INTRINSIC FOR CERTAIN PARTIALLY HYPERBOLIC DERIVED FROM ANOSOV SYSTEMS

JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´

Abstract. We show that a class of robustly transitive diffeomor- phisms originally described by Ma˜n´eare intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have constant and isomorphic unique measures of maxi- mal entropy and periodic orbits are equidistributed with respect to that measure. Additionally, we show the method applies to sev- eral classes of systems derived from Anosov, i.e., produced by an isotopy from an Anosov system, namely, a mixed Ma˜n´eexample and one obtained through a Hopf bifurcation.

1. Introduction Let f be a diffeomorphism of a M to itself. The diffeomor- phism f is transitive if there exists a point x ∈ M where + n Of (x)= {f (x)|n ∈ N} is dense in M. It is robustly transitive [3, Ch. 7] if there exists a neighborhood U of f in the space Diff1(M) of C1 diffeomorphisms such that each g in U is transitive. Since robust transitivity is an open condition, it is an important component of the global picture of dynamical systems [22]. The first examples of robustly transitive diffeomorphisms were tran- sitive Anosov diffeomorphisms: recall that a diffeomorphism is Anosov if the entire manifold is a under the action of the dif- feomorphism. Nonhyperbolic robustly transitive diffeomorphisms were first constructed by Shub [25] and Ma˜n´e[17]. These examples sat- isfy a weaker hyperbolic condition called partial hyperbolicity (see Sec. 2). It is interesting to note when results for Anosov diffeomorphisms

Date: June, 19, 2008. 2000 Subject Classification. 37C40, 37A35, 37C15. Key words and phrases. Measures of maximal entropy, topological entropy, ro- bust ergodicity, , partial hyperbolicity, intrinsic ergodicity. 1 2 JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´ continue to hold and when the properties are very different. For in- stance, C1- holds for systems with strong transversality and no others [17]. In this paper we analyze measures of maximal entropy, and a related notion of stability, for some class of non-Anosov robustly transitive diffeomorphisms based on Ma˜n´e’s ex- ample. Our method applies to some systems derived by isotopy from a C0-close Anosov system. To state our results we need to give some definitions. Dynamical entropies are measures of the of structures [5]. The topological entropy, htop(f), considers all the orbits, whereas the mea- sure theoretic entropy, hµ(f), focuses on those “relevant” to a given invariant probability measure µ. The variational principle (see for ex- ample [15, p. 181]) says that if f is a continuous self-map of a compact metrizable space and M(f) is the set of invariant probability measures for f, then htop(f) = sup hµ(f). µ∈M(f)

A measure µ ∈M(f) such that htop(f)= hµ(f)is a measure of maxi- mal entropy. By a theorem of Newhouse [20] we know that C∞ smooth- ness implies the existence of such measures (but finite smoothness does not according to Misiurewicz [19]). If there is a unique measure of max- imal entropy, then f is called intrinsically ergodic. One can then inquire about the following weakening of structural stability: Definition 1.1. We say f ∈ Diff1(M) is intrinsically stable if there exists a neighborhood U of f such that each g in U has a unique measure of maximal entropy µg and all µg define isomorphic measure-preserving transformations. The following property also holds for maximal entropy measures of Anosov systems:

Definition 1.2. Let ǫ> 0 and Perε(n) be an (ε, n)-separated subset of the set of periodic points. A diffeomorphism f is said to have equidis- tributed periodic points with respect to a measure µ if the following holds for any small enough ε> 0: 1 lim δx = µ. n→∞ |Perε(n)| x∈PerXε(n) Newhouse and Young [21] have shown that the robustly transitive diffeomorphisms constructed by Shub on T4 are intrinsically stable (and in particular intrinsically ergodic). The present work extends this to several derived from Anosov systems and in particular the robustly INTRINSIC ERGODICITY 3 transitive diffeomorphisms constructed by Ma˜n´eon T3 whose construc- tion will be recalled in Section 4. Theorem 1.3. For any d ≥ 3, there exists a non-empty open set U in Diff(Td) satisfying: • each f ∈ U is strongly partially hyperbolic, robustly transitive, and intrinsically stable (in particular the topological entropy is locally constant at f); • each f ∈ U has equidistributed periodic points; • no f ∈ U is Anosov or structurally stable. Indeed, the above result follows from a simple abstract result and we will apply it to several classes that are derived from Anosov, i.e., C0-close and obtained from a perturbation, namely: • a mixed Ma˜n´e example (see Sec. 5) • a system derived from Anosov through a Hopf bifurcation (see Sec. 6) A forthcoming work [24] of MS and CV will extend this to equilibrium states of such systems. Looking from another point of view raises the following question. Question 1.4. Is every robustly transitive diffeomorphism intrinsically ergodic? intrinsically stable? We note that examples of Kan [3, 14] suggest that the answer might be negative. These are robustly transitive systems within C1 self-maps of the compact cylinder preserving the boundary which admits two SRB measures on the boundary that are also measures of maximal entropy. In dimension three we know that every robustly transitive system is partially hyperbolic [3, p. 128]. The added structure of partial hyperbolicity could help solve the above question in the affirmative for 3-. In a follow up paper [9] of JB and TF, we analyze a set of robustly transitive diffeomorphisms on T4, based on examples of Bonatti and Viana [4] which have the weakest possible form of hyperbolicity for robustly transitive diffeomorphisms, a dominated splitting [2, 11] (see Sec. 2). We note that Hua, Saghin, and Xia [13] have also proved local con- stancy of the topological entropy, for instance in the case of partially hyperbolic diffeomorphisms C1 close to toral automorphisms with at most one eigenvalue on the unit circle have locally constant entropy. 4 JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´

