Nonexpanding Attractors: Conjugacy to Algebraic Models and Classification in 3-Manifolds

We prove a result motivated by Williams's classification of expanding attractors and the Franks-Newhouse Theorem on codimension-1 Anosov diffeomorphisms: If a mixing hyperbolic attractor has 1-dimensional unstable manifolds then it is either is expanding or is homeomorphic to a compact abelian group (a toral solenoid); in the latter case the dynamics is conjugate to a group automorphism. As a corollary we obtain a classification of all 2-dimensional basic sets in 3-manifolds. Furthermore we classify all hyperbolic attractors in 3-manifolds in terms of the classically studied examples, answering a question of Bonatti.


Introduction
In the study of hyperbolic dynamics, a major theme is that strong dynamical hypotheses impose a conjugacy between an abstract dynamical system and an algebraic, or at least highly structured, model. For instance, results of Franks and Manning established that every Anosov diffeomorphism of an infranil-manifold is conjugate to a hyperbolic infranil-automorphism [14,Theorem C ]. Among of the oldest conjectures in modern dynamics is the hypothesis that every Anosov diffeomorphism is conjugate to a hyperbolic infranil-automorphism. A partial result towards this conjecture was obtained by Franks and Newhouse for codimension-1 Anosov systems. Recall an Anosov diffeomorphism is called codimension-1 if dim(E σ ) = 1 for some σ ∈ {s, u}.
Outside the realm of global hyperbolicity, that is, when dealing with proper hyperbolic subsets Λ ⊂ M , one often sees dynamics which is not conjugate to any algebraic system. However, in the case of expanding attractors, Williams showed in [23] that the restricted dynamics f ↾ Λ is conjugate to the shift map on a generalized solenoid. Recall that by an expanding attractor we mean a hyperbolic attractor Λ such that dim(Λ) = dim(E u ↾ Λ ). Also by a generalized solenoid (or n-solenoid) we mean a topological space N (which Williams takes to be a branched n-manifold ), and a surjective map g : N → N , and define the generalized solenoid to be the inverse limit lim with the natural shift map σ. (See Section 4 for the construction of the inverse limit in a more specific setting adapted to our problem.) Theorem II ([23, Theorem A]). Assume Λ is an n-dimensional expanding attractor for f ∈ Diff(M ). Then f ↾ Λ is conjugate to the shift map of an n-solenoid.
Note that Theorem II, as originally stated in [23], required the additional hypothesis that the foliation {W s ǫ (x) | x ∈ Λ} was C 1 on some neighborhood of Λ. This was latter seen to be unnecessary (see for example [2]). While not algebraic, the conjugacy in Theorem II provides a significant insight into the topology of Λ and the dynamics of f ↾ Λ .
In this article we present a result inspired in part by the Franks-Newhouse Theorem on codimension-1 Anosov diffeomorphisms, and somewhat dual to the conjugacy between the dynamics of 1-dimensional expanding attractors and shift maps on generalized solenoids established in [21] and [22]. In particular, we study nonexpanding hyperbolic attractors Λ for an embedding f , under the assumption that dim E u ↾ Λ = 1, and show that the dynamics f ↾ Λ is conjugate to an automorphism of a compact abelian group. We take our dynamics to be generated by C r embeddings for r ≥ 1.
Theorem 1.1. Let Λ ⊂ U ⊂ M be a compact topologically mixing hyperbolic attractor for a C r embedding f : U → M such that dim E u ↾ Λ = 1. Then either Λ is expanding, or is an embedded toral solenoid (see Section 4). In the latter case, f ↾ Λ is conjugate to a leaf-wise hyperbolic solenoidal automorphism. In particular, if Λ is locally connected then Λ is homeomorphic to a torus and f ↾ Λ is conjugate to a hyperbolic toral automorphism.
Using the primary result in [11] we conclude that the only 2-dimensional toral solenoids that may be embedded in a 3-manifold are homeomorphic to T 2 . In particular, we obtain the following. Corollary 1.2. Let M be a 3-manifold, and let Λ ⊂ M be a basic set with dim(Λ) = 2. Then either Λ is a codimension-1 expanding attractor (or contracting repeller), or Λ decomposes as a disjoint union Λ = Ω 1 ∪ Ω 2 ∪ · · · ∪ Ω k where each Ω j is homeomorphic to T 2 and f k ↾ Ωj is conjugate to a hyperbolic automorphism of T 2 .
We note that the above corollary is a significantly stronger version of the main result in [7]. Indeed, in [7] the result corresponding to the second case in Corollary 1.2 requires the additional hypothesis that Λ is embedded as a subset of a closed surface in M . Our result, on the other hand, rules out the possibility that dim(E u ↾ Λ ) = 1 and W s (x) ∩ Λ is a connected 1-dimensional set that is not a manifold, for example, a Sierpinski carpet.
It should also be noted that in the conclusion of Corollary 1.2, the T 2 need not be smoothly embedded. Indeed in [12] a hyperbolic attractor is constructed as a nowhere differentiable torus embedded in a 3-manifold.
