Bernoulli equilibrium states for surface diffeomorphisms

Suppose f is a $C^{1+\alpha}$ surface diffeomorphism, and m is an equilibrium measure of a Holder continuous potential. We show that if m has positive metric entropy, then f is measure theoretically isomorphic to the product of a Bernoulli scheme and a finite rotation.

The work of Pesin and Ledrappier (see also [OW]) uses the following property of equilibrium measures of − log J u : the conditional measures on unstable manifolds are absolutely continuous [L]. This is false for general Hölder potentials [LY].
Theorem 1.1 is proved in three steps: (1) Symbolic dynamics: Any ergodic equilibrium measure on M with positive entropy is a finite-to-one Hölder factor of an ergodic equilibrium measure on a countable Markov shift (CMS).

2.
Step One: Symbolic dynamics Let G be a directed graph with a countable collection of vertices V s.t. every vertex has at least one edge coming in, and at least one edge coming out. The countable Markov shift (CMS) associated to G is the set The natural metric d(u, v) := exp[− min{|i| : u i = v i }] turns Σ into a complete 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 m [a m , . . . , a n ] := {(v i ) i∈Z ∈ Σ : v i = a i (i = m, . . . , n)} (2.1) 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[(v i ) i∈Z ] = (v i+1 ) i∈Z . Given a, b ∈ V , write a n − → b when there is a path a → v 1 → · · · → v n−1 → b in G . The left shift is topologically transitive iff ∀a, b ∈ V ∃n (a n − → b). In this case gcd{n : a n − → a} is the same for all a ∈ V , and is called the period of σ. The left shift is topologically mixing iff it is topologically transitive and its period is equal to one. See [K].
Suppose f : M → M is a C 1+α -diffeomorphism of a compact orientable smooth manifold M s.t. dim M = 2. If h top (f ) = 0, then every f -invariant measure has zero entropy by the variational principle, and Theorem 1.1 holds trivially. So we assume without loss of generality that h top (f ) > 0. Fix 0 < χ < h top (f ).
A set Ω ⊂ M is called χ-large, if µ(Ω) = 1 for every ergodic invariant probability measure µ whose entropy is greater than χ. The following theorems are in [S2]: Theorem 2.1. There exists a locally compact countable Markov shift Σ χ and a Hölder continuous map π χ : has finitely many pre-images. Theorem 2.2. Denote the set of states of Σ χ by V χ . There exists a function and v i = u for infinitely many negative i, and v i = v for infinitely many positive i, then |π −1 χ (x)| ≤ ϕ χ (u, v). Theorem 2.3. Every ergodic f -invariant probability measure µ on M such that h µ (f ) > χ equals µ • π −1 χ for some ergodic σ-invariant probability measure µ on Σ χ with the same entropy.
We will use these results to reduce the problem of Bernoullicity for equilibrium measures for f : M → M and the potential Ψ, to the problem of Bernoullicity for equilibrium measures for σ : Σ χ → Σ χ and the potential ψ := Ψ • π χ .

