JOURNALOF MODERN DYNAMICS doi: 10.3934/jmd.2011.5.593 VOLUME 5, NO. 3, 2011, 593–608

BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS

OMRI M. SARIG (Communicated by Anatole Katok)

1 ABSTRACT. Suppose f : M M is a C α (α 0) diffeomorphism on a com- → + > pact smooth orientable manifold M of dimension 2, and let µΨ be an equi- librium for a Hölder-continuous potential Ψ: M R. We show that → if µΨ has positive measure-theoretic entropy, then f is measure-theoretically isomorphic mod µΨ to the product of a and a finite rota- tion.

1. STATEMENTS 1 α Suppose f : M M is a C + (α 0) diffeomorphism on a compact smooth → > orientable manifold M of dimension two and Ψ: M R is Hölder-continuous. → An invariant probability measure µ is called an equilibrium measure, if it max- imizes the quantity h (f ) R Ψdµ, where h (f ) is the measure-theoretic en- µ + µ tropy. Such measures always exist when f is C ∞, because in this case the func- tion µ h (f ) is upper semicontinuous [12]. Let µ be an ergodic equilibrium 7→ µ Ψ measure of Ψ. We prove:

THEOREM 1.1. If h (f ) 0, then f is measure-theoretically isomorphic with µΨ > respect to µΨ to the product of a Bernoulli scheme (see §3.3) and a finite rotation (a map of the form x x 1 (mod p) on {0,1,...,p 1}). 7→ + − In the particular case of the measure of maximal entropy (Ψ 0) we can say ≡ more, see §5.1. The proof applies to certain potentials that are not necessarily Hölder-continuous, such as t log J (t R), where J is the unstable Jacobian, − u ∈ u see §5.2. The theorem is false in higher dimensions: Let f denote the product of a hyperbolic toral automorphism and an irrational rotation. This C ∞ diffeomor- phism has many equilibrium measures of positive entropy. But f cannot satisfy the conclusion of the theorem with respect to any of these measures, because f has the irrational rotation as a factor, and therefore none of its powers can have ergodic components with the K property. Bowen [4] and Ratner [20] proved Theorem 1.1 for uniformly hyperbolic dif- feomorphisms. In the nonuniformly hyperbolic case, Pesin proved that any

Received May 2, 2011; revised July 19, 2011. 2000 Mathematics Subject Classification: Primary: 37D35; Secondary: 37D25. Key words and phrases: Bernoulli, surface diffeomorphisms, equilibrium measures, count- able Markov partitions. This work was supported by ERC award ERC-2009-StG n◦ 239885.

INTHEPUBLICDOMAINAFTER 2039 593 ©2011 AIMSCIENCES 594 OMRI M.SARIG absolutely continuous ergodic invariant measure all of whose Lyapunov expo- nents are nonzero is isomorphic to the product of a Bernoulli scheme and a finite rotation [19]. By Pesin’s Entropy Formula and Ruelle’s Entropy Inequal- ity, these measures are equilibrium measures of log J . Ledrappier extended − u Pesin’s result to all equilibrium measures with nonzero exponents for the po- tential log J , including those which are not absolutely continuous [10]. These − u results hold in any dimension. The work of Pesin and Ledrappier (see also [17]) uses the following prop- erty of equilibrium measures of log J : the conditional measures on unstable − u manifolds are absolutely continuous [10]. This is false for general Hölder po- tentials [11]. Theorem 1.1 is proved in three steps: 1. : Any ergodic equilibrium measure on M with pos- itive entropy is a finite-to-one Hölder factor of an ergodic equilibrium measure on a countable Markov shift. 2. Ornstein Theory: Factors of equilibrium measures of Hölder potentials on topologically mixing countable Markov shifts are Bernoulli. 3. Spectral decomposition: The nonmixing case.

1 1 Notation. a M ± b means M − b a Mb. = ≤ ≤

2. STEPONE: SYMBOLICDYNAMICS Let G be a directed graph with a countable collection of vertices V such that every vertex has at least one edge coming in, and at least one edge coming out. The countable Markov shift associated to G is the set Z Σ Σ(G ) : {(vi )i Z V : vi vi 1 for all i}. = = ∈ ∈ → + The natural metric d(u,v) : exp[ min{ i : u v }] turns Σ into a complete = − | | i 6= i separable metric space. Σ is compact iff G is finite. Σ is locally compact iff every vertex of G has finite degree. The cylinder sets

(2.1) m[am,...,an] : {(vi )i Z Σ : vi ai (i m,...,n)} = ∈ ∈ = = form a basis for the topology, and they generate the Borel σ-algebra B(Σ). The left shift map σ: Σ Σ is defined by σ[(vi )i Z] (vi 1)i Z. Given a,b n → ∈ = + ∈ ∈ V , write a b when there is a path a v1 vn 1 b in G . The left shift −→ → →n ··· → − → n is topologically transitive iff a,b V n (a b). In this case gcd{n : a a} is ∀ ∈ ∃ −→ −→ the same for all a V , and is called the period of σ. The left shift is topolog- ∈ ically mixing iff it is topologically transitive and its period is equal to one. See [9]. Let # Σ : {(vi )i Z Σ : u,v V nk ,mk such that v m u,vn v}. = ∈ ∈ ∃ ∈ ∃ ↑ ∞ − k = k = Every σ-invariant probability measure gives Σ# full measure, because of Poin- caré’s Recurrence Theorem.

