of X with respect to T k, then E must be an (nk, ɛ-separated subset of X with respect to T. It follows that { } P n (T k, F (k, ɛ = sup f kn (x : E is (n, ɛ-separated w.r.t T k x E { } sup f kn (x : E is (nk, ɛ-separated w.r.t T x E = P kn (T, F, ɛ,

5 SUB-ADDITIVE POTENTIALS 5 from which we deduce that P (T k, F (k kp (T, F. To show the reverse inequality, for any ɛ > 0 we choose δ > 0 small enough such that d(x, y δ = max d(t i x, T i y < ɛ. (3 i k Define C = max (, sup x X f (x. Then C <. Moreover f n (x f (xf (T x... f (T n x C n for all x X and n N. Now for any given natural number n, let l be an arbitrary integer in [kn, k(n +. Observe that by (3, any (l, ɛ-separated subset of X with respect to T must be an (n, δ-separated subset of X with respect to T k. This observation together with f l (x f nk (xf l nk (T nk x C k f nk (x yields { } P l (T, F, ɛ = sup f l (x : E is (l, ɛ-separated w.r.t T x E { } sup C k f kn (x : E is (n, δ-separated w.r.t T k x E = C k P n (T k, F (k, δ. This implies kp (T, F P (T k, F (k. Lemma 2.2. Let n, k be two positive integers with n 2k. For any x X, we have n k (f n (x k C 2k2 f k (T j x, where C = max (, sup x X f (x. Proof. Let x X. For j = 0,,..., k, we have Hence f n (x f j (xf n j (T j x C j f n j (T j x. (f n (x k k k C j f n j (T j x C k2 f n j (T j x. (4 Observe that for any given j between 0 and k, ( tj ( tj f n j (T j x f k (T kl+j x f n j ktj (T ktj+j x C k f k (T kl+j x, l=0 where t j is the largest integer t so that kt + j n. Combining the above inequality with (4 yields k (f n (x k C 2k2 t j l=0 This finishes the proof of the lemma. n k f k (T kl+j x = C 2k2 l=0 f k (T j x. After a first version of this paper was completed, we were informed that an analogous inequality, (f n (x k C 3k2 n f k(t j x, was obtained by Käenmäki (see [7, Lemma 2.2].

6 6 CAO, FENG AND HUANG Lemma 2.3. Suppose {ν n } n= is a sequence in M(X, where M(X denotes the space of all Borel probability measures on X with the weak* topology. We form the new sequence {µ n } n= by µ n = n n ν n T i. Assume that µ ni converges to µ in M(X for some subsequence {n i } of natural numbers. Then µ M(X, T, and moreover lim sup log f ni (x dν ni (x F (µ. i n i Proof. The statement µ M(X, T is well-known (see [29, Theorem 6.9] for a proof. To show the desired inequality, we fix an k N. For n 2k, by Lemma 2.2 we have n where µ n,k = log f n (x dν n (x = kn n i n k n k+ kn = n k + kn log (f n (x k dν n (x n k log f k (T j x dν n (x + 2k 2 log C log f k dµ n,k + ν n T i. In particular, log f ni dν ni n i k + log f k dµ ni,k + n i k Note that for any g C(X, n g dµ n (n k + g dµ n,k = n i=n k+ (k sup x X 2k log C, n 2k log C n i. (5 g(t i x dν n (x g(x. It follows that lim n ( g dµ n g dµ n,k = 0. Since lim i µ ni = µ, we have lim i µ ni,k = µ. Notice that f k is a non-negative continuous function on X. We have lim sup log f k dµ ni,k lim lim sup log(f k + ɛ dµ ni,k i ɛ 0 i = lim log(f k + ɛ dµ ɛ 0 = log f k dµ. Combining it with (5 yields lim sup log f ni dν ni log f k dµ. i n i k Now the desired inequality follows by letting k. Lemma 2.4. Let ν M(X. Suppose ξ = {A,..., A k } is a partition of (X, B(X. Then for any positive integers n, l with n 2l, we have ( n ( n H ν T i ξ l l H ν n T i ξ + 2l log k, n where ν n = n n ν T i.

