Symbolic extensions for partially hyperbolic diffeomorphisms

We show that every partially hyperbolic diffeomorphism with a 1-dimensional center bundle has a principal symbolic extension. On the other hand, we show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing non-hyperbolic robustly transitive sets with center indecomposable bundle of dimension at least 2.


Introduction
Symbolic dynamical systems arose as a tool to code complicated dynamical systems by the use of more tractable systems (shift spaces). The existence of Markov partitions for hyperbolic systems provides an effective coding of these systems using symbolic dynamics. A natural question, that we address in the present work, is to what extent a non-hyperoblic system can be codified using symbolic systems.
A dynamical system (X, f ) has a symbolic extension if there exists a subshift (Y, σ) and a continuous surjective map π : Y → X such that π • σ = f • π: the system (Y, σ) is called an extension of (X, f ) and (X, f ) is called a factor of (Y, σ).
1.1. Existence of symbolic extensions. Every dynamical system with a symbolic extension has finite entropy. J. Auslander asked if the converse holds. Namely, that if every system with finite entropy has a symbolic extension. In fact, the existence of symbolic extensions seems to be related to the notion of asymptotically h-expansive systems, as defined by Misiurecicz [24] (in Section 2 we review all relevant notions in the introduction). A nice form of a symbolic extension is a principal one, that is an extension given by a factor map which preserves entropy for every invariant measure. Boyle, D. Fiebig, and U. Fiebig [10] were able to show the following: Theorem 1.1 ( [10]). If (X, f ) is asymptotically h-expansive, then f has a principal symbolic extension.
Using a result of Buzzi [13], that states that any C ∞ diffeomorphism is asymptoically h-expansive one gets the following: [10,13]). Every C ∞ map of a compact manifold has a principal symbolic extension.
This result shows connections between the regularity of the maps and the existence of symbolic extensions.
From now on we let Diff r (M) be the space of C r diffeomorphisms of a closed compact manifold M endowed with the usual uniform topology. When r = 1 we simply write Diff(M). [19]). If f ∈ Diff r (M) and r ≥ 2, then f has a symbolic extension.

Conjecture 1.3 (Downarowicz and Newhouse
For instance, in a recent work Downarowicz and Maass [18] have shown the above conjecture for maps of a closed interval.
Besides the regularity of the system a second ingredient for the existence of symbolic extensions seems to be hyperbolic-like properties. This is precisely the idea behind our first result.
Theorem A. Every partially hyperbolic diffeomorphism with a 1-dimensional center bundle is asymptotically h-expansive and therefore has a principal symbolic extension.
The above result supports the principle of Pugh and Shub that "a little hyperbolicity goes a long way" [25] and the suggestion made to us by Burns [12] that 1-dimensional center partially hyperbolic systems in many ways behave like the hyperbolic ones.
One important class of weakly hyperbolic dynamical systems are the robustly transitive ones. Given an open set U and a diffeomorphism f , we let Λ f (U) = n∈Z f n (Ū); we say that the set Λ f (U) is robustly transitive if Λ f (U) ⊂ U and for every g C 1 -close to f the set Λ g (U) is transitive (has a point with a forward dense orbit). When U is the whole manifold we say f is robustly transitive.
In fact, there is a "finest" dominated splitting: that is a dominated splitting [6,Theorem 4]. Additionally, some of the examples above are partially hyperbolic and have a 1-dimensional center bundle. We have the following corollary to Theorem A. Corollary 1.5. Every robustly transitive set with a partially hyperbolic splitting and a center 1-dimensional direction has a symbolic extension.
As a remark, we observe that this result shows that super-exponential growth of the number of periodic points does not preclude the existence of symbolic extensions. Specifically, the C 1 generic diffeomorphisms satisfying Corollary 1.5 have super-exponential growth of the number of periodic points [5].  [19]. They show that a generic area-preserving C 1 diffeomorphism of a surface is either Anosov or has no symbolic extension. This result relies on the existence of "persistent homoclinic tangencies." Asaoka [1] gives a simple example of diffeomorphisms in a 3-dimensional disk with C 1 persistent homoclinic tangencies, and uses the result of Downarowicz and Newhouse, to show that for any smooth manifold M with dim(M) ≥ 3, there exists an open subset of Diff(M) in which generic diffeomorphisms have no symbolic extensions. Finally, he extends this example to higher dimensions using normal hyperbolicity.