Acknowledgment. We thank Jean-Paul Allouche and Lennard Bakker for their help on Pisot numbers, Sheldon Newhouse for helpful discus- sions, and Marcelo Viana for his comments and orientation.

2. Background We now review a few facts on entropy, hyperbolicity, and partial hyperbolicity. Let X be a compact metric space and f be a continuous self-map of X. Fix ǫ> 0 and n ∈ N. Let cov(n, ǫ, f) be the minimum cardinality of a covering of X by (ǫ, n)-balls, i.e., sets of the form

{y ∈ X : d(f k(y), f k(x)) < ǫ for all 0 ≤ k ≤ n}. The topological entropy is [5] 1 htop(f) = lim(lim sup log cov(n, ǫ, f)). ǫ→0 n→∞ n Let Y ⊂ X and cov(n,ǫ,f,Y ) be the minimum cardinality of a cover of Y by (n, ǫ)-balls. Then the topological entropy of Y with respect to f is 1 htop(f,Y ) = lim lim sup log cov(n,ǫ,f,Y ). ǫ→0 n→∞ n If (X, f)and(Y,g) are continuous and compact systems and φ : X → Y is a continuous surjection such that φ◦f = g ◦φ, then htop(g) ≤ htop(f) (f is called an extension of g and g is called a factor of f). For the definition of measure theoretic entropy refer to [15, p. 169]. An invariant set Λ is hyperbolic for f ∈ Diff(M) if there exists an s u invariant splitting TΛM = E ⊕ E and an integer n ≥ 1 such that Df n uniformly contracts Es and uniformly expands Eu: for any point x ∈ Λ, n 1 s kDfx vk ≤ 2 kvk, for v ∈ Ex, and −n 1 u kDfx vk ≤ 2 kvk, for v ∈ Ex . If A ∈ GL(d, Z) has no eigenvalues on the unit circle, then the in- duced map fA of the d-torus is called a hyperbolic toral automorphism. By construction any hyperbolic toral automorphism is Anosov. If Λ is a hyperbolic set, x ∈ Λ, and ǫ> 0 sufficiently small, then the local stable and unstable manifolds at x are respectively:

s n n Wǫ (x, f)= {y ∈ M | for all n ∈ N,d(f (x), f (y)) ≤ ǫ}, and u −n −n Wǫ (x, f)= {y ∈ M | for all n ∈ N,d(f (x), f (y)) ≤ ǫ}. INTRINSIC ERGODICITY 5

The stable and unstable manifolds of x are respectively: W s(x, f)= {y ∈ M | lim d(f n(y), f n(x))=0}, and n→∞ W u(x, f)= {y ∈ M | lim d(f −n(y), f −n(x))=0}. n→∞ They can be obtained from the local manifolds as follows: s −n s n W (x, f)= n≥0 f (Wǫ (f (x), f)) , and u n u −n W (x, f)= Sn≥0 f (Wǫ (f (x), f)) . For a Cr diffeomorphism theS stable and unstable manifolds of a hyper- bolic set are Cr injectively immersed submanifolds. An ǫ-chain from a point x to a point y for a diffeomorphism f is a sequence {x = x0, ..., xn = y} such that