7 SUB-ADDITIVE POTENTIALS 7 Proof. The above result is contained implicitly in the proof of the classical variational principle given by Misiurewicz (see [29]. In the following we present a complete proof for the reader s convenience. For 0 j l, let t j denote the largest integer t so that tl + j n. Then we have n t j l T i ξ = T rl j T i ξ T m ξ r=0 m S j for any 0 j l, where S j is a subset of {0,..., n } with cardinality at most 2l. Hence ( n t j ( l H ν T i ξ H ν T rl j T i ξ + 2l log k. r=0 Summing this over j from 0 to l gives ( n l t j ( lh ν T i ξ H ν = = r=0 ( n l H ν p=0 ( n H ν p=0 n H ν T p p=0 ( l nh νn T p l T rl j l T i ξ + 2l 2 log k T i ξ + 2l 2 log k T p l T i ξ + 2l 2 log k ( l T i ξ T i ξ + 2l 2 log k, + 2l 2 log k where the last inequality has used the concavity of the function φ(x = x log x. This implies the desired inequality. 3. The proof of Theorem.. In this section we give the proof of Theorem., which is influenced by Misiurewicz s elegant proof of the classical variational principle [2]. Proof of Theorem.. We divide the proof into three small steps: Step : P (T, F h µ (T + F (µ, µ M(X, T with F (µ. Suppose µ is an element in M(X, T satisfying F (µ. Let ξ = {A,..., A k } be a partition of (X, B(X. Let α > 0 be given. Choose ɛ > 0 so that ɛk log k < α. Since µ is regular there are compact sets B j A j with µ(a j \B j < ɛ, j k. Let η be the partition {B 0, B,..., B k }, where B 0 = X\ k j= B j. Then a direct check shows H µ (ξ/η < ɛk log k < α (see [29, Page 89] for details. Now set b = min i j k d(b i, B j > 0. Pick δ > 0 so that δ < b/2. Let n N. For each C n T j η, choose some x(c Closure(C such that f n (x(c = sup y C f n (y. We claim that for each C n T j η, there are at

8 8 CAO, FENG AND HUANG most 2 n many different C s in n T j η such that d n (x(c, x( C := ( max d T j (x(c, T j (x( C 0 j n < δ. To see this claim, for each C n T j η we pick the unique index (i 0 (C, i (C,..., i n (C {0,,..., k} n such that C = B i0(c T B i(c T (n B in (C. Now fix a C n T j n η and let Y denote the collection of all C T j η with d n (x(c, x( C < δ. Then we have { # i l ( C : C } Y 2, l = 0,,..., n. (6 To see this inequality, we assume on the contrary that there exists 0 l n and C, C 2, C 3 Y such that i l ( C, i l ( C 2, i l ( C 3 are distinct. Without loss of generality, we assume i l ( C 0 and i l ( C 2 0. This implies d n (x( C, x( C ( 2 d T l (x( C, T l (x( C ( 2 d B il ( C, B i l ( C 2 b > 2δ > d n (x(c, x( C ( + d n x(c, x( C 2, which leads to a contradiction. Thus (6 is true, from which the claim follows. In the following we construct an (n, δ-separated subset E of X with respect to T such that 2 n f n (y f n (x(c. (7 y E C n T j η To achieve this purpose we first choose an element C n T j η such that f n (x(c = max C f n (x(c. n T j η n Let Y denote the collection of all C T j η with d n (x(c, x( C < δ. Then the cardinality of Y does not exceed 2 n. If the collection n T j η\y is not empty, we choose an element C 2 n T j η\y such that f n (x(c 2 = max C n T j η\y f n (x(c. Let Y 2 denote the collection of C n T j η\y with d n (x(c 2, x( C < δ. We continue this process. More precise, in step m we choose an element C m n T j η\ m j= Y j such that f n (x(c m = max C n T j η\ m j= Yj f n (x(c. Let Y m denote the collection of all C n T j η\ m j= Y j with d n (x(c m, x( C < δ. Since the partition n T j η is finite, the above process will stop at some step l. Denote E = {x(c j : j =,..., l}. Then E is (n, δ-separated and f n (y = y E l f n (x(c j j= from which (7 follows. l 2 n j= C Y j f n (x(c = 2 n C n T j η f n (x(c,