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1 α Suppose f : M M is a C + -diffeomorphism of a compact, orientable and → smooth manifold M such that dimM 2. If h (f ) 0, then every f -invariant = top = measure has zero entropy by the variational principle, and Theorem 1.1 holds trivially. So we assume without loss of generality that h (f ) 0. top > Fix 0 χ h (f ). A set Ω M is said to be χ-large, if µ(Ω) 1 for every < < top ⊂ = ergodic invariant probability measure µ whose entropy is greater than χ. The following theorems are in [23]:

THEOREM 2.1. There exists a locally compact countable Markov shift Σχ and a # Hölder-continuous map πχ : Σχ M such that πχ σ f πχ, πχ[Σχ] is χ-large, # → ◦ = ◦ and every point in πχ[Σχ] has finitely many preimages.

THEOREM 2.2. Denote the set of states of Σχ by Vχ. There exists a function ϕχ : Vχ Vχ N such that if x πχ[(vi )i Z] and vi u for infinitely many neg- × → = ∈ = 1 ative i, and v v for infinitely many positive i, then π− (x) ϕ (u,v). i = | χ | ≤ χ THEOREM 2.3. Every ergodic f -invariant probability measure µ on M such that 1 h (f ) χ equals µ π− for some ergodic σ-invariant probability measure µ on µ > b ◦ χ b Σχ with the same entropy. We will use these results to reduce the problem of Bernoullicity for equilib- rium measures for f : M M and the potential Ψ, to the problem of Bernoul- → licity for equilibrium measures for σ: Σ Σ and the potential ψ : Ψ π . χ → χ = ◦ χ

3. STEPTWO:ORNSTEIN THEORY First we describe the structure of equilibrium measures of Hölder-continuous potentials on countable Markov shifts, and then we show how this structure forces, in the topologically mixing case, isomorphism to a Bernoulli scheme.

3.1. Equilibrium measures on one-sided countable Markov shifts [7]. Sup- pose G is a countable directed graph. The one-sided countable Markov shift associated to G is N {0} Σ+ Σ+(G ) : {(vi )i 0 V ∪ : vi vi 1 for all i}. = = ≥ ∈ → + Proceeding as in the two-sided case, we equip Σ+ with the metric d(u,v) : = exp[ min{i 0 : u v }]. The cylinder sets − ≥ i 6= i (3.1) [a0,...,an 1] : {u Σ+ : ui ai (i 0,...,n 1)} − = ∈ = = − form a basis for the topology of Σ+. Notice that unlike the two-sided case (2.1), there is no left subscript; the cylinder starts at the zero coordinate. The left shift map σ: Σ+ Σ+ is given by σ :(v ,v ,...) (v ,v ,...). This → 0 1 7→ 1 2 map is not invertible. The natural extension of (Σ+,σ) is conjugate to (Σ,σ). A function φ: Σ+ R is said to be weakly Hölder-continuous if there are con- → stants C 0 and θ (0,1) such that var φ Cθn for all n 2, where > ∈ n < ≥ var φ : sup{φ(u) φ(v): u v (i 0,...,n 1)}. n = − i = i = − JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 596 OMRI M.SARIG

The following inequality holds: Ãn 1 ! X− j X∞ (3.2) varn m φ σ varj φ. + j 0 ◦ ≤ j m 1 = = + If φ is bounded, weak Hölder-continuity is the same as Hölder-continuity. The equilibrium measures for weakly Hölder potentials were described by Ruelle [21] for finite graphs and by Buzzi and the author for countable graphs [7]. Make the following assumptions: (a) σ: Σ Σ is topologically mixing. → (b) φ is weakly Hölder-continuous and supφ . (This can be relaxed [7].) < ∞ (c) P (φ) : sup{h (σ) R φdm} , where the supremum ranges over all G = m + < ∞ shift-invariant measures m such that h (σ) R φdm . The po- m + 6= ∞ − ∞ tentials we will study satisfy P (φ) h (f ) max Ψ . G ≤ top + | | < ∞ P φ(y) Define (LφF )(x) σ(y) x e F (y) for F : Σ+ R+ (“Ruelle operator“). The = = → iterates of Lφ are n 1 n X φ(y) φ(σy) φ(σ − y) (LφF )(x) e + +···+ F (y). = σn (y) x =

THEOREM 3.1 (Buzzi and Sarig). Suppose (a), (b), (c). Then φ: Σ+ R has at → most one equilibrium measure. If this measure exists, then it is equal to h dν where

1. h : Σ+ R is a positive continuous function such that L h λh; → φ = 2. ν is a Borel measure on Σ that is finite and positive on cylinder sets, L∗ ν φ = λν, and R h dν 1; = n n 3. λ expPG (φ) and λ− L 1[a] ν[a]h pointwise for every cylinder [a]. φ n = −−−−→→∞ Parts (1) and (2) continue to hold if we replace (a) by topological transitivity.

COROLLARY 3.2. Assume (a),(b),(c) and let µ be the equilibrium measure of φ. For every finite S∗ V there exists a constant C ∗ C ∗(S∗) 1 such that for every ⊂ = > m,n 1, every n-cylinder [a], and every m-cylinder [c], ≥ µ[a,c] 1. if the last symbol in a is in S∗ and [a,c] ∅, then 1/C ∗ C ∗; 6= ≤ µ[a]µ[c] ≤ µ[c,a] 2. if the first symbol of a is in S∗ and [c,a] ∅, then 1/C ∗ C ∗. 6= ≤ µ[a]µ[c] ≤ Proof. We begin with a couple of observations (see [24]).