d(f(xj−1), xj) < ǫ for all 1 ≤ j ≤ n. A standard result that applies to Anosov diffeomorphisms is the Shad- j2 owing Theorem, see for example [23, p. 415]. Let {xj}j=j1 be an j2 j ǫ-chain for f. A point y δ-shadows {xj}j=j1 provided d(f (y), xj) < δ for j1 ≤ j ≤ j2. We remark that there are much more general versions of the next theorem, but the following statement will be sufficient for the present work. Theorem 2.1. (Shadowing Theorem) If f is an Anosov diffeomor- phism, then given any δ > 0 sufficiently small there exists an ǫ > 0 j2 such that if {xj}j=j1 is an ǫ-chain for f, then there is a y which δ- j2 shadows {xj}j=j1 . If j2 = −j1 = ∞, then y is unique. If, moreover, the ǫ-chain is periodic, then y is periodic. A diffeomorphism f : M → M has a dominated splitting if there exists an invariant splitting T M = E1 ⊕··· Ek, k ≥ 2, (with no trivial subbundle) and an integer l ≥ 1 such that for each x ∈ M, i < j, and unit vectors u ∈ Ei(x) and v ∈ Ej(x), one has kDf l(x)uk 1 < . kDf l(x)vk 2 A diffeomorphism f is partially hyperbolic if there is a dominated split- n ting T M = E1 ⊕···⊕ Ek and n ≥ 1 such that Df either uniformly contracts E1 or uniformly expands Ek. We say f is strongly partially hyperbolic if there exists a dominated splitting T M = Es ⊕Ec ⊕Eu and n ≥ 1 such that Df n uniformly contracts Es and uniformly expands Eu. For f a strongly partially hyperbolic diffeomorphism we know there exist unique families F u and F s of injectively immersed submanifolds 6 JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´ such that F i(x) is tangent to Ei for i = s,u, and the families are invariant under f, see [12]. These are called, respectively, the unstable and stable laminations1 of f. For the center direction, however, there are examples where there is no center lamination [27]. For a strongly partially hyperbolic diffeomorphism with a 1-dimensional center bundle it is not known if there is always a lamination tangent to the center bundle, and that if there is a C1 center foliation, then it is structurally stable [12]. Let us quote a special case of this result: Theorem 2.2. [12, Theorems (7.1) and (7.2)] Let f be a C1 diffeomor- phism of a compact manifold M. If f is strongly partially hyperbolic with a C1 central foliation F, then any g C1-close to f also has a C1 central lamination G and there is a homeomorphism h : M → M such that for all x ∈ M, (i) the leaf Fx is mapped by h to the leaf Ghx; (ii) g(Ghx)= Gh(fx). This applies in particular to the Ma˜n´eexample.

3. A sufficient criterion to be intrinsically ergodic Let X be a compact metric space and f : X → X be an expansive homeomorphism with the specification property (see [15, p. 578] for a definition of specification). For n ≥ 1, denote by n Pern(f) := {x ∈ X : f (x)= x}, and define the sequence of f-invariants measures 1 (1) µn := δx, |Pern(f)| x∈PerXn(f) and define

(2) µ := lim µn n→∞ Bowen proved that the limit (2) in fact exists and µ is an f-, ergodic, and it is the unique measure maximizing the entropy of f. Let g : X → X be an extension of f, that means there exists π : M → M continuous and onto such that f ◦ π = π ◦ g.

The map π defines an equivalence relation: y ∼π z if and only if π(y)= π(z). For x ∈ X we denote by [x] := {y ∈ X : π(y)= π(x)} =

1A Cr foliation is a partition of the manifolds locally Cr-diffeomorphic (or home- omorphic if r = 0) to a partition of Rd into k-planes for some 0 ≤ k ≤ d. A lamination is a C0 foliation with C1 leaves. INTRINSIC ERGODICITY 7

π−1(π(x)) the equivalence class of x. We say that a class [x] is periodic if π(x) is a of f. A class [x] is trivial if its cardinality is one. Otherwise, we say the class is non-trivial. Now, we are ready to state our conditions:

(H1) htop(g, [x]) = 0 for any x ∈ X. (H2) Each periodic class [x] has a periodic point of g with the same period of π(x). (H3) µ({π(x) : [x] is trivial })=1. The condition (H2) allows one to chose one periodic orbit for each periodic class, that is, we set

−1 Pern(g) := { x ∈ π (Pern(f)) ∩ Pern(g) : g if o(x1,g) =6 o(x2,g) then o(π(x1), f) =6 o(π(x2), f)},

and define

1 (3) νn := δx, |Pern(g)| g x∈PerXn(g) g Theorem 3.1. If f,g : X → X are as above and assumptions (H1-H3) are satisfied, then

(4) ν = lim νn n→∞ exists and it is g-invariant, ergodic, and it is the unique measure maxi- mizing the entropy of g. In particular ν equidistribute the periodic class of g.

Proof. Let ν be any accumulation point of the sequence νn. We will prove that ν is the unique measure of maximal entropy, and hence ν will be the limit of νn and the result will follows. We will split the proof into several lemmas.

Lemma 3.2. It holds that π∗ν = µ.

Proof. First, note that π∗νn = µn, for all n ≥ 1. In fact, for every A ⊆ X Borelean, if x ∈ X is any point, then

−1 δx(π (A)) = δπ(x)(A). 8 JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´

Taking this last equality and (H2) into consideration, we conclude that −1 π∗νn(A) = νn(π (A))

1 −1 = δx(π (A)) |Pern(g)| g x∈PerXn(g) g1 = δπ(x)(A) |Pern(f)| π(x)∈XPern(f)

= µn(A).