9 SUB-ADDITIVE POTENTIALS 9 Recall µ M(X, T and F (µ. We have n n H µ T j η + log f n (x dµ(x n n C n T j η n log C n T j η n log 2 n f n (y y E µ(c (log f n (x(c log µ(c f n (x(c log 2 + n log P n(t, F, δ. In the second inequality above, we have used the basic inequality m i= p i(c i log p i log m i=, where c eci i R, p i 0 and m i= p i = (see [4, p. 4] for a proof. Now taking n and δ 0, we obtain h µ (T, η+f (µ log 2+P (T, F. Hence h µ (T, ξ + F (µ h µ (T, η + H µ (ξ/η + F (µ log 2 + α + P (T, F. Since ξ and α are arbitrary, we have h µ (T + F (µ log 2 + P (T, F. This holds for all continuous maps T and sub-additive potentials F. Thus we can apply it to T k and F (k to obtain k(h µ (T + F (µ = h µ (T k + F (k (µ log 2 + P (T k, F (k = log 2 + kp (T, F, where the last equality follows from Lemma 2.. Here we have used the fact that µ M(X, T k and F (k (µ = kf (µ. Since k is arbitrary, we have h µ (T + F (µ P (T, F. Step 2: If P (T, F, then for any small enough ɛ > 0, there exists a µ M(X, T such that F (µ and h µ (T + F (µ P (T, F, ɛ. Let ɛ > 0 be small enough such that P (T, F, ɛ. For any n N, let E n be an (n, ɛ-separated subset of X with respect to T such that f n (y > 0 for all y E n and log y E n f n (y log P n (T, F, ɛ. Let σ n M(X be the atomic measure concentrated on E n by the formula y E σ n = n f n (yδ y y E n f n (y, where δ y denote the Dirac measure at y. Let µ n = n n σ n T i. Since M(X is compact we can choose a subsequence {n i } of natural numbers such that lim i n i log P ni (T, F, ɛ = P (T, F, ɛ and {µ ni } converges in M(X to some µ M(X. By Lemma 2.3, µ M(X, T.

10 0 CAO, FENG AND HUANG Choose a partition ξ = {A,..., A k } of (X, B(X so that diam(a i < ɛ and µ( A i = 0 for i k. Such a partition does exist (see [29, Lemma 8.5] for a proof. Since each element of n T j ξ contains at most one point of E n, we have n H σn T j ξ + log f n dσ n = σ n ({y} (log f n (y log σ n ({y} y E n = ( σ n ({y} log f n (z y E n z E n Thus = log z E n f n (z log P n (T, F, ɛ. n n H σ n T j ξ + log f n dσ n n n log P n(t, F, ɛ n. (8 Fix l N. By Lemma 2.4, we have for n i 2l, n i H σni T j ξ l n i l H µ ni T j ξ + 2l log k. (9 n i Since µ( A i = 0 and {µ ni } converges to µ, we have l lim i l H µ ni T j ξ = l l H µ T j ξ. This together with (9 gives lim sup i Meanwhile by Lemma 2.3, n i H σni lim sup i n i T j ξ l l H µ T j ξ. n i log f ni dσ ni F (µ. Combining these two inequalities with (8 yields ( l l H µ T i ξ + F (µ P (T, F, ɛ. ( l Since H µ T i ξ [0, + and P (T, F, ɛ, we have F (µ. Taking l + yields ( l h µ (T + F (µ h µ (T, ξ + F (µ = lim l l H µ T i ξ + F (µ P (T, F, ɛ. Step 3: P (T, F = if and only if F (µ = for all µ M(X, T. By step we have P (T, F F (µ for all µ M(X, T with F (µ, which shows the necessity. The sufficiency is implied by step 2 (since if P (T, F, then by step 2 there exists some µ with F (µ. This finishes the proof the theorem.