Observation 1: φ∗ : φ logh logh σ logλ is weakly Hölder-continuous, and = + − ◦ −n if L Lφ then L∗µ µ, L1 1, and L 1[a] µ[a] pointwise. Note that φ∗ = ∗ = = −−−−→n need not be bounded. →∞ n n Proof. The convergence λ− L 1[a] hν[a] and (3.2) imply that logh is φ n −−−−→→∞ weakly Hölder-continuous, and var (logh) . It follows that φ∗ is weakly 1 < ∞ Hölder-continuous. The identities L∗µ µ,L1 1 can be verified by direct cal- n = = culation. To see the convergence L 1[a] hν[a] we argue as follows. Since −−−−→n φ has an equilibrium measure, φ is positive→∞ recurrent [7]. Positive recurrence

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS 597 is invariant under the addition of constants and coboundaries, so φ∗ is also positive recurrent. The limit now follows from [22, Theorem 1].

Observation 2: For any positive continuous functions F,G : Σ+ R+, Z Z → (3.3) F (G σn)dµ (LnF )G dµ. ◦ = n n n Proof. Integrate the identity (L F )G L (FG σ ) using L∗µ µ. = ◦ = n 1 Observation 3: Let φ∗ : φ∗ φ∗ σ φ∗ σ − , then, because of (3.2) and n = + ◦ + ··· + ◦ the weak Hölder-continuity of φ∗, M : exp(supn 1 varn 1φn∗) is finite. = ≥ + We turn to the proof of the corollary. Suppose a (a0,...,an 1) and an 1 − − R n R φ∗(=a,y) ∈ S∗. By Observation 2, µ([a,c]) L 1 dµ e n 1 (y)dµ(y). = [c] [a] = [c] φ∗(a,y) 1 φ∗(a,z) It holds that e n M ± e n for all y,z σ[an 1]. Fixing y and averag- = ∈ − ing over z σ[an 1] we obtain ∈ − µ 1 Z ¶ φ∗(a,y) 1 φ∗(a,z) e n M ± e n dµ(z) = µ(σ[an 1]) σ[an 1] µ − Z − ¶ µ ¶ 1 1 n 1 µ[a] M ± L 1[a] dµ M ± . = µ(σ[an 1]) = µ(σ[an 1]) − − Let C ∗ : max{M/µ(σ[a]) : a S∗}. Since an 1 S∗, 1 = ∈ − ∈ φn∗(a,y) 1 e (C1∗)± µ[a] for all y σ[an 1]. = ∈ − Since [a,c] ∅, σ[an 1] [c], so 6= − ⊇ Z φ∗(a,y) 1 µ[a,c] e n 1 (y)dµ(y) (C ∗)± µ[a]µ[c]. = [c] = 1 Now suppose a0 S∗ and [c,a] ∅, where c (c0,...,cm 1). As before ∈ 6= = − Z Z m φ∗ (c,y) µ[c,a] L 1[c] dµ e m dµ(y), = [a] = [a]

φ∗ (c,y) 1 µ[c] and e m M ± ( ). So µ+(σ[cm 1]) = − µ 1 ¶ M ± µ[c,a] µ[a]µ[c]. = µ(σ[cm 1]) − 1 M Since [c,a] ∅, σ[cm 1] [a0], therefore the term in the brackets is in [ , ]. 6= − ⊃ M µ[a0] 1 If we set C ∗ : max{M/µ[a]: a S∗}, then µ[c,a] (C ∗)± µ[a]µ[c]. 2 = ∈ = 2 The lemma follows with C ∗ : max{C ∗,C ∗}. = 1 2 3.2. Equilibrium measures on two-sided countable Markov shifts. We return to two-sided countable Markov shifts Σ Σ(G ). A function ψ: Σ R is said to = → be weakly Hölder-continuous if there are constants C 0 and 0 θ 1 such > < < that var ψ Cθn for all n 2, where n < ≥ var ψ : sup{ψ(x) ψ(y): x y (i (n 1),...,n 1)}. n = − i = i = − − − A function ψ: Σ R is said to be one-sided, if ψ(x) ψ(y) for every x, y → = ∈ Σ such that x y for all i 0. The following lemma was first proved (in a i = i ≥ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 598 OMRI M.SARIG different context) by Sinai. The proof given in [5] for subshifts of finite type also works for countable Markov shifts:

LEMMA 3.3 (Sinai). If ψ: Σ R is weakly Hölder-continuous and var ψ , → 1 < ∞ then there exists a bounded Hölder-continuous function ϕ such that φ : ψ ϕ = + − ϕ σ is weakly Hölder-continuous and one-sided. ◦ Notice that if ψ is bounded then φ is bounded, and that every equilibrium mea- sure for ψ is an equilibrium measure for φ and vice versa. Since φ: Σ R is one-sided, there is a function φ+ : Σ+ R such that φ(x) → → = φ+(x ,x ,...). If φ: Σ R is weakly Hölder-continuous, then φ+ : Σ+ R is 0 1 → → weakly Hölder-continuous. Any shift-invariant probability measure µ on Σ determines a shift-invariant probability measure µ+ on Σ+ through the equations

µ+[a0,...,an 1] : µ(0[a0,...,an 1]) − = − (cf. (2.1) and (3.1)). The map µ µ+ is a bijection, and it preserves 7→ and entropy. It follows that µ is an ergodic equilibrium measure for φ iff µ+ is an ergodic equilibrium measure for φ+.