3.
Step Two: Ornstein Theory First we describe the structure of equilibrium measures of Hölder continuous potentials on countable Markov shifts (CMS), 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 CMS [BS]. Suppose G is countable directed graph. The one-sided countable Markov shift associated to G is Proceeding as in the two-sided case, we equip Σ + with the metric d (u, v) 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.
Notice that φ * need not be bounded.
and (3.2) imply that log h is weakly Hölder continuous, and var 1 (log h) < ∞. It follows that φ * is weakly Hölder continuous. The identities L * µ = µ, L1 = 1 can be verified by direct calculation. To see we argue as follows. Since φ has an equilibrium measure, φ is positive recurrent, see [BS]. Positive recurrence is invariant under the addition of constants and coboundaries, so φ * is also positive recurrent. The limit now follows from a theorem in [S1].
Notice that if ψ is bounded then φ is bounded, and that every equilibrium measure for ψ is an equilibrium measure for φ and vice verse. Since Any shift invariant probability measure µ on Σ determines a shift invariant probability measure µ + on Σ + through the equations µ + [a 0 , . . . , a n−1 ] := µ( 0 [a 0 , . . . , a n−1 ]) (cf. (2.1) and (3.1)). The map µ → µ + is a bijection, and it preserves ergodicity and entropy. It follows that µ is an ergodic equilibrium measure for φ iff µ + is an ergodic equilibrium measure for φ + .
3.3. The Bernoulli property. The Bernoulli scheme with probability vector p = (p a ) a∈S is (S Z , B(S Z ), µ p , σ) where σ is the left shift map and µ p is given by µ p ( m [a m , . . . , a n ]) = p am · · · p an . If (Ω, F , µ, T ) is measure theoretically isomorphic to a Bernoulli scheme, then we say that (Ω, F , µ, T ) is a Bernoulli automorphism, and µ has the Bernoulli property. In this section we prove: Theorem 3.1. Every equilibrium measure of a weakly Hölder continuous potential ψ : Σ(G ) → R on a topologically mixing countable Markov shift s.t. P G (ψ) < ∞ and sup ψ < ∞ has the Bernoulli property.
This was proved by Bowen [B1] in the case when G is finite. See [Ra] and [W3] for generalizations to larger classes of potentials.
We need some facts from Ornstein Theory. Suppose β = {P 1 , . . . , P N } is a finite measurable partition for an invertible probability preserving map (Ω, F , µ, T ). For every m, n ∈ Z s.t. m < n, let β n m := Ornstein showed that if an invertible probability preserving transformation has a generating increasing sequence of weak Bernoulli partitions, then it is measure theoretically isomorphic to a Bernoulli scheme [O1,OF].
Proof of Theorem 3.1. First we make a reduction to the case when var 1 ψ < ∞.
To do this, recode Σ(G ) using the Markov partition of cylinders of length two and notice that var 1 of the new coding equals var 2 of the original coding. The supremum and the pressure of ψ remain finite, and the variations of ψ continue to decay exponentially. Suppose µ is an equilibrium measure of ψ : . We claim that α(V ′ ) is weak Bernoulli. This implies the Bernoulli property, because of the results of Ornstein we cited above.
To estimate these sums we use the following decomposition: for every y ∈ [c ′ , b], . By the choice of m, e φ * n+1 (a,z) = e ±δ e φ * n+1 (a,w) for all w, z ∈ [c]. Fixing z and averaging over w ∈ [c], we see that Estimate of the main term: The first bracketed sum is bounded above by µ + [a]. To bound it below, we use the assumption that a n ∈ S * to write Estimate of the error term: Since a n ∈ S * , (3.4) implies that We get that the error term is less than 5δµ(A)µ(B).
Step 3. α(V ′ ) has the weak Bernoulli property for every finite V ′ ⊂ V .
Then (X, B, µ, T ) is measure theoretically isomorphic to the product of a Bernoulli scheme and a finite rotation.
Proof (see [ASS]). Let X i := T i (X) (i = 0, . . . , p − 1). Since T is ergodic and measure preserving, µ(X i ) = 1 p for all p. Also, T p (X i ) = X i mod µ for all i.
Let (Σ, F , m, S) denote a Bernoulli scheme s.t. h m (S) = h µ (T ). The map S p : Σ → Σ is isomorphic to a Bernoulli scheme with entropy ph µ (T ). It follows that S p is isomorphic to T p : X 0 → X 0 . Let ϑ : X 0 → Σ be an isomorphism map: ϑ • T p = S p • ϑ. Define: x = T i (y) (this makes sense on a set of full measure).
It follows that Π is a measure theoretic isomorphism.
To deal with this difficulty we appeal to the spectral decomposition theorem.
Since µ is ergodic, it is carried by a topologically transitive Σ ′ = Σ(G ′ ) where G ′ is a subgraph of G . Let p denote the period of Σ ′ (see §2). The Spectral Decomposition Theorem for CMS [K,Remark 7.1.35] states that . It is not difficult to see that µ i is an equilibrium measure for σ p : Σ ′ i → Σ ′ i with respect to the potential ψ p := ψ + ψ • σ + · · · + ψ • σ p−1 . It is also not difficult to see that ψ p can be identified with a bounded Hölder continuous potential ψ i p on Σ(G ′ i ) and that P G (ψ i p ) = pP G (ψ) < ∞. By Theorem 3.1, σ p : [O1], factors of Bernoulli automorphisms are Bernoulli automorphisms. So f p : X i → X i are Bernoulli automorphisms.
In particular, f p : [O3]. By Lemma 4.1, (M, B(M ), µ, f ) is isomorphic to the product of a Bernoulli scheme and a finite rotation.