11 SUB-ADDITIVE POTENTIALS 4. Other definitions of topological pressures for sub-additive potentials. In this section, we first discuss the definitions of topological pressures given by Falconer [9] and Barreira []. Then we provide an equivalent definition via open covers, following the idea of Bowen used in his definition of topological entropy for non-compact sets. In [9], Falconer gave a definition of topological pressures on mixing repellers. Without loss of generality, we only consider one-sided sub-shifts of finite type rather than mixing repellers. Let (Σ A, σ be a one-sided sub-shift space over an alphabet {,..., m}, where m 2 (see [4]. As usual Σ A is endowed with the metric d(x, y = m n+ where x = (x k k=, y = (y k k= and n is the smallest of the k such that x k y k. For any admissible string I = i... i n of length n over the letters {,..., m}, denote [I] = {(x k Σ A : x j = i j for j n}. The [I] is called an n-cylinder in Σ A. Let F be a sub-additive sequence of functions on Σ A. In [9], Falconer defined the topological pressure of F by F P (σ, F = lim n n log F P n(σ, F and F P n (σ, F = [I] sup f n (x, x [I] where the summation is taken over the collection of all n-cylinders in Σ A. It is not so hard to see that in this special case, F P n (σ, F is identical to P n (σ, F, /m, and P n (σ, F, m k = F P n+k (σ, F for all k N. This implies F P (σ, F = P (σ, F. Now let us turn to Barreira s approach in defining pressures for sub-additive potentials via open covers. As in the previous sections, let T be a continuous map acting on a compact metric space (X, d. Let F = {log f n } n= be a sub-additive sequence of functions defined on X. Suppose that U is a finite open cover of the space X. Denote diam(u := max{diam(u : U U}. For n we denote by W n (U the collection of strings U = U... U n with U i U. For U W n (U we call the integer m(u = n the length of U and define X(U = U T U 2... T (n U n = { x X : T j x U j for j =,..., n }. We say that Γ n W n(u covers X if U Γ X(U = X. For each U W n(u, we write f(u = sup x X(U f n (x when X(U and f(u = otherwise. For s R, define M(T, s, F, U = lim n inf Γ n U Γ n e sm(u f(u, where the infimum is taken over all Γ n j n W j(u that cover X. Likewise, we define M(T, s, F, U = lim inf inf e sm(u f(u, n Γ n U Γ n M(T, s, F, U = lim sup inf e sm(u f(u, n Γ n U Γ n where the infimum is taken over all Γ n W n (U that cover X. Define P (T, F, U = inf{s : M(T, s, F, U = 0} = sup{s : M(T, s, F, U = + }, CP (T, F, U = inf{s : M(T, s, F, U = 0} = sup{s : M(T, s, F, U = + }, CP (T, F, U = inf{s : M(T, s, F, U = 0} = sup{s : M(T, s, F, U = + }.

12 2 CAO, FENG AND HUANG Define P (T, F = lim inf P (T, F, U, diam(u 0 CP (T, F = CP (T, F = lim inf CP (T, F, U, diam(u 0 lim inf CP (T, F, U. diam(u 0 Barreira named P (T, F the topological pressure, CP (T, F and CP (T, F the lower and upper topological pressures of F. In the following we prove Lemma 4.. For any finite open cover U of X, we have P (T, F, U = CP (T, F, U = CP (T, F, U. Proof. The lemma was first proved by Barreira (see [, Theorem.6] under an additional assumption that 0 < f n < Cf n+ for a constant C > 0. In the following we modify Barreira s argument to obtain the general result. Fix a finite open cover U of X. By the definitions we have immediately that P (T, F, U CP (T, F, U CP (T, F, U. Thus to prove the lemma it suffices to show P (T, F, U CP (T, F, U. To see this inequality we may assume P (T, F, U < +. Take any s R such that s > P (T, F, U. Then there exists a Γ n W n(u which covers X and N(Γ := U Γ e sm(u f(u <. Since X is compact, the Γ can be assumed to be finite. Define Γ n = {U... U n : U i Γ} and Γ = n Γn. Since F is sub-additive, we have and N(Γ := N(Γ n := U Γ n e sm(u f(u N(Γ n, U Γ e sm(u f(u N(Γ n < N(Γ. n= Define K = max{m(u : U Γ}. For any integer n 2K, define { k Λ n = U U 2... U k : U i Γ for i k, m(u i < n K i= } k m(u i. It is clear n K m(u < n for any U Λ n. Furthermore Λ n Γ and Λ n covers X. Now define Λ n = { UV : U Λ n, V W n m(u (U }. i=