We now describe another class of C 1 generic diffeomorphisms without symbolic extensions, and show how this is related with non-hyperbolic properties.
Let U be an open set in a closed manifold M. Let T (U) be the set of diffeomorphisms, f , such that Λ f (U) is robustly transitive. Let T nh (U) denote the set of diffeomorphisms, f , such that Λ f (U) is not hyperbolic. We let T nh 2 (U) be the subset of T nh (U) such that for each f ∈ T nh 2 (U) there is a neighborhood V of f such that for each g ∈ V the set Λ g (U) has a non-hyperbolic center indecomposable bundle of dimension at least 2.
Theorem B. There is a C 1 residual set R in T nh 2 (U) such that each diffeomorphism in R has no symbolic extension.
Relevant systems satisfying this theorem are the DA-diffeomorphisms of the four torus, T 4 , in [8]. In this case, T 4 = U and every non-trivial dominated splitting of the system is of the form E ⊕ F where both E and F are two-dimensional and non-hyperbolic.
We emphasize that the arguments in [19] are all two dimensional and that their translation to higher dimensions is not straightforward. Most of Asaoka's paper deals with the existence of persistent tangencies and the proof of non-existence of symbolic extensions is quite brief and refers to [19]. Moreover, it considers a very specific dynamical configuration. So we consider important to give a complete explanation of these arguments, provided in Section 4, in the general case.
Theorems A and B raise the following natural question.
Question 1.6. If a partially hyperbolic diffeomorphism has a center bundle that splits into 1-dimensional subbundles, then does it have a principal symbolic extension?

background
In this section we review some facts on subshifts, entropy, asymptotic h-expansivity, hyperbolicity, and weak forms of hyperbolicity. The existence and non-existence of symbolic extensions is related to the entropy structure of a dynamical system as defined by Downarowicz [17]. We now review some basic definitions of topological and measure theoretical entropy. Let (X, d) be a compact metric space and f be a continuous self-map of X. The d n metric on X is defined as and is equivalent to d and defined for all Let (X, B, µ) be a Lebesgue measure space with µ(X) = 1. A partition of X is a collection ξ of measurable sets, ξ = {C α ∈ B | α ∈ I}, such that For partitions ξ and ν the joint partition of ξ and ν is Let f be a measure preserving transformation of (X, B, µ). For a measurable partition ξ and n ∈ N we define the joint partition of ξ with respect to f for n to be the partition Then the entropy of f with respect to µ is We now review the definition of asymptotic h-expansivity. Given a subset Y ⊂ X we let We denote the closed ball with center at x and radius ǫ in the d n metric as B n ǫ (x). Let Then .
Hyperbolicities. We now review some definitions and facts of weak forms of hyperbolicity. For a diffeomorphism f of M a compact f -invariant set Λ has a dominated splitting if where each E i is non-trivial and Df -invariant for 1 ≤ i ≤ k and there exists an m ∈ N such that The set Λ is partially hyperbolic if it has a dominated splitting and there exists some n ∈ N such that Df n either uniformly contracts where there exists some n ∈ N such that Df n uniformly contracts E s and uniformly expands E u . A set Λ is locally maximal if there exists an open set U containing Λ such that Λ = n∈Z f n (U). A set is transitive if there exists a point in the set whose forward orbit is dense in the whole set. A hyperbolic set is a basic set if it is locally maximal and transitive. One of the important properties of hyperbolic sets is structural stability. This allows one to define the continuation of a hyperbolic sets. See [22, p. 571-572]. For a hyperbolic set Λ = Λ f of a diffeomorphism f we denote the continuation of Λ for g close to f by Λ g .