From the continuity of π∗ we have that

π∗ν = µ. 

Lemma 3.3. The measure ν is of maximal entropy, that is,

hν (g)= htop(g).

−1 Proof. Let P be a partition of X and P˜ = π (P). Then hν(g, P˜) = hµ(f, P) and so hµ(f) ≤ hν (g). On the other hand, Bowen’s formula [5] states that

htop(g) ≤ htop(f) + sup htop(g, [x]). x∈X Therefore, from (H1) and the variational principle we conclude that

htop(g) ≤ htop(f)= hµ(f) ≤ hν (g) ≤ htop(g).  We say that A is saturated if A = π−1(π(A)). In general, the saturation of A ⊆ X is defined as sat(A) := π−1(π(A)). Note that ν(sat(A)) = µ(π(A)). Lemma 3.4. For ever Borel set A we have ν(A)= ν(sat(A)). Proof. Let X˜ = {x ∈ X : [x] = {x}}. From (H3) and the fact that π∗ν = µ we have that ν(X˜) = 1. For A ⊆ X Borelean, we have ν(sat(A)) = ν(sat(A) ∩ X˜)= ν(A ∩ X˜)= ν(A). 

Corollary 3.5. The probability measure ν is ergodic. Proof. From Lemma 3.4 it follows that if P is a g-invariant subset, then ν(P )= ν(π−1(π(P )) = µ(π(P )). Since π(P ) is f-invariant and µ is ergodic we know that ν is ergodic.  INTRINSIC ERGODICITY 9

. Lemma 3.6. Let η be a g-invariant probability measure and assume that η is singular with respect to ν. Then

hη(g) < htop(g).

Proof. Let ρ = π∗η. It follows that ρ is singular with respect to µ. The Ledrappier-Walter’s formula [16] states that

−1 hη(g) ≤ hρ(f)+ htop(g, π (x))dρ(x). ZX and from (H1) it follows that

hη(g) ≤ hρ(f).

Bowen proved [7] that hρ(f) < htop(f)= htop(g) and the result follows.  Now, we can finish the proof of Theorem 3.1. Let η be any g-invariant probability measure such that hη(g)= htop(g). We can write η = αη1 + (1 − α)η2 for some α ∈ [0, 1] such that ηi are probability measures, η1 << ν and η2 is singular with respect to ν. It follows that

htop(g)= hη(g)= αhη1 (g)+(1 − α)hη2 (g) ≤ htop(g). The previous lemma implies that α = 1, that is, η is absoluty con- tinuous with respect to ν. As ν is ergodic we have that η = ν. This completes the proof of the theorem.  Remark: The condition (H2) can be removed. In fact, if [x] is a m periodic class with π(x) ∈ Pern(f), then [x] is compact and g |[x] : [x] → [x] is a homeomorphism, where m is the period of π(x). So, m there exists a g -invariant probability measure δ[x] suported on [x]. Of m course g∗δ[x] is a g -invariant probability measure supported on [π(x)] and π∗δ[x] = δπ(x). In particular, 1 m−1 gkδ m ∗ [x] Xk=0 is a g-invariant probability measure supported on the orbit of the pe- riodic class [x].

So we can define Pern(g) as the set of equivalent classes that are fixed by gn and for each periodic class we select a g-invariant probability g k measure δ[x] as above (with δ[gk(x)] = g∗ δ[x]). Now (3) can be written as 1 (5) νn := δ[x]. |Pern(g)| g [x]∈XPern(g) g 10JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´

Note that (H2) imples that δ[x] can be chosed a Dirac measure sup- ported on a periodic point of [x]. Nevertheless, as we will see in the proof of Lemma 3.2, it is enough that there exist a bijection between

Pern(f) and Pern(g) and

g h∗δ[x] = δh(x).

4. Intrinsic ergodicity for Man˜e’s´ robustly transitive Ma˜n´e’s example of a robustly transitive that is not Anosov was constructed on T3. We will use his construction for diffeomorphisms of higher dimensional tori.