COROLLARY 3.4. Suppose σ: Σ Σ is topologically mixing. If ψ: Σ R is → → weakly Hölder-continuous, supψ , var ψ , and P (ψ) then ψ has < ∞ 1 < ∞ G < ∞ at most one equilibrium measure µ. This measure is the natural extension of an equilibrium measure of a potential φ: Σ+(G ) R which satisfies assumptions → (a),(b),(c).

3.3. The Bernoulli property. The Bernoulli scheme with probability vector p Z Z = (pa)a S is (S ,B(S ),µp ,σ) where σ is the left shift map and µp is given by ∈ µ ( [a ,...,a ]) p p . If (Ω,F ,µ,T ) is measure-theoretically isomor- p m m n = am ··· an phic to a Bernoulli scheme, then we say that (Ω,F ,µ,T ) is a Bernoulli auto- morphism, and µ has the Bernoulli property. In this section we prove:

THEOREM 3.1. Every equilibrium measure of a weakly Hölder-continuous po- tential ψ: Σ(G ) R on a topologically mixing countable Markov shift such that → P (ψ) and supψ has the Bernoulli property. G < ∞ < ∞ This was proved by Bowen [4] in the case when G is finite. See [20] and [26] for generalizations to larger classes of potentials. We need some facts from Ornstein Theory. Suppose β {P ,...,P } is a finite = 1 N measurable partition for an invertible probability-preserving map (Ω,F ,µ,T ). n Wn i For every m,n Z such that m n, let βm : i m T − β. ∈ < = = DEFINITION 3.5 (Ornstein). A finite measurable partition β is said to be weak Bernoulli if ε 0 k 1 such that ∀ > ∃ > X X µ(A B) µ(A)µ(B) ε for all n 0. 0 k n | ∩ − | < > A β n B β + ∈ − ∈ k

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Ornstein showed that if an invertible probability-preserving transformation has a generating increasing sequence of weak Bernoulli partitions, then it must be measure-theoretically isomorphic to a Bernoulli scheme [13, 16]. Proof of Theorem 3.1. First we make a reduction to the case when var ψ . 1 < ∞ To do this, recode Σ(G ) using the Markov partition of cylinders of length two and notice that var1 of the new coding equals var2 of the original coding. The supremum and the pressure of ψ remain finite, and the variations of ψ con- tinue to decay exponentially. Suppose µ is an equilibrium measure of ψ: Σ(G ) R. For every V 0 V → ⊂ finite, let ( ) : © [v]: v ª ©S [v]ª. We claim that ( ) is weak α V 0 0 V 0 v V 0 0 α V 0 = ∈ ∪ 6∈ Bernoulli. This implies the Bernoulli property, because of the results of Orn- stein we cited above. We saw in the previous section that the measure µ+ on Σ+(G ) given by

µ+[a0,...,an 1] : µ(0[a0,...,an 1]) − = − n satisfies L∗µ+ µ+ and L 1[a] µ+[a], where L Lφ and φ∗ : Σ+(G ) R = −−−−→n = ∗ → is weakly Hölder-continuous. By→∞ (3.2),

sup(varn mφn∗) 0. n 1 + −−−−→m ≥ →∞ t 1 Fix 0 δ 1 so small that 1 e− ( t,t) for all 0 t δ . Fix some smaller < 0 < − ∈ 2 < < 0 0 δ δ0, to be determined later, and choose: < < ¡S ¢ a finite collection S∗ of states (vertices) such that µ a S 0[a] 1 δ; • ∈ ∗ > − a constant C ∗ C ∗(S∗) 1 as in Corollary 3.2; • = > a natural number m m(δ) such that supn 1(varn mφn∗) δ; • = ≥ + < 2 S δ/2(C ∗) a finite collection γ of m-cylinders [c] such that µ( γ) e− ; • 0 > points x(c) [c] γ; • ∈ 0 ∈ natural numbers K (c,c0) ([c],[c0] γ) such that for every k K (c,c0) • ∈ ≥ k δ (L 1 )(x(c0)) e± µ+[c] [c] = n (recall that L 1[c] µ+[c]); n −−−−→→∞ K (δ) : max{K (c,c0):[c],[c0] γ} m. • = ∈ + Step 1. Let A : n[a0,...,an] and B : k [b0,...,bn] be two nonempty cylinders = − = of length n 1. If b ,a S∗, then for every k K (δ) and every n 0, + 0 n ∈ > ≥ µ(A B) µ(A)µ(B) 2sinh(10δ)µ(A)µ(B). | ∩ − | < Proof. Let α denote the collection of all m-cylinders [c]. For every k 2m, m 0 > X (k m) µ(A B) µ( n[a,c] σ− − 0[c0,b]) ∩ = [c], [c ] α − ∩ 0 0 0 ∈ m X (k n m) µ(0[a,c] σ− + − 0[c0,b]) (shift-invariance) = [c], [c ] α ∩ 0 0 0 ∈ m X (k n m) µ+([a,c] σ− + − [c0,b]). = [c], [c ] α ∩ 0 0 0 ∈ m

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By Observation 2 in the proof of Corollary 3.2,

Z Z X k n m X k n m µ(A B) (L + − 1[a,c])dµ+ (Lφ+ − 1[a,c])dµ+. ∩ =[c], [c ] γ [c0,b] + [c], [c ] α [c0,b] 0 0 0 ∈ 0 0 0 ∈ m [c] γ or [c ] γ 0 6∈ 0 0 6∈ We call the first sum the “main term” and the second sum the “error term.” To estimate these sums we use the following decomposition: for every y [c0,b], ∈

k n m X φ∗ (a,z) φ∗ (z) (L + − 1 )(y) e n 1 e k m 1 1 (z). [a,c] = + − − [c] σk m 1z y − − =