13 SUB-ADDITIVE POTENTIALS 3 Then Λ n W n (U covers X. Observe that e sm(uv f(uv U Λ n,v W n m(u (U ( e sm(u f(u e sm(v f(v U Λ n V K j= Wj(U N(Γ <. V K j= Wj(U e sm(v f(v We have M(T, s, F, U <. Hence CP (T, F, U s. It follows CP (T, F, U P (T, F, U. This finishes the proof of the lemma. As a direct corollary, we have Corollary 4.2. P (T, F = CP (T, F = CP (T, F. Now we consider the connection between P (T, F and P (T, F. Lemma 4.3. P (T, F P (T, F. Proof. The following proof is a slightly modified version of the proof of Theorem 9.2 of [29]. For any open cover U of X, denote Q n (T, F, U = inf f(u. Γ U Γ where the infimum is taken over all Γ W n (U that cover X. It is clear that CP (T, F, U = lim sup n n log Q n(t, F, U. For any ɛ > 0, suppose U is an open cover of X with diam(u < ɛ. Let E be an (n, ɛ- separated subset of X with respect to T. Since no members of n T i U contains two elements of E we have x E f n(x Q n (T, F, U, therefore P n (T, F, ɛ Q n (T, F, U. Letting ɛ 0 we obtain P (T, F CP (T, F = P (T, F. We don t know whether the equality P (T, F = P (T, F always holds under the above general setting. However we can show the equality in several cases, see Propositions Proposition 4.4. Assume the topological entropy h(t < and the entropy map µ h µ (T is upper semi-continuous. Then P (T, F = P (T, F. Proof. We divide the proof into several small steps. Step. P (T, F sup{h µ (T + k log(f k + ɛdµ} for any ɛ > 0 and k N. Fix ɛ > 0 and k N. Define g(x = (f k (x + ɛ /k. Set G = {g n } n=, where g n (x = n g(t j x. Then by Walters variational principle (see ( and [29, Theorem 9.4], P (T, G = sup { } h µ (T + k log(f k + ɛdµ : µ M(X, T. Set C = max(, sup x X f (x. Then by Lemma 2.2, f n (x C 2k g n k (x C 2k ɛ k g n (x, x X, n N.

14 4 CAO, FENG AND HUANG It follows P (T, F P (T, G, and thus { } P (T, F sup h µ (T + k log(f k + ɛdµ, µ M(X, T. Step 2. For any k N, lim sup ɛ 0 = sup { h µ (T + { h µ (T + } k log(f k + ɛdµ, µ M(X, T }. k log f kdµ, µ M(X, T In this step, we will use the assumption h µ (T < and the entropy map is upper semi-continuous. Denote by A the value of the first limit. Then there exist a convergent sequence {µ n } in M(X, T, and {ɛ n } 0 such that lim n h µn (T equals a finite value (denoted by B and furthermore lim n k log(f k + ɛ n dµ n = A B. Let ν denote the limit of {µ n }. For any fixed ɛ > 0, lim n k log(f k + ɛ n dµ n lim n k log(f k + ɛdµ n. Therefore Letting ɛ 0, we obtain lim n k log(f k + ɛ n dµ n lim n k log(f k + ɛdν. k log(f k + ɛ n dµ n k log f kdν. On the other hand, h ν (T lim n h µn (T = B. The above two inequalities imply h ν (T + k log f kdν B + (A B = A. Thus the desired equality follows. Step 3. lim k sup{h µ (T + k log f kdµ, µ M(X, T } = sup{h µ (T +F (µ, µ M(X, T }. The direction is clear. To show the other direction, we will also use the assumption that h(t < and the entropy map is upper semi-continuous. We denote by A the value of the first limit. Then there exist {µ n } ν in M(X, T, and {k n } + such that lim n h µn (T equals a finite value (denoted by B and furthermore lim log f kn dµ n = A B. n k n Fix l N. Then by Lemma 2.2, for large n N we have and thus k n l (f kn (x l C 2l2 f l (T j x kn l 2l log C log f l (T i x log f kn (x +, x X. k n k n k n l