fA f0

q q1 q q2

Figure 1. Ma˜n´e’s construction

We fix some dimension d ≥ 3 and let A ∈ GL(d, Z) be a hyperbolic toral automorphism with only one eigenvalue inside the unit circle and all eigenvalues real, positive, simple, and irrational. Let λs be the unique modulus less than 1 and λc be the smallest of the moduli greater than 1. d c We denote the induced linear Anosov system on T by fA and let F be the foliation corresponding to the eigenvalue λc; so locally at each point F c is just a line segment in the direction of an eigenvector associ- s u ated with λc. Similarly, F and F are the foliations corresponding to an eigenvalue λs and all the eigenvalues greater than λc, respectively. Since all eigenvalues are irrational, each leaf of F s, F c, and F u is dense in Td. Such matrices can be built for any d ≥ 3 as companion matrices to the minimal polynomial over Q of a Pisot number whose algebraic INTRINSIC ERGODICITY 11 conjugates are all real. Such numbers are given by Theorem 5.2.2 in [1, p. 85] (the proof implies that the conjugates are real). The moduli are then pairwise distinct by [26]. Without loss of generality, we may assume that fA has at least two fixed points and that any unstable eigenvalue other than λc has mod- ulus greater than 3 (if not, replace A by some power). Let p and q be fixed points under the action of fA and ρ > 0 be a small number to be determined below. Following the construction in [17] we define f0 by modifying fA in a sufficiently small domain C c contained in Bρ/2(q) keeping invariant the foliation F . So there is a neighborhood U of p such that fA|U = f0|U . Inside C the fixed point q undergoes a pitchfork bifurcation in the direction of the foliation F c. The stable index of q increases by 1, and two other saddle points with the same stable index as the initial q are created. (See Figure 1.) The resulting diffeomorphism f0 is strongly partially hyperbolic with a C1 center foliation F c. According to [17], it is also robustly transitive (in fact topologically [3, p. 184]) for ρ> 0 sufficiently small. The next proposition will be helpful in the proof of Theorem 1.3.

Proposition 4.1. (Shadowing proposition) Let fA be an Anosov dif- feomorphism of the d-torus, d ≥ 3, as above. Let f ∈ Diff1(Td) satisfy the following properties: (a) there exist constants ǫ> 0 and δ > 0 such that each ǫ-chain un- der fA is δ-shadowed by an orbit under fA and 3δ is an expan- d n n sive constant for fA, (i. e. if x, y ∈ T and d(fA(x), fA(y)) < 3δ for all n ∈ Z, then x = y), and (b) each f-orbit is an ǫ-chain for fA. Then the map π : Td → Td, where π(x) is the point in Td that under the action of fA will δ-shadow the f-orbit of x, is a semiconjugacy from f to fA, i.e., it is a continuous and onto map with π ◦ f = fA ◦ π. Proof. By the shadowing theorem we know that the map π is well- defined and that π(f(x)) = fA(π(x)) and d(π(x), x) < δ. We need to see that π is continuous [25, Theorem 7.8] and surjective. It is probably folklore, but we provide a proof for the convenience of the reader. To show that π is continuous we take a sequence xn → x and show that π(xn) → π(x). Fix M ∈ N. Then there exists an N(M) ∈ N such that for each n ≥ N(M) j j d(f (xn), f (x)) < δ for all − M ≤ j ≤ M. We then have j j d(fA(π(xn)), fA(π(x))) < 3δ for all − M ≤ j ≤ M 12JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´ where n ≥ N(M). It follows that for any limit point y of the sequence {π(xn)} we have

j j (6) d(fA(y), fA(π(x))) ≤ 3δ for all j ∈ Z.

Since 3δ is an expansive constant for fA this implies that y = π(x) and π(xn) converges to π(x). We now show that π is surjective. Assume that it is not and let y∈ / π(Td). Consider the closed ball B = B(y, 3δ) and the map from the ball to its boundary r : B → ∂B as follows: for x ∈ B,r(x) is intersection of the ray (starting at y and passing through π(x)) with the boundary ∂B. The map is well defined since π(x) =6 y.

Moreover it is continuous. On the other hand r/∂B : ∂B → ∂B is isotopic to the identity (since d(π(x), x) < δ and the ball has radius 3δ). This contradicts Brouwer Theorem.2 We shall also use the following (folklore) fact:

Lemma 4.2. Let g : Td → Td be an injective continuous self-map. Let K be a compact such that the lengths of all its iterates, gn(K), n ≥ 0, are bounded by a constant L. Then h(g,K)=0.

Proof of Lemma For each n ≥ 0, there exists a subset K(ε, n) of gn(K) with cardinality at most L/ε + 1 dividing gn(K) into −k with length at most ε. Observe that 0≤k

Proof of Theorem 1.3 The strategy of the proof of Theorem 1.3 is to use the semiconjugacy πg from Proposition 4.1 and to show that for d 1 −1 each x ∈ T and each g C -close to f0, the set πg (x) is a compact −1 interval of bounded length contained in a center leaf, and πg (x) is a unique point for almost every x. These facts, together with Lemma 4.2 and Theorem 3.1 imply the result. We note that the measure of d maximal entropy for fA is Lebesgue measure, denoted µ, on T . We note that for ρ > 0 small enough, any diffeomorphism f that is 1 C close to the previously constructed diffeomorphism f0, satisfies the hypothesis of Proposition 4.1. Let r > 0 be an expansive constant for fA and fix a neighborhood U ⊂ U0 of f0 such that each g ∈ U satisfies the hypothesis of Proposi- tion 4.1 with 0 <ǫ<δ< min(r/3, ρ). For each g ∈ U we denote πg as the semiconjugacy mapping g to fA given by Proposition 4.1. Let µ be Lebesgue measure on Td and set