Concluding remarks
We discuss some additional consequences of the proof we presented in the previous sections. In what follows f : M → M is a C 1+α surface diffeomorphism on a compact smooth orientable surface. We assume throughout that the topological entropy of f is positive.
5.1. The measure of maximal entropy is virtually Markov. Equilibrium measures for Ψ ≡ 0 are called measures of maximal entropy for obvious reasons.
A famous theorem of Adler & Weiss [AW] says that an ergodic measure of maximal entropy µ max for a hyperbolic toral automorphism f : T 2 → T 2 can be coded as finite state Markov chain. More precisely, there exists a subshift of finite type σ : Σ → Σ and a Hölder continuous map π : Σ → T 2 such that (a) π •σ = f •π; (b) µ max = µ max • π −1 where µ max is an ergodic Markov measure on Σ; and (c) π is a measure theoretic isomorphism.
This was extended by Bowen [B2] to all Axiom A diffeomorphisms, using Parry's characterization of the measure of maximal entropy for a subshift of finite type [Pa]. Bowen's result holds in any dimension.
Proof. The arguments in the previous section show that µ max = µ max • π −1 where µ max 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 s.t. π • σ = f • π and such that π is finite-to-one on a set of full µ max -measure. Since x → |π −1 (x)| is f -invariant, π is n-to-one on a set of full measure for some n ∈ N.
Gurevich's Theorem [G] says that µ max is a Markov measure. Ergodicity forces the support of µ max to be a topologically transitive sub-CMS 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 u . Theorem 1.1 was stated for equilibrium measures µ of Hölder continuous functions Ψ : M → R, but the proof works equally well for any function Ψ s.t. ψ := Ψ • π χ is a bounded Hölder continuous function on Σ χ . Here χ is any positive number strictly smaller than h µ (f ), and π χ : Σ χ → M is the Markov extension described in §2.
Let M ′ denote the set of x ∈ M s.t. T x M splits into the direct sum of two one-dimensional spaces E s (x) and E u (x) so that lim sup Notice that J u (x) is only defined on M ′ . Oseledets' Theorem and Ruelle's Entropy Inequality guarantee that µ(M \ M ′ ) = 0 for every f -ergodic invariant measure with positive entropy.
The maps x → E u (x), x → E s (x) are in general not smooth. Brin's Theorem states that these maps are Hölder continuous on Pesin sets [BP,§5.3]. Therefore J u (x) is Hölder continuous on Pesin sets. We have no reason to expect J u (x) to extend to a Hölder continuous function on M .
Since f is a diffeomorphism, log(J u • π) is also globally defined, bounded and Hölder continuous.
Theorem 5.2. Suppose µ maximizes h µ (f ) − t (log J u )dµ among all ergodic invariant probability measures carried by M ′ . If h µ (f ) > 0, then f is measure theoretically isomorphic w.r.t. µ a Bernoulli scheme times a finite rotation.
The case t = 1 follows from the work of Ledrappier [L], see also Pesin [Pe].
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).
Let π χ : Σ χ → M denote the Markov extension described in §2, and let G denote the directed graph s.t. Σ χ = Σ (G ). We saw in the proof of Theorem 1.1 that every ergodic equilibrium measure µ for Ψ s.t. h µ (f ) > χ is the projection of some ergodic equilibrium measure for ψ := Ψ • π χ : Σ(G ) → R. So it is enough to show that ψ has at most countably many ergodic equilibrium measures.
Every For every subgraph H satisfying (i),(ii), and (iii) there is exactly one equilibrium 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 H 1 , H 2 like that have disjoint sets of vertices. Assume by contradiction that H 1 , H 2 share a vertex. Then H := H 1 ∪ H 2 satisfies (i), (ii), and (iii). By the discussion above, Σ(H ) carries at most one equilibrium measure for ψ. But it carries at least two such measures: one with support Σ(H 1 ) and one with support Σ(H 2 ). This contradiction shows that H 1 and H 2 cannot have common vertices.
The case Ψ = − log J u is due to Ledrappier [L] and Pesin [Pe]. The case Ψ ≡ 0 was done at [S2]. Buzzi [Bu] had shown that the measure of maximal entropy of a piecewise affine surface homeomorphism has finitely many ergodic components, and has conjectured that a similar result holds for C ∞ surface diffeomorphisms with positive topological entropy. 5.4. Acknowledgements. The author wishes to thank A. Katok and Y. Pesin for the suggestion to apply the results of [S2] 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 .