φ∗ (a,z) δ φ∗ (a,w) By the choice of m, e n 1 e± e n 1 for all w,z [c]. Fixing z and + = + ∈ averaging over w [c], we see that ∈ µ 1 Z ¶ φn∗ 1(a,z) δ φn∗ 1(a,w) e + e± e + dµ+(w) = µ+[c] [c] µ Z ¶ δ 1 n 1 e± (L + 1[a,c])dµ+(w) = µ+[c] µ ¶ δ µ+[a,c] e± = µ+[c] therefore

µ ¶ n k m δ µ+[a,c] k m 1 (3.4) (L + − 1[a,c])(y) e± (Lφ− − 1[c])(y) for y [c0,b]. = µ+[c] ∈

Estimate of the main term: Suppose k K (δ). If [c], [c0] γ and y [c0,b], > 0 0 ∈ ∈ then

k m 1 δ k m 1 (L − − 1 )(y) e± (L − − 1 )(x(c0)), by choice of m and since y,x(c0) [c0] [c] = [c] ∈ 2δ e± µ+[c], by choice of K (δ). = After plugging this into (3.4), we see that if k K (δ) then > n k m 3δ (L + − 1 )(y) e± µ+[a,c] φ [a,c] = on [c0,b]. Integrating over [c0,b], we see that for all k K (δ) the main term > equals

à !à ! 3δ X 3δ X X e± µ+[a,c]µ+[c0,b] e± µ+[a,c] µ+[c0,b] . = [c], [c ] γ 0[c] γ [c ] γ 0 0 0 ∈ ∈ 0 0 ∈ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS 601

The first bracketed sum is bounded above by µ+[a]. To bound it below, we use the assumption that a S∗ to write n ∈ X X µ+[a,c] µ+[a] µ+[a,c] = − 0[c] γ 0[c] αm γ,[a,c] ∅ ∈ ∈ à 6= Ã ! X µ+[a] 1 C ∗ µ+[c] = − 0[c] αm γ,[a,c] ∅ ∈ à 6= Ã ! X µ+[a] 1 C ∗ µ+[c] ≥ − [c] α γ 0 ∈ m à 2 ¡ δ/2(C ∗) ¢ µ+[a] 1 C ∗(1 e− ) , by choice of γ ≥ − − δ e− µ+[a], by choice of δ . ≥ 0 δ So the first bracketed sum is equal to e± µ+[a]. Similarly, the second brack- δ 5δ eted sum is equal to e± µ+[b]. Therefore the main term is e± µ+[a]µ+[b] 5δ = e± µ(A)µ(B).

Estimate of the error term: Since a S∗,(3.4) implies that n ∈ n k m δ k m 1 (L + − 1 )(y) C ∗e µ+[a](L − − 1 )(y) on [c0,b]. [a,c] ≤ [c] Therefore the error term is at most Z δ X k m 1 C ∗e µ+[a] (L − − 1[c])(y)dµ+ [c], [c ] α [c0,b] 0 0 0 ∈ m [c] γ or [c ] γ 0 6∈ 0 0 6∈ δ X (k m 1) C ∗e µ+[a] µ+([c] σ− − − [c0,b]) = [c], [c ] α ∩ 0 0 0 ∈ m [c] γ or [c ] γ 0 6∈ 0 0 6∈ µ ¶ δ X (k 1) X (k m 1) C ∗e µ+[a] µ+([c] σ− − [b]) µ+(σ− − − [c0,b]) ≤ ∩ + 0[c] αm γ [c ] α γ ∈ à 0 0 ∈ m à µ ¶ δ X X C ∗e µ+[a] µ+[d,b] µ+[c0,b] = + 0[d] αk 1 0[c0] αm γ ∈ − ∈ à 0[d0,...,dm 1] γ µ − 6∈ ¶ 2 δ X X (C ∗) e µ+[a] µ+[d]µ+[b] µ+[c0]µ+[b] ( b S∗) ≤ + ∵ 0 ∈ 0[d] αk 1 0[c0] αm γ ∈ − ∈ à 0[d0,...,dm 1] γ − 6∈ 2 δ c δ (C ∗) e µ+[a]µ+[b] 2µ[( γ) ] 2e δµ+[a]µ+[b] 5δµ+[a]µ+[b]. ≤ · ∪ ≤ < We get that the error term is less than 5δµ(A)µ(B).

5δ We see that for all k K (δ), µ(A B) (e± 5δ)µ(A)µ(B), whence > ∩ = ± 5δ 5δ µ(A B) µ(A)µ(B) µ(A)µ(B)max{e 5δ 1,1 e− 5δ}. | ∩ − | ≤ + − − + It follows that µ(A B) µ(A)µ(B) 2sinh(10δ)µ(A)µ(B). | ∩ − | ≤ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 602 OMRI M.SARIG

Step 2. For every k K (δ), for every n 0, > ≥ X µ(A B) µ(A)µ(B) 2sinh(10δ) 4δ. 0 k n | ∩ − | < + A α n ,B α + ∈ − ∈ k