For a partially hyperbolic diffeomorphism f with a splitting E s ⊕ E c ⊕ E u we know there exist unique families F u and F s of injectively immersed submanifolds such that F i (x) is tangent to E i for i = s, u, and the families are invariant under f . These are called, respectively, the unstable and stable foliations of f . For the center direction it is known, in the general case, that there is no foliation tangent to the center bundle [20]. If the center bundle is 1-dimensional, then there always exist curves tangent to the center direction through every point. Such curves are called central curves. However, these curves need not be unique due to the fact that the center bundle, in general, is not Lipschitz.

Partially hyperbolic diffeomorphisms with 1-dimensional center foliations
In this section we prove Theorem A. We will see that the existence of a 1-dimensional center bundle will allow us to use 1-dimensional arguments to compute the entropy. Before proceeding we define some notation. For a curve η in M its lenght is denoted by |η|.
The case where E s or E u is trivial is a simplification and is omitted. Furthermore, we let dim(E s ) = s. The first step of the proof is to show that there exists a "foliation chart" on a uniform scale. More precisely, define λ as the maximal expansion of the derivative of f and select constants δ 0 , δ 1 , δ 2 , > 0 satisfying the following: • for every curve γ tangent to E c with |γ| < λ δ 0 and every sdisk ∆ s ⊂ F s with s-diameter less than δ 1 one has that γ ∩ ∆ s contains at most one point; and • for every (s + 1)-disks Υ of the form where γ is a curve tangent to E c with length bounded by δ 0 and F s δ 1 (y) is the δ 1 -disk centered at y contained in F s , and every u-disk ∆ u ⊂ F u with u-diameter less than δ 2 one has that Υ ∩ ∆ u contains at most one point. We select small δ > 0 such that λ δ < δ i for i = 0, 1, 2. Fix α ∈ (0, δ/2). For each x ∈ M consider a curve γ(x) tangent to E c containing x with radius α. (So the extreme points of γ(x) are at distance α from x, where α < δ/2.) This is possible since E c is 1-dimensional. By construction, F s δ 1 (z) ∩ γ(x) consists of at most one point for all x and z. We For small τ > 0 we define For notational simplicity, we fix small α > 0 and write . For every ǫ > 0 sufficiently small and x ∈ M we have Note that this inclusion holds for all γ(x) as above. See Figure 1.
Proof: We consider the forward orbit of x and define x n = f n (x). For each n we define a central curve γ n = γ(f n (x)) = γ(x n ) such that γ n is "compatible" with f (γ n−1 ): this means we consider the components γ ± n (x n ) of γ n (x n ) \ {x n } and have either The "cubes" V γn (x n ) containing B ǫ (x n ) are defined as above. Let z ∈ Φ ǫ (x). By definition and construction, f n (z) = z n ∈ V γn (x n ) for all n ≥ 0. We let y be the unique point in V s γ(x),α (x) such that z ∈ F u α (y) and y ′ be the unique point in γ(x) such that y ∈ F s α (y ′ ). See Figure 1.
We define y n , y ′ n , and z n inductively. By construction, y ′ n ∈ γ n , otherwise, by Remark 3.1, z n / ∈ B ǫ (x n ). If z n = y n , then the uniform expansion along the unstable direction implies that there is a first n such that z n and y n can not be in the same foliation chart of radius α. To be more precise, by hypothesis, we know that y n is in V s γn (x n ) for all n. If z n = y n for some n the uniform expansion along the unstable direction implies that there is a first m such that z m ∈ V γm (x m ). Hence, z m / ∈ B ǫ (x m ), a contradiction. This ends the proof of the lemma. 2 Given a central curve η of size at most δ 0 and a point y ∈ V s η,α we let y ′ be the unique point in η such that y ∈ F s α (y ′ ). Note that the proof of Lemma 3.2 gives the following: This claim implies that |f n (Γ c (x))| < 2 δ 0 .
A folklore fact implies that the growth of an (n, δ)-spanning set in Γ c (x) is subexponential for all δ > 0, see for instance [14,Lemma 3.2].
The next lemma will show that f is asymptotically h-expansive, completing the proof of Theorem A.