(7) m = µ(B(q, 3ρ)) > 0. INTRINSIC ERGODICITY 13

The above construction implies is such that the maximum contraction in the center direction, denoted b(f), satisfies 1−m 2m (8) λc b(f) > 1 where m is defined in (7). 1−m 2m Fix γ > 0 such that (λc − γ) (b(f) − γ) > 1 . Possibly by reducing U, we may and do assume that dC1 (f0,g) <γ and that robust transitivity holds for all g ∈ U. −1 Fix g ∈ U and suppose that y1,y2 ∈ πg (x). By construction of n n πg, this implies d(g (y1),g (y2)) < 2δ for all n ∈ Z. The normal hyperbolicity of the center lamination implies that such y1 and y2 must lie in the same center leaf. By the bounded property, the c whole segment of F between y1 and y2 stays within 2δ 0 such that 1−m−σ 2m+σ (λc − γ) (b(f) − γ) > 1. 14JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´

Hence, for µ-almost every x ∈ Td, there exists some K(x) > 0 such −1 that, for all z ∈ πg (x), all k ≥ 0, and

k c 1−m−σ 2m+σ k |Dgz F | ≥ K(x)[a(g) b(g) ] 1−m−σ 2m+σ k ≥ K(x)[(λc − γ) (b(f) − γ) ] ≥ K(x)ck

−1 with c> 1. As πg (x) must keep a bounded length it must be a unique point for µ-almost every x. This shows that ν =µ ¯(mod 0) and g is intrinsically ergodic. 2

5. A Mixed Man˜e´ Example Derived from Anosov We now consider further classes of examples. Let f : Tn → Tn be a (linear) Anosov diffeomorphism, where n ≥ 4, such that T Tn = Ess ⊕ Es ⊕ Eu ⊕ Euu with dim Es = dim Eu = 1 and the rate of contraction/expansion is λss <λs < 1 <λu <λuu where λss ss is the largest modulus of an eigenvalue corresponding to E , λs is the s modulus of the eigenvalue corresponding to E , λu is the modulus of u the eigenvalue corresponding to E , and λuu is the smallest modulus of an eigenvalue corresponding to Euu. Taking a power of f, if necessary, assume that f has two different fixed points p and q. Let r> 0 be small (to be determined later) and deform f inside B(p,r) and B(q,r) similar to Ma˜ne’s derived from Anosov construction: in B(p,r) we perform a flip perturbation along Es and on B(q,r) we perform a flip bifurcation along Eu. In this way we can obtain g satisfying the following: • g is partially hyperbolic: T Tn = Ess ⊕ Ecs ⊕ Ecu ⊕ Euu which is dominated (each subbundle dominates the previous ones by a factor a< 1), and dim Ecs = dim Ecu =1. These subbundles are C0 close to the respective ones of f; • dC0 (f,g)

cs σ1 = sup{kDg|E (x)k : x ∈ M}, cu σ2 = inf{kDg|E (x)k : x ∈ M}, INTRINSIC ERGODICITY 15

m m and m ∈ N be such that σ1λs < 1 and σ2λu > 1. Let ρ> 0 satisfy µ(M \ B(j, ρ)) ≥ 1 − 1/2m, j = p, q where µ is the Bowen measure of f, where the Bowen measure is the unique measure of maximal entropy for the Anosov diffeomorphism (see for example [15, p. 618]). We assume that 2Cr < ρ/2. Let h : Tn → Tn be the semicongugacy. We will assume for the sake of simplicity that Ec = Ecs ⊕ Ecu is uniquely integrable. Notice that Ecs and Ecu are one-dimensional and hence they are integrable. Let J be a segment tangent to Ecs, we say that Ecs is uniquely integrable through J if any maximal integral curve of Ecs through any point of J must contain J. Analogously for Ecu. Lemma 5.1. Let x ∈ Tn be any point. Then, one and only one of the following hold: (1) h−1(x) consists of a single point. (2) h−1(x) is a segment tangent to Ecs of length less than 2Cr. (3) h−1(x) is a segment tangent to Ecu of length less than 2Cr. (4) h−1(x) is a square tangent to Ecs ⊕ Ecu: −1 cs −1 • for each y ∈ h (x) we have that Wǫ (y) ∩ h (x) is a central stable segment that we denote by J cs(y) and Ecs is uniquely integrable through J cs(y). Similar for Ecu. • If y and z are in h−1(x) then, ∅= 6 J cs(y)∩J cu(z) ∈ h−1(x). Proof. Assume that h−1(x) is not trivial, and let y, z ∈ h−1(x) be two different points. By a similar argument as the previous example we c cs cs −1 conclude that y ∈ Wǫ (z). And also, if z ∈ W (y) then [y, z] ⊂ h (x). cs −1 cs This means that Wǫ (y) ∩ h (x) is a segment, say J (y) whose length remains bounded in the future and in the past and, by the domination in Ecs ⊕Ecu we conclude that Ecs is uniquely integrable through J cs(y). cu Similar if z ∈ Wǫ (y). Assume also that neither (2) nor (3) hold. Consider local central cs cu integral curves Wǫ (y) and Wǫ (z) and call w the point of intersection. Although they may not have rate of expansion or contraction, a similar argument can be done so that h(w) = h(z) = h(y). Therefore, {w} = J cs(y) ∩ J cu(z) ∈ h−1(x). 