Proof. Write A n[a0,...,an] and B k [b0,...,bn]. We break the sum into: = − = 1. the sum over A,B such that a ,b S∗; n 0 ∈ 2. the sum over A,B such that a S∗; n 6∈ 3. the sum over A,B such that a S∗ and b S∗. n ∈ 0 6∈ The first sum is less than 2sinh(10δ). The second and third sums are bounded S c δ by 2µ[( a S 0[a]) ] 2(1 e− ) 2δ. ∈ ∗ < − < Step 3. α(V 0) has the weak Bernoulli property for every finite V 0 V . ⊂ Proof. Choose δ so small that 2sinh(10δ) 4δ ε and take K K (δ) as above, P + < = then 0 k n µ(A B) µ(A)µ(B) ε for all n 0. Since the partitions A α n ,B α + ∈ − ∈ k | ∩ − | < ≥ 0 k n 0 k n α(V 0) n and α(V 0)k+ are coarser than α n and αk+ , the weak Bernoulli prop- − − erty for α(V 0) follows by the triangle inequality. 

4. STEPTHREE: THENONMIXINGCASE

LEMMA 4.1 (Adler, Shields, and Smorodinsky). Let (X ,B,µ,T ) be an ergodic in- vertible probability-preserving transformation with a measurable set X0 of posi- tive measure such that p 1. T (X0) X0 modµ; = p 1 2. X0,T (X0),...,T − (X0) are pairwise disjoint modµ; 3. T p : X X equipped with µ( X ) is a Bernoulli automorphism. 0 → 0 ·| 0 Then (X ,B,µ,T ) is measure-theoretically isomorphic to the product of a Ber- noulli scheme and a finite rotation. i Proof (see [2]). Let Xi : T (X )(i 0,...,p 1). Since T is ergodic and measure- 1 = = p− preserving, µ(Xi ) p for all p. Also, T (Xi ) Xi modµ for all i. Since T is p = = p invertible, T : Xi Xi equipped with µi : µ( Xi ) is isomorphic to T : X0 → p = ·| 1 → X0. It follows that hµ (T ) are all equal. Since µ (µ0 µp 1) and since i = p + ··· + − µ h (T p ) is affine, h (T p ) h (T p ) ph (T ) for every i. 7→ µ µi |Xi = µ = µ Let (Σ,F ,m,S) denote a Bernoulli scheme such that h (S) h (T ). The m = µ map Sp : Σ Σ is isomorphic to a Bernoulli scheme with entropy ph (T ). It → µ follows that Sp is isomorphic to T p : X X . Let ϑ: X Σ be an isomor- 0 → 0 0 → phism map: ϑ T p Sp ϑ. Define: ◦ = ◦ F : {0,1,...,p 1} • p = − R : F F , R(x) x 1 (mod p) • p → p = + Π: X Σ F , Π(x) (Si [ϑ(y)],i) for the unique (y,i) X F such that • → × p = ∈ 0 × p x T i (y) (this makes sense on a set of full measure). = Π is an isomorphism from (X ,B,µ,T ) to (Σ F ,F 2F0 ,m c,S R), where × p ⊗ × × c is 1 the counting measure on F : p × p i 1 i 1. Π is invertible: The inverse function is (z,i) T (ϑ− [S− (z)]). 7→ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS 603

i 2. Π T (S R) Π: Suppose x X , and write x T (y) with (y,i) X0 Fp . ◦ = × ◦ i 1 ∈ = ∈ × If i p 1, then T (x) T + (y) with (y,i 1) X F , so < − = + ∈ 0 × p i 1 i Π[T (x)] (S + [ϑ(y)],i 1) (S R)(S [ϑ(y)],i)) (S R)[Π(x)]. = + = × = × p 1 p p If i p 1, then T (x) T [T − (y)] and (T (y),0) X F . Since ϑ T = − = ∈ 0 × p ◦ = Sp ϑ on X , ◦ 0 Π[T (x)] (ϑ(T p y),0) (Sp [ϑ(y)],R(p 1)) (S R)(Si [ϑ(y)],i) (S R)[Π(x)]. = = − = × = × In all cases, Π T (S R) Π. 1 ◦ = × ◦ 3. µ Π− m c: For every Borel set E Σ and i F , ◦ = × ⊂ ∈ p 1 1 i 1 i (µ Π− )(E {i}) µ[ϑ− S− (E)] µ(X )µ(ϑ− S− (E) X ) ◦ × = = 0 | 0 1 i 1 i 1 µ(X0)(µ0 ϑ− )(S− E) m(S− E) m(E) (m c)(E {i}). = ◦ = p = p = × × It follows that Π is a measure-theoretic isomorphism. 

Proof of Theorem 1.1. Suppose µ is an equilibrium measure with positive en- tropy for f and the Hölder potential Ψ: M R. Fix some 0 χ h (f ). By → < < µ Theorems 2.1 and 2.3, there exists a countable Markov shift σ: Σ Σ, a Hölder- → continuous map π: Σ M, and a shift-invariant ergodic probability measure µb 1→ on Σ such that µ π− µ and h (σ) h (f ). In particular, if ψ : Ψ π, then b ◦ = µb = µ = ◦ h (σ) R ψdµ h (f ) R Ψdµ. µb + b = µ + For any other ergodic shift-invariant probability measure mb , there is a set of full measure Σb Σ such that π: Σb M is finite-to-one (Theorem 2.2). There- ⊂ → 1 fore the f -invariant measure m : m π− has the same entropy as m, whence = b ◦ b Z Z Z Z h (σ) ψdm h (f ) Ψdm h (f ) Ψdµ h (σ) ψdµ. mb + b = m + ≤ µ + = µb + b It follows that µ is an equilibrium measure for σ: Σ Σ and ψ. b → We wish to apply Theorem 3.1. The potential ψ is Hölder-continuous and bounded, and P (ψ) h (f ) R Ψdµ . But σ: Σ Σ may not be topologi- G = µ + < ∞ → cally mixing. To deal with this difficulty we appeal to the Spectral Decomposi- tion Theorem. Since µ is ergodic, it is carried by a topologically transitive Σ0 Σ(G 0) where b = G 0 is a subgraph of G . Let p denote the period of Σ0 (see §2). The Spectral Decomposition Theorem for countable Markov shifts [9, Remark 7.1.35] states that

