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 ® (® 0) diffeomorphism on a com- ! Å È pact smooth orientable manifold M of dimension 2, and let ¹ª be an equi- librium measure for a Hölder-continuous potential ª: M R. We show that ! if ¹ª has positive measure-theoretic entropy, then f is measure-theoretically isomorphic mod ¹ª to the product of a Bernoulli scheme and a finite rota- tion. 1. STATEMENTS 1 ® Suppose f : M M is a C Å (® 0) diffeomorphism on a compact smooth ! È orientable manifold M of dimension two and ª: M R is Hölder-continuous. ! An invariant probability measure ¹ is called an equilibrium measure, if it max- imizes the quantity h (f ) R ªd¹, where h (f ) is the measure-theoretic en- ¹ Å ¹ tropy. Such measures always exist when f is C 1, because in this case the func- tion ¹ h (f ) is upper semicontinuous [12]. Let ¹ be an ergodic equilibrium 7! ¹ ª measure of ª. We prove: THEOREM 1.1. If h (f ) 0, then f is measure-theoretically isomorphic with ¹ª È respect to ¹ª 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! Å ¡ In the particular case of the measure of maximal entropy (ª 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 2 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 1 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. Symbolic dynamics: 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. Æ · · 2. STEPONE: SYMBOLIC DYNAMICS 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 § §(G ) : {(vi )i Z V : vi vi 1 for all i}. Æ Æ 2 2 ! Å The natural metric d(u,v) : exp[ min{ i : u v }] turns § into a complete Æ ¡ j j i 6Æ i separable metric space. § is compact iff G is finite. § is locally compact iff every vertex of G has finite degree. The cylinder sets (2.1) m[am,...,an] : {(vi )i Z § : vi ai (i m,...,n)} Æ 2 2 Æ Æ form a basis for the topology, and they generate the Borel σ-algebra B(§). The left shift map σ: § § is defined by σ[(vi )i Z] (vi 1)i Z. Given a,b n ! 2 Æ Å 2 2 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 8 2 9 ¡! ¡! the same for all a V , and is called the period of σ. The left shift is topolog- 2 ically mixing iff it is topologically transitive and its period is equal to one. See [9]. Let # § : {(vi )i Z § : u,v V nk ,mk such that v m u,vn v}. Æ 2 2 9 2 9 " 1 ¡ k Æ k Æ Every σ-invariant probability measure gives §# full measure, because of Poin- caré’s Recurrence Theorem. JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS 595 1 ® Suppose f : M M is a C Å -diffeomorphism of a compact, orientable and ! smooth manifold M such that dimM 2. If h (f ) 0, then every f -invariant Æ top Æ 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 È Fix 0  h (f ). A set ­ M is said to be Â-large, if ¹(­) 1 for every Ç Ç top ½ Æ ergodic invariant probability measure ¹ whose entropy is greater than Â. The following theorems are in [23]: THEOREM 2.1. There exists a locally compact countable Markov shift §Â and a # Hölder-continuous map ¼Â : §Â M such that ¼Â σ f ¼Â, ¼Â[§Â] is Â-large, # ! ± Æ ± and every point in ¼Â[§Â] has finitely many preimages. THEOREM 2.2. Denote the set of states of §Â by VÂ. There exists a function ' : V V N such that if x ¼Â[(vi )i Z] and vi u for infinitely many neg- £ ! Æ 2 Æ 1 ative i, and v v for infinitely many positive i, then ¼¡ (x) ' (u,v). i Æ j  j ·  THEOREM 2.3. Every ergodic f -invariant probability measure ¹ on M such that 1 h (f )  equals ¹ ¼¡ for some ergodic σ-invariant probability measure ¹ on ¹ È b ±  b §Â 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 ª, to the problem of Bernoul- ! licity for equilibrium measures for σ: § § and the potential à : ª ¼ .  ! Â Æ ±  3. STEP TWO: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} §Å §Å(G ) : {(vi )i 0 V [ : vi vi 1 for all i}. Æ Æ ¸ 2 ! Å Proceeding as in the two-sided case, we equip §Å with the metric d(u,v) : Æ exp[ min{i 0 : u v }]. The cylinder sets ¡ ¸ i 6Æ i (3.1) [a0,...,an 1] : {u §Å : ui ai (i 0,...,n 1)} ¡ Æ 2 Æ Æ ¡ form a basis for the topology of §Å. 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: §Å §Å is given by σ :(v ,v ,...) (v ,v ,...). This ! 0 1 7! 1 2 map is not invertible. The natural extension of (§Å,σ) is conjugate to (§,σ). A function Á: §Å R is said to be weakly Hölder-continuous if there are con- ! stants C 0 and θ (0,1) such that var Á Cθn for all n 2, where È 2 n Ç ¸ var Á : sup{Á(u) Á(v): u v (i 0,...,n 1)}. n Æ ¡ i Æ i Æ ¡ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 596 OMRI M. SARIG The following inequality holds: Ãn 1 ! X¡ j X1 (3.2) varn m Á σ varj Á. Å j 0 ± · j m 1 Æ Æ Å If Á 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: § § is topologically mixing. ! (b) Á is weakly Hölder-continuous and supÁ . (This can be relaxed [7].) Ç 1 (c) P (Á) : sup{h (σ) R Ádm} , where the supremum ranges over all G Æ m Å Ç 1 shift-invariant measures m such that h (σ) R Ádm . The po- m Å 6Æ 1 ¡ 1 tentials we will study satisfy P (Á) h (f ) max ª .