Proof: Since f is uniformly contracting in the stable direction we know that any exponential growth of the spanning sets for Φ ǫ (x) occurs along Γ c (x). This means that we can focus on the growth of (n, δ)-spanning set in Γ c (x). More precisely: Fact 3.5. Let Y be an (n, δ/2)-spanning set in Γ c (x). Then Y is a (n, δ)-spanning set in V s Γ c (x),α for all n sufficiently large.
By the comments above, this growth is subexponential and the proof of the theorem is complete.

Diffeomorphisms with no symbolic extensions
This section consists of three parts. In the first one, in the context of non-hyperbolic robustly transitive sets (and diffeomorphisms), we explain how the existence of a center indecomposable bundle of dimension at least 2 generates persistent homoclinic tangencies, see Proposition 4.2. To get the non-existence of symbolic extensions we will use the 2-dimensional arguments in [19]. For that we need to consider special saddles in the robustly transitive set (saddles with real multipliers) and to see that the tangencies occur in a local normally hyperbolic surface. In the second one, we explain how the non-existence of symbolic extensions follows from this property. Finally, in the last section we prove Theorem B.

4.1.
Persistence of homoclinic tangencies. In this section we consider diffeomorphisms in T nh 2 (U), that is, the subset of Diff(M) of diffeomorphisms f such that Λ f (U) = n∈Z f n (Ū ) is a robustly transitive set whose finest dominated splitting has some (indecomposable) center bundle with dimension at least 2. We begin by reviewing the constructions in [4,6,9,2].
Define N (U) as the set of diffeomorphisms f ∈ Diff(M) such that Λ f (U) is robustly transitive and contains periodic points of different indices. Here the index of a periodic point is the dimension of its stable bundle. We let s − (f ) and s + (f ) be the minimum and maximum of the indices of periodic points of Λ f (U).
The set N (U) is open and dense in the set T nh (U) of diffeomorphisms f such that Λ f (U) is non-hyperbolic and robustly transitive, see [9].
We are ready to state a key result about persistence of tangencies.
Suppose that p f ∈ Λ f (U) is a saddle of index d 1 + · · · + d l−1 + 1 such that its continuation is defined for every g in a neighborhood U f ⊂ V(U) ∩ T nh 2 (U) of f . Then there is a dense subset T of U f consisting of diffeomorphisms with homoclinic tangencies associated to p g .
Note that if f ∈ V(U)∩T nh 2 (U) then there exists some l ∈ [α, β] such that d l ≥ 2 and therefore there is a saddle p f as in Proposition 4.2.
In what follows, we fix a component C(U) of V(U) ∩ T nh 2 (U) where the maps in Remark 4.1 are constants and let r = d 1 + · · · + d l−1 + 1.
We need to introduce some notation. Given a periodic point q we denote by τ (q) its period and by λ 1 (q), . . . , λ n (q) the eigenvalues of Df τ (q) (q) counted with multiplicity and ordered in non-decreasing modulus. We say that λ i (q) is the i-th eigenvalue of q.
We define Per R (f, U) as the set of saddles q ∈ Λ f (U) such that all eigenvalues of q are real and different in modulus. We let Per R (f, U) j be the subset of Per R (f, U) of saddles of index j. We also consider the subset Per(f, U) j n of Per R (f, U) j of points with period n. We define Per(f, U) r,r+1 C as the set of saddles q ∈ Λ f (U) such that |λ r−1 (q)| < |λ r (q)| = |λ r+1 (q)| < 1 < |λ r+2 (q)| and λ r (q) and λ r+1 (q) are non-real. Note that q has index r + 1.  An immediate and standard consequence of Hayashi's Connecting Lemma [21] is the next remark.

Remark 4.4.