15 SUB-ADDITIVE POTENTIALS 5 Therefore 2l log C kn l + log f kn dµ n + log f l dµ n. k n k n k n l Taking n + yields A B l log f ldν. Then letting l + we obtain A B F (ν. Since the entropy map is upper semi-continuous, we have h ν (T lim n h µn (T = B. Thus A h ν (T + F (ν. This finishes the proof of step 3. Step 4. P (T, F = P (T, F. By Lemma 4.3 it suffices to prove P (T, F P (T, F. To see this, by step -3, we have P (T, F sup{h µ (T + F (µ : µ M(X, T }. Since h(t <, by Theorem. we have P (T, F = sup{h µ (T + F (µ : µ M(X, T }, from which P (T, F P (T, F follows. This finishes the proof of the proposition. Proposition 4.5. Assume the dynamical system (X, T satisfies the following doubling property: there exists a sequence {r n } N such that lim n log r n /n = 0, and for any n N, x X and ɛ > 0, the Bowen ball B n (x, ɛ := {y X : d(t i x, T i y < ɛ for i n } can be covered by at most r n many Bowen balls B n (y i, ɛ/2, i =,..., r n. Then P (T, F = P (T, F. Proof. For any open cover U of X, denote Q n (T, F, U = inf Γ f(u. where the infimum is taken over all Γ W n (U that cover X. It is easily checked that Q n+m (T, F, U Q n (T, F, UQ m (T, F, U. Thus we have U Γ P (T, F, U = CP (T, F, U = inf n n log Q n(t, F, U. In the following we show that for any n N and ɛ > 0, there exists an open cover U of X with diam(u ɛ such that Q n (T, F, U r 4 np n (T, F, ɛ. (0 Since lim n log r n /n = 0, the above inequality implies P (T, F P (T, F. By Lemma 4.3, we have P (T, F = P (T, F. To show (0, we choose an (n, ɛ-separated subset E of X with maximal cardinality. Then the family of Bowen balls {B n (x, ɛ : x E} covers X. For any x E, pick y(x Closure(B n (x, ɛ such that f n (y(x = sup f n (y. y B n(x,ɛ Define U = {B(T i x, ɛ : x E, i n }, where B(x, δ := {y X : d(x, y < δ}. Then U covers X and diam(u ɛ. Since we have X = B n (x, ɛ = x E n x E T i B(T i x, ɛ, Q n (T, F, U x E f n (y(x. (

16 6 CAO, FENG AND HUANG To prove (0 we first claim that for any x E there are at most rn 4 many x E such that d n (y(x, y(x ɛ. To show this, let x,..., x l be l many different points in E such that d n (y(x i, y(x ɛ for i l. Then B n (x, 4ɛ l i= B n(x i, ɛ. By the assumed doubling property, B n (x, 4ɛ can be covered by at most rn 4 many Bowen balls B n (z, ɛ/4 s. Since any B n (z, ɛ/4 intersects at most one of the Bowen balls B n (x i, ɛ/4, we obtain l rn. 4 Now we pick z E such that f n (y(z = max{f n (y(x : x E}. Then we get a set E by removing all those points x in E such that B n (y(z, y(x ɛ. If E, we pick z 2 E such that f n (y(z 2 = max{f n (y(x : x E }. Then we get a set E 2 by removing all those points x in E such that B n (y(z 2, y(x ɛ. Continuing this process (which will stop at some step p, we obtain an (n, ɛ-separated subset {y(z i : i p} and p i= f n (y(z i r 4 n f n (y(x. x E Combining it with ( yields (0. This finishes the proof. Proposition 4.6. Let (X, T be a dynamical system such that for any ɛ > 0, there exists a partition U = {U,..., U k } of X such that diam(u < ɛ and all U i are open. Then P (T, F = P (T, F. Proof. Assume U = {U,..., U k } is an open partition of X. Then all U i are both open and closed. Set γ = min d(u i, U j. i<j k Then γ > 0. For any 0 < δ < γ and l N, one has inf e sm(u f(u e sl P l (T, F, δ, Γ l U Γ l where the infimum is taken over all Γ l W l (U that cover X. This implies CP (T, F, U P (T, F, δ. Hence one has P (T, F, U = CP (T, F, U P (T, F, δ, moreover P (T, F, U P (T, F. Since the diameter of U can be arbitrarily small, we have P (T, F P (T, F. This finishes the proof. If we put some bounded distortion assumptions on F, then we can also show the equivalence of these two definitions. Proposition 4.7. Assume that F = {log f n } n= satisfies the following additional assumptions: (i f n (x > 0 for all x X and n N. (ii log γ n (F, U lim sup lim sup = 0, diam(u 0 n n where for any finite open cover U, γ n (F, U is defined by γ n (F, U = sup{f n (x/f n (y : x, y X(U for some U W n (U}. Then we have P (T, F = P (T, F. Proof. We take some arguments similar to that in the proof of Proposition 4.5. Under the above assumptions, instead of (0 we may show that for any finite open cover U of X and n N, Q n (T, F, U γ n (F, UP n (T, F, ɛ.