The motivation for this work was initially to answer a question by Bonatti [1] which can be paraphrased as follows: Do there exist examples of hyperbolic attractors in 3-manifolds besides the classical examples? We answer this question in the negative. Theorem 1.3. Let M be a 3-manifold, and let Λ ⊂ U ⊂ M be a topologically mixing, hyperbolic attractor for a C r embedding f : U → M . If dim Λ = 0: then Λ is an attracting fixed point for f ; dim Λ = 1: then dim E u ↾ Λ = 1 and Λ is conjugate to the shift map on a generalized 1-solenoid as classified by Williams ([22]); dim Λ = 2: then we have 1 ≤ dim E u ↾ Λ ≤ 2 and if dim E u ↾ Λ = 1: then Λ is homeomorphic to T 2 and f ↾ Λ is conjugate to a hyperbolic toral automorphism; dim E u ↾ Λ = 2: then Λ is a codimension-1 expanding attractor studied by Plykin ([17], [18]); dim Λ = 3: then Λ = M ∼ = T 3 and f is conjugate to a hyperbolic toral automorphism.
We remark that in the case of 1-dimensional topologically mixing attractors (which are necessarily expanding), the proof of Theorem 1.1 provides a mechanism to determine if the attractor is algebraic, that is, if f ↾ Λ is conjugate to a solenoidal automorphism. In particular the presence or absence of a global product structure as described in Section 5.3.2 determines whether or not a 1-dimensional attractor is algebraic. See Proposition 5.30.

Hyperbolic dynamics
We begin with background material in hyperbolic dynamics and attractors. Let M be a smooth manifold endowed with a Riemannian metric. Given U ⊂ M and a C r embedding f : U → M , r ≥ 1, we say a subset Λ ⊂ U is invariant if f (Λ) = Λ. A compact invariant set Λ is said to be hyperbolic if there exist a Riemannian metric on M (called the adapted metric), a constant κ < 1, and a continuous Df -invariant splitting of the tangent bundle T x M = E s (x) ⊕ E u (x) over Λ so that for every x ∈ Λ and n ∈ N When Λ is hyperbolic, there exists an ǫ > 0 such that the sets , f −n (y)) < ǫ, for all n ≥ 0} are C r embedded open disks, called the local stable and unstable manifolds. Furthermore, if d is the distance on M induced by the adapted metric, there are λ < 1 < µ so that for x ∈ Λ, y ∈ W s ǫ (x), z ∈ W u ǫ (x) and n ≥ 0 we have Note that (1) and (2) . For x ∈ Λ we also have the sets called the global stable and unstable manifolds. Both W u (x) and W s (x) are C r injectively immersed submanifolds. Note that in the case that f is invertible (that is, when An invariant set Λ is said to be topologically transitive under f if it contains a dense orbit. Alternatively, a compact invariant subset Λ ⊂ M is topologically transitive if for all pairs of nonempty open sets U, V ⊂ Λ, there is some n such that A hyperbolic set Λ is a hyperbolic attractor if there is some open neighborhood Λ ⊂ V such that n∈N f n (V ) = Λ. Alternatively, if Λ is a hyperbolic set, then it is an attractor if and only if W u (x) ⊂ Λ for all x ∈ Λ. When Λ is a hyperbolic attractor, the set y∈Λ W s (y) is called the basin of Λ. Note that if Λ is a topologically mixing hyperbolic attractor, then for each x ∈ Λ, W u (x) is dense in Λ.
We recall from the introduction that a hyperbolic attractor Λ is called expanding if the topological dimension of Λ equals the dimension of the unstable manifolds. (For an introduction to topological dimension, see [10].) Alternatively, Λ is expanding if for every x ∈ Λ the set W s ǫ (x) ∩ Λ is totally disconnected.
2.1. Local product structure and Markov partitions. Recall that given a compact hyperbolic set, we may find 0 < δ < η so that d(x, y) < δ implies the intersection W u η (x) ∩ W s η (y) is a singleton. We say that a hyperbolic set Λ has local product structure if for η, δ above, d(x, y) < δ implies W u η (x) ∩ W s η (y) ⊂ Λ. A compact hyperbolic set Λ is called locally maximal if there exists an open set Λ ⊂ V such that Λ = n∈Z f n (V ). For compact hyperbolic sets, local maximality is equivalent to the existence of a local product structure [13]; in particular, hyperbolic attractors have local product structure. Definition 2.1. Given a set Λ with local product structure and δ and η as above, we say a closed set R ⊂ Λ is a rectangle or a local product chart if (1) sup{d(x, y) | x, y ∈ R} < δ; (2) R is proper, that is, R is equal to the closure of its interior (in Λ); Given a hyperbolic set Λ with local product structure we say a collection of rectangles R = {R j } is a Markov partition if Ri (x)) ⊂ R j . We note that every locally maximal hyperbolic set admits a Markov partition; in particular, hyperbolic attractors admit Markov partitions. Also note that if R is a rectangle and R is a Markov partition, then f (R) is a rectangle and f (R) := {f (R j )} is a Markov partition. In particular, we have the following. Claim 2.3. If Λ is locally maximal, then given any set K ⊂ W σ (x) ∩ Λ, compact in the internal topology of W σ (x), there is a rectangle containing K.

2.2.
Disintegration of the measure of maximal entropy. For a hyperbolic set with local product structure we define a canonical isomorphism between subsets of the stable and unstable manifolds.
Definition 2.4 (Canonical Isomorphism). Let Λ be a locally maximal hyperbolic set, R a rectangle, and Similarly, we may define a canonical isomorphism between subsets of local unstable manifolds.