There is a residual subset of C(U) such that for every g and any pair of saddles q g and q ′ g in Λ g (U) of indices r+1 and r one has that W s (q g ) and W u (q ′ g ) have some transverse intersection. Moreover, there are h arbitrarily close to g such that If we omit the condition of the persistence of the tangencies, this proposition is just Theorem F in [9]. We now explain how the persistence is obtained. This follows from standard arguments in the C 1 -topology that can be found in several papers. We now recall the main steps. Fix a saddle p g of index r as in the proposition. By Remark 4.3, after a perturbation we can assume that Λ g (U) contains a saddle q g ∈ Per C (g, U) r,r+1 of index r + 1. By Remark 4.4, we can assume that W u (p g ) and W s (q g ) has some transverse intersection. Therefore, we can assume that W u (p g ) spiralizes around W u (q g ). By Remark 4.4, after a perturbation, we can get an intersection between W s (p g ) and W u (q g ) (obtaining a cycle associated to p g and q g ). Unfolding this cycle, one gets a homoclinic tangency assoicated to p g . These arguments are depicted in Figure 2. See for instance Section 2.2 [4] for details.

Downarowicz-Newhouse construction.
To begin this section we review the condition given in [19] that guarantees the non-existence of symbolic extensions, see Theorem 4.6. Using Proposition 4.2 we will then verify that there is a residual set R in T nh 2 (U) such that each f ∈ R satisfies this condition, and therefore every f ∈ R has no symbolic extension.
As we will examine invariant measures for a system (X, f ), we review some basic facts of invariant measures. Let M(f, X) be the set of invariant measures and M e (f, X) be the set of ergodic invariant measures for (X, f ). If f is a homeomorphism of a compact metric space then M(f, X) is nonempty and satisfies the following (see [11,Section 4.6]): • M(f, X) is convex and compact for the weak * -topology, and • M e (f, X) is precisely the extreme points of M(f, X).
A sequence of partitions {α k } is essential if • diam(α k ) → 0 as k → 0, and • µ(∂α k ) = 0 for any µ ∈ M(f, X), where ∂α k is the union of the boundaries of the elements in the partition α k .
A simplicial sequence of partitions is a sequence {α k } of nested partitions (each α k is a refinement of α k−1 ) whose diameters go to zero and such that for each α k the partition is given by a smooth triangulation of M.
Then a symbolic extension for f does not exist.
The construction involves measures supported on hyperbolic basic sets. By [27] these measures can be obtained as limits of periodic measures defined as follows. Let f ∈ Diff(M) and p be a hyperbolic periodic point for f with period τ (p). Then the periodic measure for p is Using Proposition 4.2 we will construct E as the closure of a certain set of periodic measures.
Note that it is enough to prove Theorem B for the set V(U) ∩ U f , see Remark 4.1 and Proposition 4.2. Therefore the number r = d 1 + · · · + d l−1 + 1 in Propostion 4.2 remains fixed from now on.
Let λ j (q) be the j-th eigenvalue of a saddle q. We let χ j (q) be the j-th Lyapunov exponent of q, i.e., log(|λ j (q)|), and define For an ergodic measure µ we let χ j (µ) be its j-th Lyapunov exponent. A hyperbolic set Λ is subordinate to a partition ξ if there exist compact sets Λ 1 , ..., Λ j for some j ≥ 1 such that  (a) there exists a 0-dimensional hyperbolic basic set Λ(q, n) for f such that Λ(q, n) ∩ ∂α n = ∅ and Λ(q, n) ⊂ Λ(f, U), and (d) for every µ ∈ M e (f, Λ(q, n)) we have Let f ∈ O(U) (see Remark 4.3) and let τ r (f, U) be the minimum period of points in Per R (f, U) r . We let R r m be the set of diffeomorphisms in O(U) with τ r (f, U) = m.