17 SUB-ADDITIVE POTENTIALS 7 To show this, as in the proof of Proposition 4.5, we have Q n (T, F, U f n (y(x γ n (F, U f n (x γ n (F, UP n (T, F, ɛ. (2 x E x E This finishes the proof. In the remaining part of this section, we will follow Bowen s idea in [3] to give an equivalent definition of topological pressure via open covers. Let (X, T be a compact dynamical system. Let C X denote the set of all finite Borel covers of X. Let F = {log f n } n= be a sub-additive potential. For any open cover U of X, we define Pn (T, F, U = min α C X, α n sup f n (x, T i U x A A α where α n T i U means that for each A α, there exists U W n (U such that A X(U. Furthermore define P (T, F, U = inf log Pn (T, F, U, n n and P (T, F = sup P (T, F, U. (3 U where U ranges over all open covers of X. Proposition 4.8. P (T, F = P (T, F. Proof. We divide the proof into several small parts: Step. P (T, F = lim diam(u 0 P (T, F, U. It comes from the fact that P (T, F, U P (T, F, V, whenever U and V are two open covers of X and diam(v is less than the Lebesgue number of U. Step 2. P (T, F, U = lim n n log Pn (T, F, U. It follows from the fact that Pn (T, F, U is sub-multiplicative, i.e, Pn+m(T, F, U Pn (T, F, UPm (T, F, U for all n, m N. To show this fact, observe that for any α, β C X with α n T i U and β m T i U, we have α T n β C X, α T n β n+m T i U, and furthermore sup f n+m (x sup f n (x sup f m (y. x C x A y B A α B β C α T n β Step 3. For any ɛ > 0, if U is an open cover of X with diam(u < ɛ, then P (T, F, U P (T, F, ɛ. To show this fact, it suffices to note that for any α C X with α n T i U and any (n, ɛ-separated set E of X, each e E is contained exactly in one element A α. Step 4. For any n N and ɛ > 0, there exists an open cover U of X such that diam(u ɛ and P n (T, F, ɛ Pn (T, F, U. To prove this statement, let n N and ɛ > 0. Pick x X with f n (x = sup x X f n (x, pick x 2 X\B n (x, ɛ such that f n (x 2 = sup x X\Bn(x,ɛ f n (x, pick x 3 X\ 2 i= B n(x i, ɛ such that f n (x 3 = sup x X\ 2 i= Bn(xi,ɛ f n(x, and so on. We obtain an (n, ɛ-separated set {x, x 2,, x l } of maximal cardinality. Let α be the partition { B n (x, ɛ, B n (x 2, ɛ\b n (x, ɛ, B n (x 3, ɛ } 2 B n (x i, ɛ,.... i=