Σ0 Σ0 Σ10 Σ0p 1 = ] ]···] − p where every Σi0 is a union of states of Σ, σ(Σi0 ) Σi0 1 (mod p), and σ : Σi0 p = + → Σ0 is topologically mixing. Each σ : Σ0 Σ0 is topologically conjugate to the i i → i countable Markov shift Σ(Gi0) where Gi0 is the directed graph with vertices (v0,v1,...,vp 1) where v0 vp 1 is a path in G 0 which starts • − → ··· → − at one of the states in Σi0 , and edges (v0,...,vp 1) (w0,...,wp 1) iff vp 1 w0. • − → − − = JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 604 OMRI M.SARIG

: Let µbi µb( Σi0 ). It is not difficult to see that µbi is an equilibrium measure for p = ·| p 1 σ : Σ0 Σ0 with respect to the potential ψ : ψ ψ σ ψ σ − . It is also i → i p = + ◦ +···+ ◦ not difficult to see that ψp can be identified with a bounded Hölder-continuous i i potential ψp on Σ(Gi0) and that PG (ψp ) pPG (ψ) . p = < ∞ By Theorem 3.1, σ : Σ0 Σ0 equipped with µ is isomorphic to a Bernoulli i → i bi scheme. p Let Xi : π(Σi0 ). Since π σ f π, f (Xi ) Xi 1 (mod p). Each Xi is f - = p ◦ = ◦ = + p invariant, and f : X X equipped with µ : µ( X ) is a factor of σ : Σ0 i → i i = ·| i i → Σi0 . By Ornstein’s Theorem [13], factors of Bernoulli automorphisms are also Bernoulli automorphisms. So f p : X X are Bernoulli automorphisms. i → i In particular, f p : X X are ergodic. Since X X is f p -invariant, ei- i → i i ∩ j ther Xi X j or Xi X j ∅modµ. So there exists q p such that M X0 = ∩ = | q = ] Xq 1 modµ. Since q p, f (Xi ) Xi 1 (mod q), and f : X0 X0 is a root ···] − | = + → of f p : X X . Since f p is Bernoulli, f q is Bernoulli [15]. By Lemma 4.1, 0 → 0 (M,B(M),µ, f ) is isomorphic to the product of a Bernoulli scheme and a finite rotation. 

5. CONCLUDINGREMARKS We discuss some additional consequences of the proof we presented in the 1 α previous sections. In what follows f : M M is a C + diffeomorphism on a → compact smooth orientable surface. We assume throughout that the topologi- cal entropy of f is positive. 5.1. The measure of maximal entropy is virtually Markov. Equilibrium mea- sures for Ψ 0 are called measures of maximal entropy for obvious reasons. ≡ A famous theorem of Adler and Weiss [1] says that an ergodic measure of maximal entropy µ for a hyperbolic toral automorphism f : T2 T2 can be max → coded as a finite state . More precisely, there exists a subshift of finite type σ: Σ Σ and a Hölder-continuous map π: Σ T2 such that (a) → 1 → π σ f π; (b) µ µ π− where µ is an ergodic Markov measure on ◦ = ◦ max = bmax ◦ bmax Σ; and (c) π is a measure-theoretic isomorphism. This was extended by Bowen [5] to all diffeomorphisms, using Parry’s characterization of the measure of maximal entropy for a subshift of finite type [18]. Bowen’s result holds in any dimension. 1 α In dimension two, we have the following generalization to general C + sur- face diffeomorphisms with positive :

THEOREM 5.1. If µmax is an ergodic measure of maximal entropy for f , then there exists a topologically transitive countable Markov shift σ: Σ Σ and a → 1 Hölder-continuous map π: Σ M such that (a) π σ f π; (b) µ µ π− → ◦ = ◦ max = bmax◦ where µ is an ergodic Markov measure on Σ; and (c) Σ0 Σ of full measure bmax ∃ ⊂ such that π is n-to-one. |Σ0 1 Proof. The arguments in the previous section show that µ µ π− where max = bmax◦ µbmax is an ergodic measure of maximal entropy on some topologically transitive countable Markov shift Σ(G ) and π: Σ(G ) M is Hölder-continuous map such → JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS 605 that π σ f π and such that π is finite-to-one on a set of full µbmax-measure. ◦ = 1◦ Since x π− (x) is f -invariant, π is n-to-one on a set of full measure for some 7→ | | n N. ∈ Gurevich’s Theorem [8] says that µbmax is a Markov measure. Ergodicity forces the support of µbmax to be a topologically transitive countable Markov subshift of Σ(G ).

The example mentioned in the introduction shows that the theorem is false in dimension larger than two.