Corollary 5.2. Conditions (H1) and (H2) are satisfied for g. Proof. We need only to check (H1) in case (4) above. By the above notations, we observe that g(J cs(y)) = J cs(g(y)) and g(J cu(y)) = J cu(g(y)). 16JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´ cs Wǫ (y) F c(x)

w b W cu(z) b z ǫ

b y

Figure 2

Therefore, the product structure is invariant and it is not difficult to see that the maximal cardinality of a (n, ǫ)-separated set in the equivalent class has at most polynomial growth. We can say a bit more on the structure of the iterates of an equivalent class. Let y be in the class and consider J cu(y). Since the length remains bounded in the future, we claim that kDgn|Ecu(y)k ≤ a−1/2 for any n large enough. Otherwise, for any w ∈ J cu(y) we have that kDgn|Ecu(w)k≥kDgn|Ecu(y)ka1/4 ≥ a−1/4 and therefore the segment J cu(y) will be, for n large, larger than 2Cr, a contradiction, and the claim is proved. Now, by domination, we conclude that kDgn|Ecs(y)k ≤ a1/2 for every n large enough, and in fact the same holds for any w ∈ J cu(y). Hence, the length of J cs(w) will decrease exponentially fast. We remark that a similar property holds in the past: the cu segments are contracted exponentially fast. Then, it is not difficult to see that for any ǫ> 0 the cardinality of a maximal (n, ǫ)-separated set in the class of y is the same as the cardinality of a maximal (n, ǫ)-separated set in J cu(y) for n large enough. Thus, htop(g, [y])= 0. Condition (H2) holds trivially in the first three cases, and also in the fourth one by use, for instance, of Brouwer’s fixed point theorem. Nevertheless, by the structure given above, case (4) can not be a periodic class.  Finally, for condition (H3), a similar proof can be done as for the first Ma˜n´eexample.

6. A Derived From Anosov Through Hopf Bifurcation 6.0.1. Example 4: Derived from Anosov through a Hopf bifurcation. This example is the one treated in [18]. Explicit formulas and details can be founded there. This example, obtained from a linear Anosov INTRINSIC ERGODICITY 17

gn gn

cs cs cs

cu cu cu h−1(x)

Figure 3

through a Hopf bifurcation is not generic and does not include the examples in [10]. A sharper analysis should be done to study the latter. Let f : T3 → T3 be a linear Anosov diffeomorphism, T T3 = Es ⊕ Eu with dim Es = 2 and the f has complex eigenvalues in Es(p), where p is a fixed point. We will deform f inside a small ball B(p,r) to obtain a diffeomorphism g : T3 → T3 with the following features: • p is a repeller for g; • g is partially hyperbolic T T3 = Ecs ⊕ Eu; • Dg uniformly contracts Ecs outside B(p,r); cs u • kDg/E (x)k ≤ 1 for any x∈ / Wloc(p); • dC0 (g, f)

Es(p) Ecs(p)