Recall that a point x ∈ M is said to be nonwandering if for every neighborhood U of x, there is an n so that f n (U ) ∩ U = ∅. Let NW(f ) denote the nonwandering points of f . Recall that given an Axiom-A diffeomorphism, (respectively a locally maximal hyperbolic set Λ = n∈Z f n (V )) we have a partition, called the spectral decomposition, of the nonwandering points NW(f ) = Ω 1 ∪ · · · ∪ Ω k (respectively NW(f ↾ Λ ) = Ω 1 ∪ · · · ∪ Ω k ) where each Ω j is a transitive hyperbolic set for f (see [13], [20]). Given a spectral decomposition, we call the partition elements Ω j above basic sets. That is, a compact hyperbolic set Ω ⊂ NW(f ) is a basic set, if Ω is open in NW(f ) and f is topologically transitive on Ω. Clearly topologically mixing hyperbolic attractors are basic sets.
Given a basic set, there is a canonical disintegration of the measure of maximal entropy as a product of measures supported on the stable and unstable manifolds. The following is adapted from [19].
Theorem 2.5 (Ruelle, Sullivan [19]). Let Ω be a basic set for f . Let h be the topological entropy of f ↾ Ω . Then there is an ǫ > 0 so that for each x ∈ Ω there is a measure µ u x on W u ǫ (x) and a measure µ s x on W s ǫ (x) such that: x × µ s x is locally equal to Bowen's measure of maximal entropy. By 2.5(b) we drop the subscript and simply write µ σ . By additivity, we may extend the definition of µ σ to any set K ⊂ W σ (x) for σ ∈ {s, u}. The following properties of µ σ are corollaries to the proof of Theorem 2.5 in [19].
Furthermore, in the case dim E u ↾ Λ = 1, we have the following.
Corollary 2.7. Let Λ be a hyperbolic attractor such that dim E u ↾ Λ = 1. Then for any connected set K ⊂ W u (x) (that is, an interval) we have µ u (K) < ∞ if and only if its closure K in W u (x) is compact in the internal topology of W u (x).
Proof. If K is not compact in W u (x), then K passes through some rectangle a countable number of times which implies that µ u (K) = ∞.

Limits of directed and inverse systems
We review basic constructions and properties of the direct and inverse limit objects in algebra and topology.
3.1. Direct limits. Given a topological space X and an injective continuous map f : X → X we construct the direct limit as follows. Endow N with the discrete topology and introduce the equivalence relation on X × N generated by the relation (x, k) ∼ (f (x), k + 1). Then we define , whence it is natural to refer to τ f as the left shift on lim − → (X, f ). We present an alternate, more explicit, construction of the set lim − → (X, f ). For every j ∈ N define a homeomorphism h j : X j → X and consider the inclusion i j : X j ֒→ X j+1 given by When the map f : X → X is open, the inclusions i j induce a nested inclusion of topologies, the union of which correctly reconstructs the topology of the direct limit. Given ξ ∈ lim − → (X, f ), we have that ξ ∈ X j for some j whence we may define . One verifies this definition of τ f coincides with that above.
By the second construction, we see that if X is a C r manifold and f a C r embedding, then lim − → (X, f ) can be endowed with a C r differential structure under which τ f is a C r diffeomorphism.
Given a group G and a homomorphism h : as follows. Let i j : G j → G be a group isomorphism and define h j : . Let N be the normal subgroup of k∈N G k generated by the elements The following proposition is straightforward from the Van Kampen theorem.
Proposition 3.1. Let X be a connected manifold, f : X → X an embedding, and The construction above allows us to embed every hyperbolic attractor as an attractor for an ambient diffeomorphism.
Note that in constructing the direct limit, we assumed the map f : X → X was injective to avoid pathological topological properties in the limiting object.

3.2.
Inverse limits. Let f : X → X be a continuous map (which we typically take to be surjective). We then define the inverse limit to be the subset of X N := i∈N X satisfying hence it is natural to call σ f a right shift map. We will call the topological object lim ← − (X, f ) a generalized solenoid.
Note that even in the case that X is a manifold and f is a smooth map, we do not expect lim ← − (X, f ) to have a manifold structure. Indeed in the case that f is a C ∞ covering with degree greater than 1, the limit lim ← − (X, f ) will locally be the product of a Cantor set and a manifold.
If G is a group, and h a homomorphism we define

Toral solenoids
We give a brief introduction to toral solenoids, the compact abelian groups obtained as the algebraic models in the conclusion of Theorem 1.1. For more detailed exposition, see, for example, [3]. For an explicit construction of toral solenoids embedded as hyperbolic attractors for differentiable dynamics, see [8].
Let A ∈ Mat(k, Z) have non-zero determinant. Then considering the standard torus T k := R k /Z k as a compact abelian group, the map A : R k → R k induces an endomorphism A : T k → T k . We define a toral solenoid S A to be the topological group obtained via the inverse limit Note the above inverse limit is taken both as a limit of topological and algebraic objects, and that S A inherits the right shift automorphism σ A : S A will fail to be a manifold in the case when | det(A)| > 1, in which case we will call S A proper. Let C ξ denote the path component of ξ in S A . Then, even in the case | det(A)| > 1, we can endow the path components {C ξ } with the smooth Euclidean structure pulled back from the projection to the zeroth coordinate S A → T k . With respect to this Euclidean structure the map σ A : C ξ → C σA(ξ) is smooth. Furthermore, in the case when A has no eigenvalues of modulus 1, the map σ A : C ξ → C σA(ξ) is hyperbolic with respect to the pull-back metric, whence we say σ A is leaf-wise hyperbolic.