18 8 CAO, FENG AND HUANG Then A α sup x A f n (x = l i= f n(x i P n (T, F, ɛ. Let U = {B(T j x i, ɛ : 0 j n, i l}. We have diam(u ɛ, α n T i U and P n (T, F, ɛ Pn (T, F, U. By the statements in step and step 3, we have P (T, F P (T, F. Whilst by the statements in step 2 and step 4, for any ɛ > 0, there exists an open cover U with diameter not exceeding ɛ such that P (T, F, U P (T, F, ɛ. This fact, together with the result in step, gives P (T, F P (T, F when we let ɛ go to 0. This finishes the proof of the proposition. Acknowledgement The authors thank Prof. Yingfei Yi for some helpful discussions. They thank Prof. Ka-Sing Lau for bringing us an early version of the reference [22]. Cao was partially supported by NSFC(05730, NCET and SRFDP of China; Feng was also partially supported by the RGC grant and the direct grant in CUHK, Fok Ying Tong Education Foundation and NSFC( Huang was partially supported by NSFC( REFERENCES [] L. M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergodic Theory Dynam. Systems, 6 (996, [2] L. Barreira, Nonadditive thermodynamic formalism: equilibrium and Gibbs measures. Discrete Contin. Dyn. Syst., 6 (2006, [3] R. Bowen, Topological entropy for noncompact sets. Trans. Amer. Math. Soc., 84 (973, [4] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture notes in Math. No. 470, Springer-Verlag, 975. [5] R. Bowen, Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math., No. 50, (979, 25. [6] I. Daubechies and J.C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions. SIAM J. Math. Anal., 22 (99, [7] I. Daubechies and J.C. Lagarias, Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals. SIAM J. Math. Anal., 23 (992, [8] I. Daubechies and J.C. Lagarias, On the thermodynamic formalism for multifractal functions. The state of matter (Copenhagen, 992, , Adv. Ser. Math. Phys., 20, World Sci. Publishing, River Edge, NJ, 994. [9] K.J. Falconer, A subadditive thermodynamic formalism for mixing repellers. J. Phys. A, 2 (988, no. 4, L737 L742. [0] K.J. Falconer, The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc., 03 (988, [] K.J. Falconer, Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Chichester, 990. [2] D.J. Feng, Smoothness of the L q -spectrum of self-similar measures with overlaps. J. London Math. Soc., 68 (2003, [3] D.J. Feng, Lyapunov exponent for products of matrices and multifractal analysis. Part I: Positive matrices. Israel J. Math., 38 (2003, [4] D.J. Feng, The variational principle for products of non-negative matrices. Nonlinearity, 7 ( [5] D.J. Feng, The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math., 95 (2005, [6] D.J. Feng and K.S. Lau, The pressure function for products of non-negative matrices, Math. Res. Lett., 9 (2002, [7] A. Käenmäki, On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math., 29 (2004, [8] G. Keller, Equilibrium states in ergodic theory. Cambridge University Press, 998. [9] S. P. Lalley, Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution. J. London Math. Soc., 57 (998,

19 SUB-ADDITIVE POTENTIALS 9 [20] K.S. Lau, Multifractal structure and product of matrices. Analysis in Theory and Applications, 9 ( [2] M. Misiurewicz, A short proof of the variational principle for a Z N + action on a compact space. Asterisque, 40 (976, [22] A. Mummert, The thermodynamic formalism for almost-additive sequences. Discrete Contin. Dyn. Syst., 6 (2006, [23] Ya. B. Pesin, Dimension theory in dynamical systems. Contemporary views and applications. University of Chicago Press, 997. [24] Ya. B. Pesin and B. S. Pitskel, Topological pressure and the variational principle for noncompact sets. Functional Anal. Appl., 8 (984, [25] D. Ruelle, Statistical mechanics on a compact set with Z v action satisfying expansiveness and specification. Trans. Amer. Math. Soc., 87 (973, [26] D. Ruelle, Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics. Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co., Reading, Mass., 978. [27] D. Ruelle, Repellers for real analytic maps. Ergodic Theory Dynamical Systems, 2 (982, [28] P. Walters, A variational principle for the pressure of continuous transformations. Amer. J. Math., 97 (975, [29] P. Walters, An introduction to ergodic theory. Springer-Verlag, 982. [30] M. Zinsmeister, Thermodynamic formalism and holomorphic dynamical systems. Translated from the 996 French original by C. Greg Anderson. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, address: ylcao@suda.edu.cn; djfeng@math.cuhk.edu.hk; wenh@mail.ustc.edu.cn