5.2. Equilibrium measures for t log J . Theorem 1.1 was stated for equilib- − u rium measures µ of Hölder-continuous functions Ψ: M R, but the proof → works equally well for any function Ψ such that ψ : Ψ π is a bounded Hölder- = ◦ χ continuous function on Σχ. Here χ is any positive number strictly smaller than h (f ), and π : Σ M is the Markov extension described in §2. µ χ χ → We discuss a particular example which appears naturally in hyperbolic dy- namics (see e.g., [3, 10,4]). Let M 0 denote the set of x M such that T M splits into the direct sum of ∈ x two one-dimensional spaces E s(x) and E u(x) such that

1 n s limsup log d fx v f n (x) 0 for all v E (x) {0}, n n k k < ∈ à →∞ and 1 n u n limsup log d fx− v f − (x) 0 for all v E (x) {0}. n n k k < ∈ à →∞ It is well-known that if the spaces E s(x), E u(x) exist, then they are unique, and d f [E u(x)] E u(f (x)), d f [E s(x)] E s(f (x)). x = x = DEFINITION 5.1. The unstable Jacobian is J (x) : det(d f u ) (x M 0). u = | x |E (x) | ∈ Equivalently, Ju(x) is the unique positive number such that d f (v) J (x) v for all v E u(x). k x kf (x) = u k kx ∈ Notice that Ju(x) is only defined on M 0. Oseledets’ Theorem and Ruelle’s Entropy Inequality guarantee that µ(M M 0) 0 for every f -ergodic invariant à = measure with positive entropy. The maps x E u(x), x E s(x) are in general not smooth. Brin’s Theorem 7→ 7→ states that these maps are Hölder-continuous on Pesin sets [3, §5.3]. Therefore Ju(x) is Hölder-continuous on Pesin sets. We have no reason to expect Ju(x) to extend to a Hölder-continuous function on M. Luckily, the following holds [23, Proposition 12.2.1]: For the Markov exten- sion π : Σ M, E u(π(u)),E s(π(u)) are well-defined for every u Σ, and the χ χ → ∈ maps u E u(u),u E s(u) are Hölder-continuous on Σ . As a result J π is a 7→ 7→ χ u ◦ globally defined bounded Hölder-continuous function on Σχ. Since f is a diffeomorphism, log(J π) is also globally defined, bounded and u ◦ Hölder-continuous.

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 606 OMRI M.SARIG R THEOREM 5.2. Suppose µ maximizes h (f ) t (log J )dµ among all ergodic µ − u invariant probability measures carried by M 0. If h (f ) 0, then f is measure- µ > theoretically isomorphic with respect to µ a Bernoulli scheme times a finite rota- tion.

The case t 1 follows from the work of Ledrappier [10], see also Pesin [19]. =

5.3. How many ergodic equilibrium measures with positive entropy?

THEOREM 5.3. A Hölder-continuous potential on M has at most countably many ergodic equilibrium measures with positive entropy.

Proof. Fix Ψ: M R Hölder-continuous (more generally a function such that → ψ defined below is Hölder-continuous). Given 0 χ h (f ), we show that Ψ has at most countably many ergodic < < top equilibrium measures µ such that h (f ) χ. µ > Let π : Σ M denote the Markov extension described in §2, and let G de- χ χ → note the directed graph such that Σ Σ(G ). We saw in the proof of Theorem χ = 1.1 that every ergodic equilibrium measure µ for Ψ such that h (f ) χ is the µ > projection of some ergodic equilibrium measure for ψ : Ψ π : Σ(G ) R. So = ◦ χ → it is enough to show that ψ has at most countably many ergodic equilibrium measures. Every ergodic equilibrium measure µ on Σ(G ) is carried by Σ(H ) where (i) H is a subgraph of G , (ii) σ: Σ(H ) Σ(H ) is topologically transitive, and (iii) → Σ(H ) carries an equilibrium measure for ψ: Σ(G ) R. Take the subgraph with → vertices a such that µ( [a]) 0 and edges a b such that µ( [a,b]) 0. 0 6= → 0 6= For every subgraph H satisfying (i),(ii), and (iii) there is exactly one equi- librium measure for ψ on Σ(H ). The support of this measure is Σ(H ), see Corollary 3.4 and Theorem 3.1. So every ergodic equilibrium measure sits on Σ(H ) where H satisfies (i), (ii), and (iii), and every such Σ(H ) carries exactly one measure like that. As a result, it is enough to show that G contains at most countably many subgraphs H satisfying (i), (ii), and (iii). We do this by showing that any two different subgraphs H1, H2 like that have disjoint sets of vertices. Assume by contradiction that H1,H2 share a ver- tex. Then H : H H satisfies (i), (ii), and (iii). By the discussion above, = 1 ∪ 2 Σ(H ) carries at most one equilibrium measure for ψ. But it carries at least two such measures: one with support Σ(H1) and one with support Σ(H2). This contradiction shows that H1 and H2 cannot have common vertices.

The case Ψ log J is due to Ledrappier [10] and Pesin [19]. The case Ψ 0 = − u ≡ was done at [23]. Buzzi [6] had shown that the measure of maximal entropy of a piecewise affine surface homeomorphism has finitely many ergodic compo- nents, and has conjectured that a similar result holds for C ∞ surface diffeomor- phisms with positive topological entropy.

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Acknowledgments. The author wishes to thank A. Katok and Y. Pesin for the suggestion to apply the results of [23] to the study of the Bernoulli property of surface diffeomorphisms with respect to measures of maximal entropy and equilibrium measures of t log J . − u

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OMRI M.SARIG : Faculty of Mathematics and Computer Sci- ence, The Weizmann Institute of Science, POB 26, Rehovot, Israel

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