4.1. R k and Z k actions. We now define an R k action and an induced Z k action on S A . (Compare to the immersions of R k and Z k into S A constructed in [3]).
We then define the Z k action ϑ : Z k × S A → S A to be the restriction of θ to the subgroup Z k ⊂ R k .
Let p 0 : S A → T k denote the projection in the zeroth coordinate.
Claim 4.2. The action θ has the following properties. a) For each ξ ∈ S A , the θ-orbit of ξ is dense.
b) The homomorphism σ A is θ-equivariant; that is, and Here Let p : R k → T k denote the canonical projection. Given an m ∈ Z k , we may find some curve γ : . This motivates the construction in the next section.

4.2.
A covering space for (S A , σ A ). Define the topological group S to be the product Σ × R k . The group action ϑ of Z k on S A induces a group action ϑ of Z k on Σ. We define an embedding α of Z k as a subgroup of S by where e is the identity element of Σ. Then α naturally defines a Z k action on S by We also define maps σ : S → S given by and q : S → S A given by We check that σ is an injective endomorphism. Furthermore, q is seen to be a homomorphism by Claim 4.2(d).
We have the following properties of the above construction.
In particular, v ∈ Z k . Furthermore Now σ : S → S is an injective homomorphism, but will fail to be surjective whenever | det(A)| > 1. We define the topological group S as the direct limit and defineσ : S → S to be the left shift automorphism (τ σ in the notation of Section 3.1) induced by σ : S → S ; that is, where the second equality holds for l ≥ 1. Furthermore, we define the torsion-free abelian group commutes whence we may extend the embedding α : which is seen to be well defined by Claim 4.2(b). As above, the embeddingα of Z k [A −1 ] as a subgroup of S defines a natural Z k [A −1 ] action on S. We also define a group homomorphismq : . We enumerate properties of the above constructions.
To see 4.4(c), note that for any s ∈ S and l ∈ N we havẽ We thus have thatq : S → S A is a covering map and thatσ lifts σ A .

4.3.
Metrization of S A . We conclude this section with the construction of a canonical metric on S A with respect to which S A behaves (metrically) like a hyperbolic set for σ A when A is hyperbolic.
Firstly, let ρ denote the standard metric on R k . Given a curve γ : [0, 1] → S A , there is a unique curve γ ′ : [0, 1] → R n with γ ′ (0) = 0 such that γ(t) = θ γ ′ (t) (γ(0)). If x, y ∈ S A lie in the same path component, define Γ(x, y) to be the set of all curves γ : [0, 1] → S A with γ(0) = x and γ(1) = y. Then define Secondly, for any x, y ∈ S A we define Note that for any . Given x, y ∈ S A , let Ξ(x, y) be the set of all sequences ξ = (x 0 , y 0 , x 1 , y 1 , . . . , x l , y l ) such that there is a curve γ j ⊂ S A with endpoints x j and y j for 0 ≤ j ≤ l. Then define is a hyperbolic matrix, E + and E − denote the expanding and contracting subspaces, Since we are only concerned with the topology of Λ and the dynamics f ↾ Λ in Theorem 1.1, by Claim 3.2 we assume without loss of generality that f : M → M is a diffeomorphism. To prove Theorem 1.1 we first present some preliminary observations and constructions that will enable us to build the essential dichotomy. Let g be a Riemannian metric on M . Note that for any liftf of f , Λ is a hyperbolic set under the pull-back metric π * (g). For x ∈ Λ and σ ∈ {s, u} denote by W σ (x) the σ-manifold of x under the dynamicsf in the metric π * (g). Note also that W σ (x) is the connected component of π −1 W σ (π(x)) containing x. In addition note that for g ∈ G different from the identity, W σ (g(x)) ∩ W σ (x) = ∅. Given a subset X ⊂ Λ we will write W σ (X) := x∈X W σ (x).
x to be the distance function on W s (x) induced by restricting the metric π * (g) to W s (x). For simplicity we shall suppress the dependence on x and simply write d s (x, y) := d s x (x, y) whenever y ∈ W s (x). Note that B admits a codimension-1 foliation W s by the stable manifolds of Λ. Since π 1 ( B) = {1}, the foliation W s is transversely orientable. (Indeed, one can always make W s and B orientable by passing to double covers.) Fix a transverse orientation for W s . Note that neither G nor any lift of f is assumed to preserve this transverse orientation.
Given a compact, oriented, C 1 curve γ ⊂ B that is everywhere transverse to the foliation W s , we define a signed length l u (γ). Let {V i } be a cover of Λ by local s-product neighborhoods (see Definition 2.1) and let { V ij } = π −1 ({V i }) where for each j, the set V ij is homeomorphic to V i . Letf be a lift of f . Then we may find some n > 0 so that γ ⊂f −n ( ij V ij ).