For m ≤ n let D r m,n be the set of R r m of diffeomorphisms satisfying property S r n . Proof: The proof of the above lemma follows from the proof of Lemma 5.1 in [19]. Since the sets Λ(q, n) are hyperbolic the sets D r m,n are open (see [19, page 471]). Thus the crucial point is density and for that we use Proposition 4.2.
q q c π π Figure 3. The creation of Λ(q, n) in the planar region π Fix n and m, with m ≤ n, and consider a diffeomorphism g ∈ R r m and a saddle q ∈ Per R (g, U) r n . By Proposition 4.2, after arbitrarily small perturbation we can assume that g has a homoclinic tangency associated to q. Since the r-th and (r + 1)-th multipliers of q satisfy |λ r−1 (q)| < |λ r (q)| < 1 < |λ r+1 (q)| < |λ r+2 (q)|, after a new perturbation, we can assume that this tangency occurs in a "normally hyperbolic surface" π: so there are curves γ s and γ u contained in W s (q) ∩ π and W u (q) ∩ π having a quadratic homoclinic tangency. See for instance [4,Affirmation] and Figure 3. Therefore, from now on the arguments are "2-dimensional" and we can follow [19].
More precisely, let c be the tangency point. After a perturbation, for each large n, we get a hyperbolic set Λ(q, n) of f subordinate to the partition α n and contained in a small neighborhood of the orbit of c such that h top (f, Λ(q, n)) is close to χ r+1 (q), see [19, equation (43), page 484]. By construction, the set Λ(q, n) is contained in Λ f (U). This completes the sketch of the proof of Lemma 4.8. 2 Remark 4.9. By construction, each periodic pointq ∈ Λ(q, n) satisfies Proposition 4.2 and belong to Per R (f, U) r . Therefore, we can apply the previous construction to these saddles.

4.3.
Proof of Theorem B. Note that it is enough to prove a local version of this theorem for diffeomorphisms g ∈ U f ∩ V f ∩ S, where S is the residual set in Proposition 4.5. Recall that for these diffeomorphisms the functions α, β, s − , and s + are constant. The proof is similar to the proof of Theorem 1.3 in [19, pages 471-2]. Recalling equatiom (3) define (4) ρ 0 = χ r+1 (g, U) 2 and let E 1 (g) = {µ q : q ∈ Per R (g, U) r and χ r+1 (q) > ρ 0 }, and set E(g) as the closure of E 1 (g).
It is sufficient to verify the conditions of Theorem 4.6 for µ q ∈ E 1 (g). More precisely, from Lemma 4.8 we know that for each n sufficiently large there exists a periodic hyperbolic basic set Λ(q, n) that is subordinate to α n and all ergodic measures of Λ(q, n) are 1 n close to µ q . Moreover, from (d) in Definition 4.7, χ r+1 (µ) > n − 1 n χ r+1 (q) and from (c) in Definition 4.7 for every ν n ∈ M e (g, Λ(q, n)) it holds that (5) h νn (g) > n − 1 n χ r+1 (q).
Proof: Just recall that every ergodic measure supported on a hyperbolic basic set is the weak * -limit of periodic measures for periodic points in the set, see [27]. Moreover, by Remark 4.9, the periodic points in Λ(q, n) belong to Per R (g, U) r . 2 Lemma 4.11. Let ν n ∈ M e (g, Λ(q, n)). Then h νn (α n , g) = 0.
Proof: Recall that (see equations (1) and (2) Since Λ(q, n) is subordinate to α n the sum is constant for each j, see [19,464]. Therefore this limit goes to 0 as j → ∞. 2 Fix k 0 ∈ N and a measure µ ∈ E(g). By Theorem 4.6 it is enough to prove that there exists a sequence ν n → µ such that (6) lim sup n→∞ h νn (g) − h νn (α k 0 , g) > ρ 0 .
Fix a sequence η n ∈ E(g) converging to µ. By definition of E(g), we know that for each η n there is is a periodic measure µ q ∈ E 1 (g) arbitrarily close η n . Then by (d) in Definition 4.7 and Lemma 4.10 there exists ν n ∈ M e (g, Λ(q, n)) ⊂ E(g) which is close to µ q . Take large n ≥ k 0 , the α n is a refinement of α k 0 , we have from Lemma 4.11 that h νn (α k 0 , g) = 0. Then, from (5) and the choice of ρ 0 in equation (4), for n sufficiently large we have h νn (g) − h νn (α k 0 , g) > n − 1 n χ r+1 (q) > χ r+1 (g, U) 2 = ρ 0 .
Since ν n converges to µ we get (6), ending the proof of the theorem. 2