b b p p Eu(p) Eu(p) A g

Figure 4 18JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´

We will prove that g in the above conditions has a unique probability measure of maximal entropy. Denote by h the semiconjugation between f and g. It is not difficult to see that h is injective on each W u(y,g) for any y and moreover, h(W u(y,g)) = W u(h(y), f). Indeed, h preserves the unstable foliation and the central stable foliation h(W cs(y,g)) = W s(h(y), f). Therefore, an equivalent class is contained in a central . Lemma 6.1. If x∈ / W cs(p), then diam(gn[x]) → 0. Proof. Since x∈ / W cs(p) there are infinitely many n ≥ 0 such that gn(x) ∈/ B(p,r) (and we may assume that without loss of generality n u n that g (x) ∈/ Wloc(p)). Therefore, kDg/Ecs(x)k→ 0 and the same holds for any y ∈ [x] (and uniformly on y.) The conclusion follows.  Corollary 6.2. Conditions (H1), (H2), and (H3) hold. Proof. First notice that the class [p] is a closed disc, with p as a repeller and the boundary attracts everything on the disk but p. Therefore, cs htop(g, [p]) = 0. If [x] ⊂ W (p) and [x] =6 [p] then the class [x] is attracted by the invariant circle and so htop(g, [x]) = 0. Now, if [x] ∈/ W cs(p), then diam(gn[x]) → 0 and, therefore, for any ǫ and any n large enough the cardinality of any (n, ǫ)-separated set in [x] is bounded, and hence htop(g, [x])=0. We have proved that (H1) holds. For (H2), let [x] be a periodic class. Notice that [x] ∩ W cs(p) = ∅ and therefore diam(gn[x]) → 0. But since [x] is periodic this means that [x]= x and so x is periodic and (H2) holds trivially. Condition (H3) can be proved with similar methods as the previous examples.  References [1] M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux- Delefosse, and J. P. Schreiber. Pisot and Salem numbers. Birkhauser, Basel, 1992. [2] C. Bonatti, L. J. D´ıaz, and E. Pujals. A C1 generic dichotomy for diffeomor- phisms; weak forms of hyperbolicity or infinitely many sinks or sources. Annals of Math., 158:355–418, 2003. [3] C. Bonatti, L. J. D´ıaz, and M. Viana. Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective, volume 102 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. [4] C. Bonatti and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math., 115:157–193, 2000. [5] . Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc., 153:401–414, 1971. [6] Rufus Bowen. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc., 154:377–397, 1971. INTRINSIC ERGODICITY 19

[7] Rufus Bowen. Some systems with unique equilibrium states. Math. Systems Theory, 8(3):193–202, 1974/75. [8] Rufus Bowen and . The ergodic theory of Axiom A flows. Invent. Math., 29(3):181–202, 1975. [9] J. Buzzi and T. Fisher. Measures of maximal entropy for certain robustly transitive diffeomorphisms that are not partially hyperbolic. in preparation. [10] M. Carvalho. Sinai-Ruelle-Bowen measures for N-dimensional [N dimen- sions] derived from Anosov diffeomorphisms. Ergodic. Theory Dynam. Systems, 13(1):21–44, 1993. [11] L. J. D´ıaz, E. Pujals, and R Ures. Partial hyperbolicity and robust transitivity. Acta Math., 183:1–43, 1999. [12] M. W. Hirsch, C. Pugh, and M. Shub. Invariant Manifolds, volume 583 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1977. [13] Y. Hua, R. Saghin, and Z. Xia. Topological entropy and partially hyperbolic diffeomorphisms. Ergod. Th. Dynamic. Systems, 28:843–862, 2008. [14] I. Kan. Open sets of diffeomorphisms having two each with an ev- erywhere dense basin. Bull. Amer. Math. Soc., 31:68–74, 1994. [15] A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1995. [16] F. Ledrappier and P. Walters. A relativized variational principle for continuous transformaitons. J. London Math. Soc., 16:568–576, 1977. [17] R. Ma˜n´e. Contributions to the stability conjecture. Topology, 17:383–396, 1978. [18] P. McSwiggen. Diffeomorphisms of the torus with wandering domains. Proc. Amer. Math. Soc., 117(4):1175–1186, 1993. [19] M. Misiurewicz. Diffeomorphism without any measures with maximal entropy. Bull. Acad. Polon. Sci. S`er. Sci. Math. Astronom. Phus., 21:903–910, 1973. [20] S. Newhouse. Continuity properties of entropy. Ann. of Math. (2), 129:215–235; Corrections: Ann. of Math. (2), 131: 409–410, 1990, 1989. [21] S. Newhouse and L.-S. Young. Dynamics of certain skew products, volume 1007 of Lecture Notes in Math., pages 611–629. Springer, Berlin, 1983. [22] J. Palis. A global perspective for non-conservative dynamics. Ann. Inst. H. Poincar´eAnal. Non Lin´eaire, 22(4):485–507, 2005. [23] C. Robinson. Dynmical Systems Stability, , and Chaos. CRC Press, 1999. [24] Martin Sambarino and Carlos V´asquez. Equilibrium states for Ma˜nes’s derived from Anosov diffeomorphims. In preparation. [25] M. Shub. Global Stability of Dynamical Systems. Springer- Verlag, New York, 1987. [26] C. Smyth. The conjugates of algebraic integers, Advanced Problem 5931. Amer. Math. Monthly, 82:86, 1975. [27] A. Wilkinson. Stable ergodicity of the time-one map of a geodesic flow. Ergodic. Theory Dynam. Systems, 18(6):1545–1587, 1998.

C.N.R.S. & Departement´ de Mathematiques,´ Universite´ Paris-Sud, 91405 Orsay, France

Department of Mathematics, Brigham Young University, Provo, UT 84602 20JER´ OMEˆ BUZZI, TODD FISHER, MARTIN SAMBARINO, CARLOS VASQUEZ´

CMAT-Facultad de Ciencias, U. de la Republica, Montevideo, Uruguay

Instituto de Matematica,´ Pontificia Universidad Catolica´ de Val- para´ıso, Valpara´ıso, Chile E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]