Definition 5.2. Given γ and n as above, we define the signed unstable length l u (γ) as follows. We first define l u on a connected component γ ′ of γ ∩f −n ( V ij ) . Let if γ is positively orientated with respect to W s , −1, if γ is negatively orientated with respect to W s , and h is the topological entropy of f ↾ Λ . We may then extend l u to all of γ additively. Theorem 2.5 shows that l u is well defined, independent of all choices above. Furthermore, given any piecewise-smooth oriented curve γ ⊂ B we may partition γ into a family of curves {γ i }, each of which is everywhere tangent to W s or everywhere transverse to W s ; in the former case we define l u (γ i ) = 0 while in the latter we use Definition 5.2. Thus we may extend the definition of l u to all piecewisesmooth oriented curve in B. Note that by Corollary 2.6, l u (γ) is non-zero on any curve γ transverse to W s and l u ({x}) = 0. Now piecewise-smooth curves generate the group of piecewise-smooth simplicial 1-chains. Thus we may extend the function l u to a piecewise-smooth simplicial 1-cochain denoted by α u . Corollary 5.4. Given x, y ∈ B and two oriented piecewise-smooth curves γ 1 , γ 2 with end points x and y then |l u (γ 1 )| = |l u (γ 2 )|.
Corollary 5.5. For each pair x, y ∈ Λ, the intersection W s (x) ∩ W u (y) contains at most one point.
Proof. If not we could find a piecewise smooth 1-cycle γ with |α u (γ)| > 0, a contradiction since α u is exact.
The above corollaries motivate the following definitions.
Definition 5.6. We say a subset V ⊂ Λ is a product chart if x, y ∈ V implies W u (x) ∩ W s (y) is non-empty and W u (x) ∩ W s (y) ⊂ V .
Definition 5.7. For x ∈ Λ and x ′ ∈ W u (x) let l u (x, x ′ ) := l u (γ xx ′ ) where γ xx ′ is the unique oriented curve in W u (x) from x to x ′ . For x ∈ Λ and L ∈ R, let x + u L denote the unique point x ′ ∈ W u (x) with l u (x, x ′ ) = L. Definition 5.8. Given x, y ∈ B we define the pseudometric d u (x, y) := |l u (γ)| for any piecewise smooth curve γ with endpoints x and y. Furthermore we define a metric on leaves of the foliation W s by d u (W s (x), W s (y)) := d u (x, y).
Note that Corollary 5.4 guarantees d u is well defined, and Corollaries 2.6 and 2.7 guarantees that the restriction of d u to W u (x) defines a complete metric consistent with the topology on W u (x). Definition 5.9. For x, y ∈ Λ we define Ξ(x, y) to be the set of sequences ξ = (x = x 0 , y 0 , . . . , where d s is the distance induced by the Riemannian metric in Definition 5.1 and d u is the pseudometric constructed from the measure µ u in Definition 5.8. Clearly d defines a metric on Λ consistent with the ambient topology.

Global product relation. We now define a binary relation on points in Λ.
Definition 5.10 (Global Product Relation). For x, y ∈ Λ we say x ∼ y if y ∈ W s (x) and Claim 5.11. ∼ is an equivalence relation.
Proof. Clearly ∼ is reflexive. To see that ∼ is symmetric suppose x ∼ y, and that there exists some y ′ ∈ W u (y) such that W s (y ′ ) ∩ W u (x) = ∅. Set L = l u (y, y ′ ) and x ′ = x + u L. Then l u (y, W u (y) ∩ W s (x ′ )) = L hence y ′ = W u (y) ∩ W s (x ′ ) contradicting the assumptions on y ′ . Thus ∼ is symmetric. A similar argument shows that ∼ is transitive.
We let [x] denote the equivalence class of x under the relation ∼.
Remark. The equivalence class [x] represents the maximal subset of Λ ∩ W s (x) with global product structure, that is, admitting the canonical homeomorphism and W u ([x]), with the quotient topology, is homeomorphic to R.
We enumerate a number of properties of the equivalence classes of ∼. . e) Let y ∈ W s (x) be such that C s (y) contains points arbitrarily close to C s (x). Then Proof. 5.12(a) and 5.12(b) are trivial. To see 5.12(c) let Let C ⊂ W u (x) be a compact connected set containing x and x ′ . Then there is some rectangle V ⊂ Λ containing π(C) by Claim 2.3. But then there is a product chart V ⊂ π −1 (V ) containing C and x j for a sufficiently large j contradicting the assumption that W s (x ′ ) ∩ W u (x j ) = ∅ for all j. Hence 5.12(c) holds.
Fix an L > 0. Clearly the set . By a similar argument as above we see that V is closed hence C s (x) ⊂ V . Since L was arbitrary 5.12(d) follows. For 5.12(e), let y satisfy the hypotheses and suppose y ′ ∈ W u (y) is such that W s (y ′ ) ∩ W u (x) = ∅ and let L = l u (y, y ′ ). Since Λ is compact we may find some δ > 0 so that for every z ∈ Λ there is some rectangle V (z, L) containing both the sets W s δ (z) ∩ Λ and π ({z + u r | r ≤ |L|}) wherez is some lift of z to Λ. By assumption we may find a w ∈ C s (y) and x ′ ∈ C s (x) so that d s (w, x ′ ) < δ; setting w ′ = w + u L we may find a product chart containing w, w ′ , and x ′ , hence W s (w ′ ) ∩ W u (x ′ ) = ∅. By 5.12(a) w ′ ∈ [y ′ ] and by 5.12 We now define a metric on the quotient Λ/∼ and study its induced topology.  Clearly d Ω defines a metric on Ω.
Corollary 5.16. The group G = π 1 (B) acts via isometries on ( Ω, d Ω ). Furthermore the dynamics on Ω induced by the dynamicsf : Λ → Λ, which we also denote byf : Ω → Ω, acts conformally: Proof. The pseudometric d u is preserved under G. Since d s Ω is defined via d u , it is also preserved. Furthermore, we have that d u transforms according to Theorem 2.5.
Note that since G acts via invertible isometries on ( Ω, d Ω ), it acts via homeomorphisms on ( Ω, d Ω ) despite the fact that the metric topology may not coincide with the quotient topology. Then Clearly U is open as a subset of W s (x) ∩ Λ since for any y ∈ U we may find an open product chart containing y and y ± u  The proof of Theorem 1.1 will follow from considering two cases. In the first case we will assume that G acts properly discontinuously on Ω and deduce that Λ is expanding. We will then show that if G fails to act properly discontinuously on Ω then Λ has a product structure which will be used to obtain a conjugacy between f ↾ Λ and an automorphism of a toral solenoid. 5.2. Case 1: G acts properly discontinuously on Ω. The goal of this section is to prove the following proposition.
Proposition 5.18. Suppose G acts properly discontinuously on Ω. Then Λ is expanding.
Given a metric space (X, ρ) and a subset Y ⊂ X we call For any x ∈ X and Y ⊂ X we write Definition 5.19. Let {A n } be a countable sequence of subsets in a metric space (X, ρ). We define the Kuratowski limit supremum by Clearly the Kuratowski limit supremum is a closed set for any collection {A n }.
Lemma 5.20. Let (X, ρ) be a proper metric space; that is, one for which any closed ball {y ∈ X | d(x, y) ≤ R} is compact. Let {A n } ⊂ X be a countable sequence of subsets such that (1) lim n→∞ diam(A n ) is finite; (2) each A n is connected; (3) there exists a Cauchy sequence {x n } with x n ∈ A n . Then lim n→∞ A n is connected.
Note that the result need not hold if assumption 3 is omitted.
Proof. By assumption 3, we may fix x := lim n→∞ x n ∈ lim n→∞ A n . By assumption 1 we may find an L and N so that for all n ≥ N , the inclusion A n ⊂ B(x, L) holds.
By assumption 3 we may find an M such that for all n ≥ M , we have A n ∩ N 1 = ∅. Furthermore, we may find an infinite subsequence {n j } such that A nj ∩ N 2 = ∅. Since A nj is connected we have A nj ∩ ∂(N 1 ) = ∅. For each j ∈ N pick some a j ∈ A nj ∩ ∂(N 1 ). Then since ∂(N 1 ) is compact we may find some y ∈ ∂(N 1 ) that is an accumulation point of {a j }. But this implies that y ∈ lim n→∞ A n whence lim n→∞ A n ∩ ∂(N 1 ) = ∅, a contradiction. Proof. We prove the lemma for the pull-back of the function r : Λ → R + ∪ {∞}. Clearly the lemma holds at x ∈ Λ if r(x) = ∞. We assume otherwise.
The function r is clearly continuous along unstable leaves. Consequently, we need only show that forx i ∈ W s ǫ (x), ifx i → x then r(x) ≥ lim i→∞ r(x i ). Passing to a subsequence {x n } ⊂ {x i } we may assume that lim n→∞ r(x n ) = lim i→∞ r(x i ).
If r(x) < ∞ but the lemma failed at x, we could find ǫ > 0 and K so that for all n > K we have r(x n ) > r(x) + ǫ and d s (x, Let Ξ = lim n→∞ C s (x n ). By Lemma 5.20, Ξ is connected and hence we have Ξ ⊂ C s (x). On the other hand, the assumption on r( contradicting the definition of r(x). Proof. Suppose first that the range of r does not contain ∞. Then by upper semicontinuity, r is globally bounded. Let M = max{r(x) | x ∈ Λ}. By hyperbolicity of f on Λ and boundedness of r we find an m ∈ N so that f m (π(C s (x))) ⊂ W s ǫ (f m (x)) (wherex is a lift of x), hence r(f m+1 (x)) ≤ λr(f m (x)) for all x ∈ Λ. On the other hand, since f is a homeomorphism, we should have But then M = λM which implies M = 0. Now if r(x) = ∞ then r(y) = ∞ for all y ∈ W u (x). Indeed, letx be a lift of x, y the lift of y contained in W u (x), and L = l u (x,ỹ). Let U be a cover of C s (ỹ). Then for every z ∈ C s (ỹ) there is an ǫ(z) > 0 so that W s ǫ(z) (z) ⊂ U for some U ∈ U and the set V (z) := {z ′ + u l | z ′ ∈ W s ǫ(z) (z), |l| ≤ |L|} is a product chart. But then {V (z)} covers C s (x), whence we conclude that U admits a finite subcover.
Thus if r(x) = ∞ for some x ∈ Λ, then r(y) = ∞ for all y ∈ W u (x). Since W u (x) is dense in Λ, the upper semicontinuity of r implies r ≡ ∞.
We thus establish that Λ is expanding under the assumption that G acts properly discontinuously on Ω.
Proof of Proposition 5.18. Let Ω = Ω/G be the orbit space. Note that since G acts properly discontinuously, Ω is Hausdorff. Denote the canonical projections by π : Λ → Λ, q : Λ → Ω, π ′ : Ω → Ω. Consider the diagram Since the equivalence classes of ∼ are G-invariant, the G-orbit of q(y) is equivalent to the G-orbit of q(g(y)) for any g ∈ G and y ∈ Λ. Thus we may find a map q ′ so that the diagram Since Λ is compact and Ω is Hausdorff, q ′ is proper, whence q is proper. Hence the equivalence classes of ∼ must be compact subsets of Λ. By Claim 5.12(d) and Corollary 5.22 this implies r ≡ 0; hence the connected components of Λ ∩ W s (x) are singletons and Λ is expanding. 5.3. Case 2: G fails to act properly discontinuously on Ω. In the case that G fails to act properly discontinuously at some point in Ω, we show that Λ is homeomorphic to a toral solenoid and f ↾ Λ is conjugate to a solenoidal automorphism.
Note we have W u (z j ) ∩ W s (x) = ∅ for sufficiently large j by 5.23(a).
For [x i ] and [y i ] as above, denote by H([ from which we obtain , H([y j ])) = 0 for all 0 ≤ j ≤ k − 1, hence applying (6) recursively one obtains (7). In particular Hence, by setting y = z j and letting R → ∞ in (8) Note that g i (x + u L) = g i (x) ± u L depending of whether g i preserves the transverse orientation on W s . However for g such that l u [x], g([x]) = t and g can not reverse the orientation since otherwise we would have a contradiction unless g is the identity. Thus we may assume that for g i in 5.23(c), Now let L ∈ R be given. By forgetting initial terms and invoking 5.23(a) and 5.23(b), we may assume that for all i Then by definition of r s we have As above we have  Endowing G with the discrete topology, we have that ζ is continuous. Since the metric topology on Ω is weaker than the quotient topology, the quotient map q : Λ → Ω induces a continuous map q * ζ : G × Λ → [0, ∞). Now q * ζ(g, x) = q * ζ(g, g ′ (x)) for all g, g ′ ∈ G hence q * ζ induces a continuous map ζ : G × Λ → [0, ∞).
As a result we have, Proof. Assume the conclusion fails for some fixed ǫ. Passing to an infinite subsequence, we may assume the conclusion fails for all i ∈ N. Let {y i } ⊂ Λ be such that ζ(g i , y i ) > ǫ and, again passing to a subsequence, let z ∈ Λ be a limit point of , g i ([z])) > ǫ/2 for all i, contradicting Corollary 5.26.

5.3.2.
Global product structure. We now establish that when G fails to act properly discontinuously, the set Λ has a global product structure; that is, for all x, y ∈ Λ we have W u (x) ∩ W s (y) = ∅. Then, as in (8) we have But then by (6) we have is defined on N and by the same argument as above Thus we must have Λ = W u ([x]) since otherwise B would not be connected.
The following is immediate from Lemma 5.28.
Corollary 5.29. When G fails to act properly discontinuously on Ω then Λ admits a global product structure.
We now shift out attention back to Λ, under the assumption that Λ admits a global product structure. Our objective is to prove the following.
Proposition 5.30. Assume Λ has a global product structure. Then f : Λ → Λ is conjugate to a leaf-wise hyperbolic automorphism of a toral solenoid (see Section 4).
Given a metric space (X, d), a homeomorphism f : X → X is called expanding if there is some µ > 1 so that for all x, y ∈ X, d(f (x), f (y)) ≥ µd(x, y). A sequence Given an L-pseudo orbit {x j } we say a point x ∈ X shadows {x j } if there is some δ so that d(f j (x), x j ) ≤ δ for all j.
Lemma 5.31 (Global Shadowing). Let h : Ω → Ω be a product homeomorphism h : (x, ξ) → (h 1 (x), h 2 (ξ)). Assume that h 1 : R k → R k is expanding with respect to the metric ρ. Furthermore assume that h 2 : Υ → Υ is asymptotically exponentially contracting on bounded sets with respect to each metric d x ; that is, given an R > 0, ξ ∈ Υ, and x ∈ R k there are c > 0 and λ < 1, depending continuously on (x, ξ) ∈ Ω, so that if d x (ξ, ζ) ≤ R then Additionally assume that Ω admits a properly discontinuous action by a subgroup G of the group of isometries of (Ω, d) such that h preserves G-orbits, and the quotient Ω/G is compact. We have the following. a) Given a C > 0 there is a K > 0 so that if d(x, y) ≤ C for any x, y ∈ Ω, then d π1(y) π 2 (y), π 2 (x) ≤ K. b) Given an R > 0 there are c > 0 and λ < 1 (depending only on R) so that for any ξ, ζ ∈ Υ and Remark 5.32. Note that Lemma 5.31 applies to Λ in the case that Λ has global product structure by choosing an x ∈ Λ and taking Υ = Λ ∩ W s (x), k = 1, ρ = d u in Definition 5.8, and d x the metrics d s x in Definition 5.1. Then the metric in Definition 5.9 corresponds to (9).
Furthermore, assuming the linear map A : R n → R n is hyperbolic, we have that the cover S constructed in Section 4 and endowed with the metricd constructed in Section 4.3 satisfies the hypotheses of Lemma 5.31 with Υ = E − × Σ and k = dim E u .
Proof of Lemma 5.31(a) and 5.31(b). The hypotheses of Lemma 5.31 guarantee that the numbers K, c, and λ can be chosen pointwise on Ω. Since the group G acts via isometries, we may assume they are constant on G-orbits. Since the quotient Ω/G is compact, we may choose uniform K, c, and λ.
Then one easily verifies that x := (y, z) shadows the L-pseudo orbit {x j }.
Fix an isomorphism Φ : H → Z k and let A : Z k → Z k be the endomorphism Φ • f * • Φ −1 . Considering Z k as embedded in R k , A : Z k → Z k induces a linear automorphism on R k and a surjective endomorphism on the quotient T k = R k /Z k , also denoted by A. Let S A and S be the solenoid and its cover constructed in Section 4, and let σ A andσ be the respective shift automorphisms.
Fix an identification G = lim − → (H, f * ) whence f * : G → G is identified with the shift map τ f * ↾H . We have that the diagram Proof of Proposition 5.30. Fix a liftf of f . Let S A , σ A , S, andσ be as above. Let D be a fundamental domain for the cover S → S A ; note that D will be compact. Let Φ be the isomorphism between G and Z k [A −1 ] above. We let Z k [A −1 ] act by addition on S via the actionα in Section 4.2.
Hence the claim holds.
Since Ψ is proper, we have that A is hyperbolic. Thus as in Remark 5.32, Lemma 5.31 applies to S. Hence, given a fundamental domain D ⊂ Λ, and x ∈ Λ, we choose {g i } so thatf i (x) ∈ g i (D). Then as above we define Ψ ′ (x) to be the unique point ξ ∈ S so that ξ shadows the pseudo orbit where e is the identity in S. We thus obtain a map Ψ ′ : Λ → S.
Thus the homeomorphism Ψ : S → Λ induces a homeomorphism h : S A → Λ such that the diagram Proof of Theorem 1.1. By Proposition 5.18, Corollary 5.29, and Proposition 5.30 if Λ is not an expanding attractor, then Λ is homeomorphic to a toral solenoid, and f ↾ Λ is conjugate to a solenoidal automorphism.
Furthermore we have W s (x) ∩ Λ is perfect for every x ∈ Λ. Thus if W s (x) ∩ Λ is locally connected, Λ cannot be expanding, which by the above implies Λ is homeomorphic to a solenoid. However, the only locally connected toral solenoids are in fact tori, that is, S A for det A = ±1.
Lemma 6.1. Let Λ ⊂ M be a compact hyperbolic set for a diffeomorphism g : M → M . The points y ∈ Λ with the property that E σ (y) = {0} for some σ ∈ {s, u} are periodic and isolated in Λ. In particular if Λ is transitive and contains such a point then Λ is finite.
We now prove the remaining results from the introduction.
Proof of Corollary 1.2. By [16,Theorem 3] and by passing to the inverse if necessary we may assume that Λ is an attractor for f . We thus have dim E u ↾ Λ ≤ 2. Furthermore, by Lemma 6.1, dim E u ↾ Λ = 0 would imply that dim(Λ) = 0, whence we have dim E u ↾ Λ ≥ 1. By the spectral decomposition and by passing to an appropriate iterate f n we may assume that Λ is topologically mixing for f n .
If dim E u ↾ Λ = 2 then Λ is a codimension-1 expanding attractor. If dim E u ↾ Λ = 1 then by Theorem 1.1 we have that Λ is an embedded toral solenoid. By [11,Theorem 1], no proper 2-dimensional solenoid may be embedded in a closed orientable 3-manifold.
If needed, we first argue on a double cover in the case M is non-orientable. Also if needed we may pass to a compact manifold with boundary N containing Λ, and glue a second copy of N along the boundary to obtain a closed manifold containing Λ. We may then apply [11,Theorem 1] and thus obtain that Λ is locally connected, hence Λ ∼ = T 2 and f n is conjugate to a toral automorphism.
Proof of Theorem 1.3. We note that for a hyperbolic attractor we always have dim E u ↾ Λ ≤ dim Λ.
If dim Λ = 0 we have that dim E u ↾ Λ = 0 hence Lemma 6.1 implies that every x ∈ Λ is periodic and isolated; hence we must have Λ = {x} in order for Λ to be topologically mixing. If dim Λ = 1 then Lemma 6.1 implies dim E u ↾ Λ = 1; hence Λ is an expanding 1-dimensional attractor and Λ is conjugate to the shift map on a generalized 1-solenoid by Theorem II.
If dim Λ = 2, Lemma 6.1 implies 1 ≤ dim E u ↾ Λ ≤ 2. When dim E u ↾ Λ = 1 the fact that Λ is topologically mixing implies Λ is connected, whence Λ is homeomorphic to T 2 and f ↾ Λ is conjugate to a hyperbolic toral automorphism by Corollary 1.2. When dim E u ↾ Λ = 2 then Λ is a codimension-1 expanding attractor by definition.